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Nonlinear Fokker–Planck–Kolmogorov equations V. I. Bogachevab  , S. V. Shaposhnikovab a Lomonosov Moscow State University b National Research University "Higher School of Economics"
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    1. IntroductionThe classical linear stationary Fokker–Planck–Kolmogorov equation with respect to a probability density ϱ has the form where Δ is the Laplace operator and b is a smooth vector field on Rd, called the drift. A nonlinear equation arises if the drift can depend on the solution ϱ, for example, if in the one-dimensional case b=ϱ. The nonlinearity makes this equation non-trivial even on the real line, where a linear equation is solved in the explicit form in an elementary way. A more general nonlinear equation with a diffusion matrix A=(aij), which can also depend on the solution, has the form A typical example of a similar nonlinear parabolic Fokker–Planck–Kolmogorov equation is the equation Throughout, the measure ϱtdx is identified with its density ϱt with respect to Lebesgue measure. Such equations are often solved with respect to measures, rather than with respect to functions; the precise definitions are given below. In some problems it is important to indicate separately the dependence of coefficients on the values of the density ϱ(x) at points in the elliptic case (or on ϱt(x) and also on t and the measure ϱt in the parabolic case). A typical example is the drift coefficient b of the formA great number of papers is devoted to the investigation of nonlinear Fokker–Planck–Kolmogorov equations; this is connected with numerous applications in physics and biology, social and economic areas. Even a simple search in the MathSciNet database of the American Mathematical Society for the keywords ‘Fokker-Planck’ produces thousands of papers; moreover, this does not include a large number of papers of the physical and applied nature. Constructions and discussions of models on the basis of Fokker–Planck–Kolmogorov equations can be found in [62], [63], [65], [77], [78], [99], and [100]. For example, in biological models for a swarm (see [99]) a population density ϱ satisfies the one-dimensional (in x) stationary equation with unit diffusion coefficient and drift coefficient b of the form where c0, ca, cr, and a,r∈R are some constants, In [79] another equation popular in biology was considered: In socio-economic models (see [65]) and in traffic models (see [112]) one considers linear and nonlinear Fokker–Planck–Kolmogorov equations on the half-line or an interval with diffusion coefficients degenerate on the boundary. Such equations are reduced by a change of coordinates to the form indicated, with coefficients growing strongly at infinity.In recent years there were intensive studies (see [48], [87], and [109]) of equations of the form where In such equations local and non-local nonlinearities are also combined, that is, the equation contains both nonlinear terms depending on the value of ϱt at the point x and terms depending on ϱt by means of the convolution with a kernel ∇W.Note also the recent papers [3], [7]–[12], which study equations of the form and prove the existence and uniqueness of solutions and their convergence to the stationary solution in L1 as t→+∞, and discuss probabilistic interpretations of solutions. The expression Δβ(ϱt) and a special form of the drift coefficient bring substantial specific features. Probabilistic representations of solutions to linear and nonlinear Fokker–Planck–Kolmogorov equations were also considered in [13] and [30].The problems of the existence, uniqueness, and convergence of probability solutions (that is, non-negative solutions whose integral over Rd equals one) to the stationary solution for the equation with a non-local nonlinearity of the form were studied in [1], [64], and [93], and some recent results and a survey of literature can be found in [27]–[29], [59], [96], and [115]. In [28], [29], [38], and [115] a nonlinear equation is written as a perturbation of the linear equation obtained after substituting the stationary solution into the coefficients. Then, assuming some smallness of the nonlinear part, one proves convergence to the stationary solution as t→+∞. This approach enables one to justify the convergence of solutions of the parabolic equation to solutions of the stationary equation in the case of irregular and strongly growing coefficients. In [2] the equation of the form was studied, where The book [4] develops functional methods of investigation of nonlinear elliptic and parabolic equations, in particular, by employing nonlinear semigroups. A Trotter-type formula for nonlinear Fokker–Planck–Kolmogorov equations was obtained in [5]. In [81] solutions to some nonlinear equations were represented as gradient flows for suitable functionals. Considerable attention has been paid to Vlasov-type equations (see [54], [37], [83], [84], and [114]).In most papers on nonlinear Fokker–Planck–Kolmogorov equations a very important role is played by specific features of the equations under consideration. It is hardly possible to create a general theory covering all types of equations, however, there are papers of a fairly general nature. For example, in [110] and [23] the existence of solutions is established with the aid of Schauder’s fixed point theorem under broad assumptions. The books [76] and [80] study equations of a rather general form, including nonlinear Fokker–Planck–Kolmogorov equations. Linear and nonlinear Fokker–Planck–Kolmogorov equation have interesting connections with other directions of research, including mean field games (see [41], [40], [42], and [106]) and Monge–Kantorovich optimal transportation problems (see [16] and [20]). For stochastic versions of such equations, see, for example, [6], [51], and [66]. Investigation of nonlinear Fokker–Planck–Kolmogorov equations on infinite-dimensional spaces was initiated in [17] and continued in [88] and [53], but so far these problems remain poorly studied. At the end of § 6 we mention an interesting special class of infinite-dimensional Fokker–Planck–Kolmogorov equations on spaces of measures, which arose in connection with finite-dimensional stochastic Itô equations whose coefficients can depend on distributions of solutions. We survey investigations of the last decade on nonlinear Fokker–Planck–Kolmogorov equations of elliptic and parabolic types and present some new results. We discuss problems of the existence and uniqueness of solutions, various estimates for solutions, and the convergence of solutions of parabolic equations to stationary solutions. In § 3 we briefly recall some basic facts about linear equations, since the nonlinear equations under consideration look as linear equations with coefficients depending on solutions. In addition, in § 4 we present some estimates for various distances between solutions to linear equations, obtained by different methods. These estimates play an important role in the study of nonlinear equations. In § 5 we begin the discussion of nonlinear equations from the elliptic case. Here we present results on the existence of solutions obtained with the aid of two types of fixed point theorems: the Schauder–Tychonoff theorem and its multivalued version, the Kakutani–Ky Fan theorem, and also with the aid of the contracting mapping theorem, which employs the estimates from § 4 and also gives conditions for the uniqueness of solutions. The last section, § 6, is concerned with nonlinear parabolic equations and contains results on the existence and uniqueness of solutions and on the convergence of solutions of parabolic equations to solutions of stationary equations. In all sections we give illustrative examples. The vast bibliography in this survey includes only the works most close to the problems we consider and covers a small part of all papers in this very intensively developing area. Of course, under reasonable restrictions on the size of the bibliography it was not possible to present all authors working in this field. A simple search on the basis of the sources included in the bibliography and the papers citing these works increases many times the number of papers and the number of authors on this subject. 2. Notation and terminologyThe standard inner product on Rn is denoted by ⟨x,y⟩ and the Euclidean norm by |x|. The operator norm of an operator A on Rn is denoted by ‖. For a Lebesgue measurable set E in \mathbb{R}^n the symbol L^p(E) denotes the standard Banach space of equivalence classes of functions whose absolute value is integrable over E to power p\geqslant 1. For an open set U\subset \mathbb{R}^n, let W^{p,k}(U), where p\geqslant 1 and k\in\mathbb{N}, be the Sobolev space of functions f in L^p(U) having generalized derivatives \partial_{x_{i_1}}\cdots \partial_{x_{i_m}}f\in L^p(U), m=1,\dots,k. This space is Banach with the norm
\begin{equation*}
\|f\|_{p,k}=\|f\|_{L^p(U)}+\sum \|\partial_{x_{i_1}}\cdots \partial_{x_{i_m}}f\|_{L^p(U)}.
\end{equation*}
\notag
Let W^{p,k}_{\rm loc} denote the set of all functions on \mathbb{R}^n whose restriction to every ball U belongs to W^{p,k}(U). The symbol C_0^\infty(U) denotes the set of all infinitely differentiable functions with compact support in U. A Borel measure on a topological space X is a bounded (possibly, signed) measure on the \sigma-algebra \mathcal{B}(X) of Borel sets. Such a measure \mu is decomposed into the difference \mu=\mu^+-\mu^- of its positive and negative parts, the sum |\mu|=\mu^+ +\mu^- of which is called the total variation of the measure \mu. The variation, or variation norm, of the measure \mu is defined by the equality \|\mu\|=|\mu|(X). Below we mainly deal with Borel measures on \mathbb{R}^d and \mathbb{R}^d\times [0,T]. The set of all Borel probability measures on \mathbb{R}^d, that is, measures \mu\geqslant 0 such that \mu(\mathbb{R}^d)=1, is denoted by \mathcal{P}(\mathbb{R}^d). We also use the larger set \mathcal{SP}(\mathbb{R}^d) of subprobability Borel measures \mu\geqslant 0 such that \mu(\mathbb{R}^d)\leqslant 1. The space of all bounded Borel measures on \mathbb{R}^d is equipped with the Kantorovich–Rubinshtein norm
\begin{equation*}
\|\sigma\|_{\rm KR}=\sup\biggl\{\int_{\mathbb{R}^d} f\, d\sigma\colon \, f\in \operatorname{Lip}_1, \, |f|\leqslant 1\biggr\},
\end{equation*}
\notag
where \operatorname{Lip}_1 is the set of all 1-Lipschitz functions on \mathbb{R}^d, that is, functions f for which |f(x)-f(y)|\leqslant |x-y|. This norm generates the Kantorovich–Rubinshtein metric which makes the sets \mathcal{SP}(\mathbb{R}^d) and \mathcal{P}(\mathbb{R}^d) (but not the whole space of signed measures) complete separable metric spaces (see [15]). The topology on \mathcal{SP}(\mathbb{R}^d) and \mathcal{P}(\mathbb{R}^d) generated by this metric is the weak topology, for which the convergence of a sequence of measures \mu_j to a measure \mu is the convergence of the integrals
\begin{equation*}
\int_{\mathbb{R}^d} f\, d\mu_j\to \int_{\mathbb{R}^d} f\, d\mu
\end{equation*}
\notag
for every bounded continuous function f. In the case where \mu_j,\mu\in \mathcal{P}(\mathbb{R}^d) this convergence is equivalent to the convergence of the integrals of functions from C_0^\infty(\mathbb{R}^d). Let \mathcal{P}_k(\mathbb{R}^d) denote the subspace of \mathcal{P}(\mathbb{R}^d) consisting of the measures with finite kth moment (that is, such that the function |x|^k is integrable). This space is equipped with the Kantorovich k-metric W_k defined by the formula
\begin{equation*}
W_k^k(\mu,\nu)=\inf\int\!\!\!\int_{\mathbb{R}^{d}\times \mathbb{R}^{d}} |x-y|^k\, \sigma(dx\, dy),
\end{equation*}
\notag
where the infimum is taken over all measures \sigma\in \mathcal{P}(\mathbb{R}^{d}\times \mathbb{R}^{d}) with projections \mu and \nu onto the factors.
3. Linear Fokker–Planck–Kolmogorov equationsThe linear elliptic (stationary) Fokker–Planck–Kolmogorov equation on the Euclidean space \mathbb{R}^d arises if on \mathbb{R}^d we are given a mapping with values in the space of symmetric nonnegative definite operators, which is called the diffusion coefficient, and a vector field called the drift coefficient or drift, for which the elements a^{ij} and b^i are Borel functions. These objects generate a second-order elliptic operator
\begin{equation*}
L_{A,b}f=\sum_{i,j\leqslant d} a^{ij}\partial_{x_i}\partial_{x_j}f+ \sum_{i\leqslant d} b^{i}\partial_{x_i}f.
\end{equation*}
\notag
A bounded (possibly, signed) Borel measure \mu on \mathbb{R}^d is called a solution to the stationary Fokker–Planck–Kolmogorov equation with the operator L_{A,b} if the functions a^{ij} and b^i are integrable on compact sets with respect to the measure |\mu| and the following identity is fulfilled:
\begin{equation}
\int_{\mathbb{R}^d} L_{A,b}f(x)\, \mu(dx)=0,\qquad f\in C_0^\infty(\mathbb{R}^d).
\end{equation}
\tag{3.2}
In the theory of partial differential equations such equations are called double divergence form equations. In the sense of distributions identity (3.2) can be written as
\begin{equation*}
\partial_{x_i}\partial_{x_j}(a^{ij}\mu)-\partial_{x_i}(b^{i}\mu)=0,
\end{equation*}
\notag
where summation over repeated indices is meant and a^{ij}\mu and b^{i}\mu are distributions, being locally finite measures. However, one should bear in mind that in the case of non-smooth coefficients the operator L_{A,b}^* is not defined at all distributions. With the aid of this identity the equation can also be defined for measures on domains or on Riemannian manifolds, moreover, one can admit measures finite on compact sets, but for the purposes of this survey we confine ourselves to globally bounded measures. Detailed surveys of the properties of solutions are given in [22] and [23] (see also the recent survey [31]), we only note here that in case of nonnegative solutions the measure (\det A)^{1/d}\mu possesses a density with respect to Lebesgue measure, so that for a non-degenerate diffusion matrix the solution itself has a density \varrho. If the matrix A is non-degenerate, its elements belong to the local Sobolev class W^{p,1}_{\rm loc} for some p>d, and the function |b^i|^p is locally integrable with respect to Lebesgue measure or the measure |\mu|, then \mu has a continuous density in the same Sobolev class, and in the case of a positive measure \mu this density has no zeros. For solutions with densities in Sobolev classes and coefficients a^{ij} from Sobolev classes (3.1) reduces to a divergence-form equation with respect to the density:
\begin{equation}
\partial_{x_i}(a^{ij}\partial_{x_j}\varrho+\partial_{x_j}a^{ij}\varrho)- \partial_{x_i}(b^{i}\varrho)=0.
\end{equation}
\tag{3.3}
In this case a solution can fail to have second Sobolev derivatives, but if they exist and the functions a^{ij} have second Sobolev derivatives, while the functions b^i have first Sobolev derivatives, then this equation becomes a classical elliptic equation. However, not every classical equation can be obtained from an equation of the form (3.1). One can also consider Fokker–Planck–Kolmogorov equations with a potential term c, which is a Borel function, locally integrable with respect to the solution \mu, which now satisfies the equation understood similarly to (3.2) for the operator L_{A,b,c}f=L_{A,b}f+c f.Various conditions are known for the existence and uniqueness of probability solutions. Most convenient ones in applications employ Lyapunov functions. Theorem 3.1. Let V\in C^2(\mathbb{R}^d) be a quasicompact function, so that \mathbb{R}^d is the union of the compact sets \{V\leqslant c_k\} for some increasing sequence of numbers c_k, and
\begin{equation*}
L_{A,b}V(x)\leqslant -1 \quad \textit{if}\ \ |x|\geqslant R
\end{equation*}
\notag
for some R>0. Suppose that A is continuous and either b is also continuous, or b is locally bounded and \det A>0. Then there exists a probability solution to equation (3.1). It is unclear whether there is always a probability solution in the case where a Lyapunov function exists if A and A^{-1} are locally bounded and the drift b is locally integrable to power p>d with respect to Lebesgue measure. Let us also mention the following result on the uniqueness of solutions. Theorem 3.2. If a^{ij}\in W^{p,1}_{\rm loc} for some p>d, the continuous version of the mapping A (which exists by the Sobolev embedding theorem) is non-degenerate and b^i\in L^{p}_{\rm loc}, then a probability solution \mu is unique, provided that the functions
\begin{equation*}
|a^{ij}(x)|(1+|x|)^{-2} \quad {and}\quad |b^{i}(x)|(1+|x|)^{-1}
\end{equation*}
\notag
belong to L^1(\mu). In particular, this is true if the coefficients have an at most linear growth. In terms of Lyapunov functions the following condition holds: there are no different probability solutions if there exists a function V\in C^2(\mathbb{R}^d) such that V(x)\to +\infty as |x|\to\infty and the estimate L_{A,b}V\leqslant C+CV holds for some C>0. Without additional conditions on the drift b even when it is smooth the equation with the unit diffusion matrix can have different probability solutions for d>1. The linear parabolic Fokker–Planck–Kolmogorov equation is also associated with a pair of mappings, the diffusion coefficient A(x,t)=(a^{ij}(x,t))_{i,j\leqslant d} and the drift coefficient b(x,t)=(b^i(x,t))_{i\leqslant d}, which are now defined on \mathbb{R}^d\times [0,T] for some T>0 and take values in the space of symmetric non-negative definite operators and in \mathbb{R}^d, respectively. As in the elliptic case, we assume that they are Borel measurable. Solutions to parabolic equations will be considered in the class of (possibly, signed) measures of the form \mu(dx\, dt)=\mu_t(dx)\, dt for some family of Borel measures on \mathbb{R}^d depending on t Borel measurably, that is, the functions t\mapsto \mu_t(B) must be Borel measurable for Borel sets B, the function t\mapsto \|\mu_t\| must be Lebesgue integrable on [0,T], and the expression indicated means that for any bounded Borel function f on \mathbb{R}^d\times [0,T] we have the equality
\begin{equation*}
\int_{\mathbb{R}^d\times [0,T]} f(x,t)\, \mu(dx\, dt)= \int_0^T\int_{\mathbb{R}^d} f(x,t)\, \mu_t(dx)\, dt.
\end{equation*}
\notag
A bounded Borel measure \mu on \mathbb{R}^d\times (0,T) of the form \mu(dx\, dt)=\mu_t(dx)\, dt satisfies the Fokker–Planck–Kolmogorov equation if the functions a^{ij} and b^i are integrable with respect to |\mu| on compact sets in \mathbb{R}^d\times (0,T) and the following equality holds for every function \varphi\in C_0^\infty (\mathbb{R}^d\times (0,T)):
\begin{equation}
\int_{\mathbb{R}^d\times (0,T)}[\partial_t\varphi+ L_{A,b}\varphi]\, \mu(dx\, dt)=0.
\end{equation}
\tag{3.5}
This is actually the same equation as in the elliptic case, but with an elliptic operator with two arguments which is completely degenerate in t. An additional difference is the form of the measure. Equation (3.4) can also be written in the form without the index t for the measure, when it is meant that \mu has the form \mu(dx\, dt)=\mu_t(dx)\, dt indicated above. This agrees with the convention that in the case where a density \varrho(x,t) exists the equation is written as \partial_t\varrho=L_{A,b}^* \varrho. The measure \mu is often identified with the family of measures (\mu_t)_{t\in (0,T)}. The Cauchy problem for the Fokker–Planck–Kolmogorov equation with the operator L_{A,b} is posed by adding an initial condition, that is, an initial measure \nu. Here, however, several different (inequivalent in the general case) technical formulations are possible. We define a solution to the Cauchy problem with initial measure \nu as a family of Borel measures (\mu_t)_{t\in [0,T]} on the space \mathbb{R}^d, Borel measurably depending on t, such that \mu_0=\nu, the measure |\mu_t|\, dt on \mathbb{R}^d\times [0,T] is finite and for every function \varphi\in C_0^\infty(\mathbb{R}^d) the equality
\begin{equation}
\int_{\mathbb{R}^d}\varphi\, d\mu_t-\int_{\mathbb{R}^d}\varphi\, d\nu =\int_0^t \int_{\mathbb{R}^d} L_{A,b}\varphi\, d\mu_s\, ds
\end{equation}
\tag{3.7}
holds almost everywhere on [0,T]. The measure \mu(dx\, dt)=\mu_t(dx)\, dt on \mathbb{R}^d\times [0,T] is also called a solution. A probability solution is a solution in which almost all the \mu_t are probability measures. Note that any solution to the Cauchy problem also satisfies the Fokker–Planck–Kolmogorov equation (3.4). This is first verified for functions with smooth functions \varphi_1 and \varphi_2 with compact support, by multiplying (3.7) by the function \varphi_2'(t), integrating over [0,T], and then integrating by parts on the right.There is a Borel set E\subset [0,T] of Lebesgue measure zero such that (3.7) is true for all t\not\in E and all \varphi\in C_0^\infty (\mathbb{R}). Indeed, there exists a countable system of functions \varphi_n\in C_0^\infty (\mathbb{R}^d) with the following property: for every function \varphi\in C_0^\infty (\mathbb{R}^d) there is a subsequence \{\varphi_{n_j}\} such that the functions \varphi_{n_j} have support in some common ball containing the support of \varphi and converge uniformly to \varphi so that their first and second derivatives also converge uniformly. The set E of points t for which (3.7) is violated for some function \varphi=\varphi_n is Borel and has measure zero. Let t\not\in E and \varphi\in C_0^\infty (\mathbb{R}^d). Let us take the corresponding subsequence \{\varphi_{n_j}\} with the properties indicated above. Then the functions L_{A,b}\varphi_{n_j} converge to L_{A,b}\varphi in L^1 with respect to the measure |\mu_t|\, dt, and the integrals of the functions \varphi_{n_j} against the measures \mu_t and \nu converge to the integrals of \varphi. Hence equality (3.7), valid for \varphi_n, remains true for \varphi. In this connection the question arises: why do not we redefine the solution at the points of the set E of measure zero in order to obtain the equality at all points? As the next proposition shows, this is in principle possible, but an important nuance appears: if all measures \mu_t of the solution we wish to redefine are probability measures (as desired in many problems), then this property can be lost after the redefinition. There are examples showing that even in the case of smooth coefficients a probability solution (\mu_t)_{t\geqslant 0} cannot be redefined in such a way that all measures \mu_t become probability measures and (3.7) is true for all t. Proposition 3.3. Suppose that we are given a family of bounded Borel measures \mu_t on \mathbb{R}^d, t\in [0,T], Borel measurably depending on t, such that and \nu is a bounded Borel measure on \mathbb{R}^d. Suppose also that Borel functions a^{ij} and b^i on \mathbb{R}^d are such that
\begin{equation*}
\int_0^T \int_B \bigl[|a^{ij}|+|b^i|\bigr]\, d|\mu_s|\, ds <\infty
\end{equation*}
\notag
for every ball B, and for every function f\in C_0^\infty (\mathbb{R}^d) the following equality holds for almost all t\in [0,T]:
\begin{equation*}
\int_{\mathbb{R}^d} f\, d\mu_t-\int_{\mathbb{R}^d} f\, d\nu= \int_0^t \int_{\mathbb{R}^d} L_{A,b}f\, d\mu_s\, ds, \qquad f\in C_0^\infty(\mathbb{R}^d).
\end{equation*}
\notag
Then there exist bounded measures \widetilde{\mu}_t Borel measurably depending on t, for which \sup_t\|\widetilde{\mu}_t\|\leqslant M, \widetilde{\mu}_t=\mu_t for almost all t, and the above equality is fulfilled for all t\in [0,T]. Proof. Let us take the Borel set E of Lebesgue measure zero indicated above and for t\in E redefine the measure \mu_t as follows. Fix the open ball B_k of radius k\in\mathbb{N} with centre at the origin and on the space of infinitely differentiable functions with support in B_k consider the linear functional
\begin{equation*}
L_{t,k}f=\int_0^t \int_{\mathbb{R}^d} L_{A,b}f\, d\mu_s\, ds+ \int_{\mathbb{R}^d} f\, d\nu.
\end{equation*}
\notag
There is a sequence of points t_n\to t, t_n\in E. Then L_{t_n,k}f\to L_{t,k}f for all f\in C_0^\infty(\mathbb{R}^d) and The right-hand side does not exceed M\sup_x |f(x)| in absolute value. Hence By Riesz’s theorem there exists a Borel measure \sigma_{t,k} on B_k for which the functional L_{t,k} is the integral against the measure \sigma_{t,k}; moreover, \|\sigma_{t,k}\|\leqslant M. The restriction of the measure \sigma_{t,k+1} to B_k coincides with \sigma_{t,k+1} since both measures assign equal integrals to smooth functions with support in B_k. Therefore, we have a measure \widetilde{\mu}_t on \mathbb{R}^d with restrictions \sigma_{t,k} to B_k and variation at most M. For points in [0,T]\setminus E we retain the original measures \mu_t. The family obtained satisfies our integral identity for all t. This yields its Borel measurability in t, since the right-hand side of the integral identity is continuous in t. \Box In the case of subprobability solutions there is a useful similar interpretation of solutions as elements of the space L^0([0,T],\mathcal{SP}(\mathbb{R}^d)) of equivalence classes of Lebesgue measurable mappings from [0,T] to the space \mathcal{SP}(\mathbb{R}^d) of subprobability Borel measures on \mathbb{R}^d equipped with the Kantorovich–Rubinshtein metric d_{\rm KR}, making it a complete separable metric space. Thus, also L^0([0,T],\mathcal{SP}(\mathbb{R}^d)) becomes a complete separable metric space with the metric of convergence in measure where some representatives of equivalence classes are used (the integral does not depend on their choice).For the existence and uniqueness of solutions to the Cauchy problem broad sufficient condition are also known, expressed in terms of estimates for coefficients and their global integrability with respect to solutions or in terms of Lyapunov functions (see [23], Chaps. 6 and 9). Only in the one-dimensional case, in the recent paper [21] the uniqueness of probability solutions was established without such conditions (but for drifts independent of time). In addition, there are results about the properties of solution densities (see [23], Chaps. 7 and 8). In particular, the very existence of a density is ensured by the condition \det A(x,t)>0, as in the elliptic case. Parabolic equations are solvable under much broader conditions than elliptic ones. For example, for A={\rm I} and b=0 the elliptic equation has no non-zero solutions, while the parabolic equation has the standard Gaussian solution. There is an inverse implication: under some restrictions on A and b the existence of a probability solution to the elliptic equation L_{A,b}^*\mu=0 implies the existence of a subprobability (but not always a probability) solution to the Cauchy problem with an arbitrary initial probability distribution. 4. Estimates of distances between solutions to linear equationsIn the study of nonlinear Fokker–Planck–Kolmogorov equations it is useful to have estimates for various distances between solutions to linear elliptic and parabolic equations with distinct coefficients. This subject deserves a separate survey; here we only present several results from [25] and [26] used below. First we give a number of estimates from [26] (see also [18] and [19]) for elliptic equations. Suppose that two measures
\begin{equation*}
\mu=\varrho_{\mu}\,dx \qquad \text{and}\qquad \sigma=\varrho_{\sigma}\,dx
\end{equation*}
\notag
are probability solutions to the equations L_{A_{\mu}, b_{\mu}}^{*}\mu=0 and L_{A_{\sigma},b_{\sigma}}^{*}\sigma=0 the coefficients of which satisfy the following conditions: the elements of the diffusion matrices belong to the Sobolev class W^{p, 1}_{\rm loc} for some p>d, the continuous versions of these matrices are positive definite, and the components of the drifts belong to L^p_{\rm loc}. Let us introduce the following notation:
\begin{equation*}
\begin{alignedat}{2} h_\mu&=(h_\mu^i)_{i=1}^d, &\qquad h_{\mu}^i&=b^i_{\mu}-\sum_{j=1}^d\partial_{x_j}a_{\mu}^{ij}, \\ h_\sigma&=(h_\sigma^i)_{i=1}^d, &\qquad h_{\sigma}^i&=b^i_{\sigma}-\sum_{j=1}^d\partial_{x_j}a_{\sigma}^{ij}. \end{alignedat}
\end{equation*}
\notag
Set
\begin{equation*}
\Phi=\frac{(A_{\mu}-A_{\sigma})\nabla\varrho_{\sigma}}{\varrho_{\sigma}}+ h_{\sigma}-h_{\mu}.
\end{equation*}
\notag
Observe that \Phi=b_{\sigma}-b_{\mu} if A_{\mu}=A_{\sigma}. Throughout, for shorter notation, in place of L_{A_{\mu}, b_{\mu}} and L_{A_{\sigma}, b_{\sigma}} we write L_{\mu} and L_{\sigma}, respectively. Let We denote by W^{p,1}(\mu) the weighted Sobolev class obtained by the completion of the space C_0^\infty(\mathbb{R}^d) with respect to the Sobolev norm \|f\|_{p,1,\mu} which differs from the standard norm by using the measure \mu in place of Lebesgue measure. As the density of \mu is continuous and positive, functions in this class do not differ locally from functions in W^{p,1}_{\rm loc}. Hence W^{p,1}(\mu) consists of functions of class W^{p,1}_{\rm loc} with finite norm \|\cdot\|_{p,1,\mu}.Theorem 4.1. Suppose that |A_{\mu}^{-1/2}\Phi|\in L^2(\sigma) and at least one of the following conditions is fulfilled: (i) the functions (1+|x|)^{-2}a^{ij}_{\mu} and (1+|x|)^{-1}|b_{\mu}| belong to L^1(\mu); (ii) there exists a function V\in C^2(\mathbb{R}^d) such that \lim_{|x|\to\infty}V(x)=+\infty, V\geqslant 0, for all x and some M>0, and also the inclusion holds. Then
\begin{equation*}
\int_{\mathbb{R}^d}\frac{|A_{\mu}^{1/2}\nabla v|^2}{v}\,d\mu\leqslant \int_{\mathbb{R}^d}|A^{-1/2}_{\mu}\Phi|^2\,d\sigma.
\end{equation*}
\notag
If A_{\mu}\geqslant \alpha\cdot {\rm I}, then it follows from this estimate that \sqrt{v}\in W^{2,1}(\mu). Given a number C_{\rm S} and a Borel measurable matrix-valued mapping A, we say that a probability measure \mu satisfies the logarithmic Sobolev inequality with constant C_{\rm S} and matrix A if
\begin{equation*}
\operatorname{Ent}_{\mu}f^2:=\int_{\mathbb{R}^d}f^2\log (f^2)\,d\mu- \int_{\mathbb{R}^d}f^2\,d\mu\log\int_{\mathbb{R}^d}f^2\,d\mu\leqslant C_{\rm S}\int_{\mathbb{R}^d}|A^{1/2}\nabla f|^2\,d\mu
\end{equation*}
\notag
for every function f\in C_0^\infty(\mathbb{R}^d). Under our assumptions this inequality extends to all functions f\in W^{2,1}(\mu) if A is bounded. Corollary 4.2. If, in addition to the hypotheses of Theorem 4.1, it is known that the measure \mu satisfies the logarithmic Sobolev inequality with constant C_{\rm S} and matrix A_{\mu}, then the following assertions are true. (i) The entropy estimate
\begin{equation*}
\operatorname{Ent}_{\mu}v\leqslant \frac{C_{\rm S}}{4}\int_{\mathbb{R}^d}|A^{-1/2}_{\mu}\Phi|^2\,d\sigma
\end{equation*}
\notag
holds. (ii) If the measures \mu and \sigma have finite second moments, then the estimate for the Kantorovich 2-metric
\begin{equation*}
W_2(\mu,\sigma)^2\leqslant \frac{C_{\rm S}^2}{4}\int_{\mathbb{R}^d} |A^{-1/2}_{\mu}\Phi|^2\,d\sigma
\end{equation*}
\notag
is fulfilled. (iii) The estimate for the variation holds.Example 4.3. Suppose that A_{\mu}=A_{\sigma}={\rm I} and for some positive number \kappa the inequality
\begin{equation*}
\langle b_{\mu}(x)-b_{\mu}(y),x-y\rangle\leqslant -\kappa|x-y|^2
\end{equation*}
\notag
is fulfilled. Then the measure \mu satisfies the logarithmic Sobolev inequality with constant 2/\kappa (see [23], Theorem 5.6.36) and the estimates in (i)–(iii) above hold with this constant in place of C_{\rm S}. Moreover, the integral in the right-hand side of these inequalities equals \|b_{\mu}-b_{\sigma}\|_{L^2(\sigma)}^2 in this case. Given a number C_{\rm P} and a Borel measurable matrix-valued mapping A, we say that a probability measure \mu satisfies Poincaré’s inequality with constant C_{\rm P} and matrix A if
\begin{equation*}
\int_{\mathbb{R}^d}\biggl|f-\int_{\mathbb{R}^d}f\,d\mu\biggr|^2\,d\mu\leqslant C_{\rm P}\int_{\mathbb{R}^d}|A^{1/2}\nabla f|^2\,d\mu
\end{equation*}
\notag
for every function f\in C_0^\infty(\mathbb{R}^d). Under our assumptions this inequality extends to all functions f\in W^{2,1}(\mu) if A is bounded. We recall that the Hellinger integral for two measures \mu=\varrho_{\mu}\, dx and \nu=\varrho_{\sigma}\, dx is the quantity
\begin{equation*}
H(\mu, \sigma)=\int_{\mathbb{R}^d}\sqrt{\varrho_{\mu}\varrho_{\sigma}}\,dx.
\end{equation*}
\notag
Corollary 4.4. If, in addition to the hypotheses of Theorem 4.1, it is known that the measure \mu satisfies Poincaré’s inequality with constant C_{\rm P} and matrix A_{\mu}, then the following inequalities are valid: andAnother method of obtaining estimates for the total variation distance between solutions consists in using solutions to Poisson’s equation L_{\mu}u=\psi, where the integral of \psi against the measure \mu vanishes. Let us give an example of a result proved in this way. Theorem 4.5. Suppose that there exist a positive function V\in C^2(\mathbb{R}^d) with the property \lim_{|x|\to \infty}V(x)=+\infty, a positive number \gamma, and a ball Q of radius R with centre at the origin such that
\begin{equation*}
L_{A_{\mu},b_{\mu}}V(x)\leqslant -\gamma V(x) \quad \textit{if} \ \ x\in\mathbb{R}^d\setminus Q.
\end{equation*}
\notag
Suppose also that the functions
\begin{equation*}
\begin{gathered} \, V^2, \quad (1+|x|)^{-1}|\Phi|V^2, \quad |A_\mu^{-1/2}\Phi|^2V^2, \\ (1+|x|)^{-1}|b_{\mu}^i|V^2, \quad\textit{and}\quad (1+|x|)^{-2}|a_{\mu}^{ij}|V^2 \end{gathered}
\end{equation*}
\notag
are integrable with respect to the measure \sigma on the whole space \mathbb{R}^d. Then one has the estimate
\begin{equation*}
\|V(\mu-\sigma)\|\leqslant C\biggl(\int_{\mathbb{R}^d} |A_\mu^{-1/2}\Phi|^2\,d\sigma\biggr)^{1/2}\biggl(\int_{\mathbb{R}^d} \bigl[V^2+|A_\mu^{-1/2}\Phi|^2V^2\bigr]\,d\sigma\biggr)^{1/2},
\end{equation*}
\notag
where C depends on d, \gamma, R, \|a^{ij}\|_{W^{p, 1}(Q_1)}, \|b^i\|_{L^p(Q_1)}, \sup_{x\in Q_1}\|A(x)^{-1}\|, and also on the minimum of the function V on Q_1 and the maxima of the function V and of the absolute values of its first and second derivatives on Q_1, where Q_1 is the ball of radius R+1 with centre at the origin. Example 4.6. Let A_{\mu}=A_{\sigma}={\rm I} and
\begin{equation*}
\langle b_{\mu}(x), x\rangle\leqslant-\gamma|x|^2, \quad \text{where}\ \ \gamma>0.
\end{equation*}
\notag
Suppose that |b_{\mu}(x)|+|b_{\sigma}(x)|\leqslant C_0(1+|x|)^m for some numbers m\geqslant 1 and C_0>0. If the measure \sigma has a finite moment of order 2m+2, then the hypotheses of Theorem 4.5 are fulfilled and the estimate
\begin{equation*}
\|(1+|x|)^2(\mu-\sigma)\|\leqslant C\|b_{\mu}-b_{\sigma}\|_{L^2(\sigma)} \bigl(\|(1+|x|)^2\|_{L^2(\sigma)}+ \|(1+|x|)^2(b_{\mu}-b_{\sigma})\|_{L^2(\sigma)}\bigr)
\end{equation*}
\notag
holds, where C depends on \gamma, C_0, and m. In [60] an estimate for the quadratic Kantorovich distance W_2 between stationary distributions was obtained under the assumption that the coefficients of the first equation belong to the Sobolev class with weight \varrho_\mu, the measures \mu= \varrho_\mu\, dx and \nu=\varrho_\nu\, dx are mutually absolutely continuous, and solutions of the Cauchy problem for the first operator converge exponentially rapidly to the stationary distribution \varrho_\mu in the metric W_2. In [25] an estimate for distances between solutions to parabolic Fokker–Planck–Kolmogorov equations was obtained. Consider a time-dependent elliptic operator
\begin{equation*}
L_{A,b}u=\sum_{i,j=1}^d a^{ij}\partial_{x_i}\partial_{x_j} u+ \sum_{i=1}^d b^i \partial_{x_i}u,
\end{equation*}
\notag
where A(x,t)=(a^{ij}(x,t))_{i,j\leqslant d} is a positive definite symmetric matrix with Borel elements and b(x,t)=(b^i(x,t))_{i=1}^d \colon \mathbb{R}^d\times [0,T]\to \mathbb{R}^d is a Borel mapping; moreover, b is locally bounded, that is, for every ball U\subset \mathbb{R}^d there is a number B=B(U)\geqslant 0 such that
\begin{equation*}
|b(x,t)|\leqslant B(U) \quad \forall\, x\in U,\ t\in [0,T],
\end{equation*}
\notag
and the matrix A is locally Lipschitz in x and positive, that is, satisfies the following condition: (H) for every ball U\subset \mathbb{R}^d there exist numbers \lambda=\lambda(U)\geqslant 0, \alpha=\alpha(U)>0, and m=m(U)>0 such that
\begin{equation*}
|a^{ij}(x, t)-a^{ij}(y, t)|\leqslant \lambda|x-y| \quad\text{and}\quad \alpha\cdot {\rm I}\leqslant A(x, t)\leqslant m\cdot {\rm I}\quad \forall\, x,y\in U, \ t\in[0, T].
\end{equation*}
\notag
Let us consider the Cauchy problem where \nu\in \mathcal{P}(\mathbb{R}^d). As above, we consider solutions of the form \mu(dx\,dt)=\mu_t(dx)\,dt on \mathbb{R}^d\times [0,T] given by families of probability measures (\mu_t)_{t\in[0,T]} on \mathbb{R}^d, Borel measurably depending on t.Suppose now that there are two solutions \mu=(\mu_t)_{t\in [0, T]} and \sigma=(\sigma_t)_{t\in [0, T]} to the Cauchy problem (4.2) with coefficients A_{\mu}, b_{\mu} and A_{\sigma}, b_{\sigma}, respectively, and a common initial condition \nu. So far we are speaking of linear equations, hence the indices \mu and \sigma do not mean any dependence of coefficients on solutions, but only label equations. Suppose that
\begin{equation*}
\text{the matrices $A_{\mu}$ and $A_{\sigma}$ satisfy condition (H)}
\end{equation*}
\notag
and
\begin{equation*}
\text{the vector fields $b_{\mu}$ and $b_{\sigma}$ are locally bounded.}
\end{equation*}
\notag
Let \mu=\varrho_{\mu}(x,t)\,dx\,dt and \sigma=\varrho_{\sigma}(x,t)\,dx\,dt. Set
\begin{equation*}
v(x,t)=\frac{\varrho_{\sigma}(x,t)}{\varrho_{\mu}(x,t)}\,,\quad \text{that is,}\quad \sigma=v\cdot\mu.
\end{equation*}
\notag
In the estimates below we use the mappings
\begin{equation*}
h_\mu=(h_\mu^i)_{i=1}^d, \quad h_\sigma=(h_\sigma^i)_{i=1}^d, \quad h_{\mu}^i=b^i_{\mu}-\sum_{j=1}^d\partial_{x_j}a_{\mu}^{ij}, \quad h_{\sigma}^i=b^i_{\sigma}-\sum_{j=1}^d\partial_{x_j}a_{\sigma}^{ij},
\end{equation*}
\notag
and
\begin{equation*}
\Phi=\frac{(A_{\mu}-A_{\sigma})\nabla\varrho_{\sigma}}{\varrho_{\sigma}}- (h_{\mu}-h_{\sigma}).
\end{equation*}
\notag
The distance between \mu_t and \sigma_t can be estimated in terms of the L^2(\sigma)-norm of the vector field A_\mu^{-1/2}\Phi. In the case of equal diffusion matrices \Phi=b_\sigma-b_\mu. Recall that for two measures \mu_1 and \mu_2 in \mathcal{P}(\mathbb{R}^d) such that \mu_1=w\cdot \mu_2 the entropy H(\mu_1\,|\,\mu_2) is defined by the formula provided that w\log w \in L^1(\mu_2). If \mu_1 and \mu_2 are given by positive densities \varrho_1 and \varrho_2 such that \varrho_1\log (\varrho_1/\varrho_2)\in L^1(\mathbb{R}^d), then H(\mu_1\,|\,\mu_2) is the integral of \varrho_1\log(\varrho_1/\varrho_2).Theorem 4.7. Suppose that |A_{\mu}^{-1/2}\Phi|\in L^2(\mathbb{R}^d\times[0,T],\sigma) and at least one of the following two conditions is fulfilled: (a) (1+|x|)^{-2}|a_{\mu}^{ij}|, (1+|x|)^{-1}|b_{\mu}|, (1+|x|)^{-1}|\Phi|\in L^1(\mathbb{R}^d\times[0,T],\sigma); (b) there exists a non-negative function V\in C^2(\mathbb{R}^d) and M\geqslant 0 such that
\begin{equation*}
\lim_{|x|\to\infty}V(x)=+\infty, \quad L_{A_{\mu},b_{\mu}}V\leqslant MV, \quad\textit{and} \quad \frac{\langle\Phi,\nabla V\rangle}{1+V}\in L^1(\mathbb{R}^d\times[0,T],\sigma).
\end{equation*}
\notag
Then Corollary 4.8. Under the hypotheses of the theorem, for every non-negative Borel function \varphi on \mathbb{R}^d\times[0,T] we have
\begin{equation}
\begin{aligned} \, \nonumber \|\varphi(\mu_t-\sigma_t)\|^2&\leqslant \bigl(1+\log\alpha(t)\bigr) \\ &\qquad\times\int_0^t\int_{\mathbb{R}^d} \biggl|\frac{A_{\mu}^{-1/2}(A_{\mu}-A_{\sigma})\nabla\varrho_{\sigma}} {\varrho_{\sigma}}-A_{\mu}^{-1/2}(h_{\mu}-h_{\sigma})\biggr|^2\,d\sigma_s\,ds, \end{aligned}
\end{equation}
\tag{4.4}
where
\begin{equation*}
\alpha(t):=\int_{\mathbb{R}^d}e^{\varphi^2(x, t)}\,\mu_t(dx).
\end{equation*}
\notag
If A_{\mu}=A_{\sigma}=A, then
\begin{equation*}
\|\varphi(\mu_t-\sigma_t)\|^2\leqslant \bigl(1+\log \alpha(t)\bigr) \int_0^t\int_{\mathbb{R}^d} \bigl|A^{-1/2}(b_{\mu}-b_{\sigma})\bigr|^2\,d\sigma_s\,ds.
\end{equation*}
\notag
If b_{\mu}=b_{\sigma}=b and A_\mu and A_\sigma do not depend on x, then
\begin{equation*}
\|\varphi(\mu_t-\sigma_t)\|^2\leqslant \bigl(1+\log \alpha(t)\bigr) \int_0^t\int_{\mathbb{R}^d}\frac{\bigl|(A_{\mu}^{1/2} A_{\sigma}^{-1/2} -A_{\mu}^{-1/2}A_{\sigma}^{1/2})A_{\sigma}^{1/2} \nabla\varrho_{\sigma}\bigr|^2}{\varrho_{\sigma}}\,dx\,ds.
\end{equation*}
\notag
Finally, for \varphi=1 these estimates are fulfilled with 1 in place of 1+\log \alpha(t). Estimates of distances between solutions to Fokker–Planck–Kolmogorov equations with dissipative drift coefficients were obtained in [89] with the aid of expressions generalizing the classical Kantorovich metric, and estimates of distances between solutions to equations with partially degenerate diffusion matrices, when degeneracy occurs only in part of the variables, were obtained in [91]. Note that such equations arise in the study of the system of stochastic Langevin equations. There are many papers on estimates of the rate of convergence of solutions of linear parabolic Fokker–Planck–Kolmogorov equations to stationary solutions as t\to+\infty (see, for example, [35], [58], [59], [75], and also the subsequent works citing these papers). 5. Nonlinear elliptic equationsNonlinear elliptic Fokker–Planck–Kolmogorov equations differ from linear ones by a dependence of coefficients on solutions. A typical example of such a dependence is supplied by expressions of the form The interest in nonlinear elliptic Fokker–Plank–Kolmogorov equations is explained in the first place by the fact that solutions of nonlinear parabolic Fokker–Plank– Kolmogorov equations approach to solutions of such equations as t\to+\infty. The key role in obtaining sufficient conditions for the existence and uniqueness of solutions is played by estimates of solutions of linear equations with the aid of the method of Lyapunov functions, by estimates of distances between solutions of two different linear equations, and by sufficient conditions for the existence of continuous solution densities with respect to Lebesgue measure. Let us proceed to precise formulations.Let
\begin{equation*}
V\in C^2(\mathbb{R}^d), \quad V\geqslant 0, \quad \text{and} \quad \lim_{|x|\to\infty}V(x)=+\infty.
\end{equation*}
\notag
For \alpha>0 we denote by \mathcal{P}_{\alpha,V}(\mathbb{R}^d) the set of probability measures \mu on \mathbb{R}^d satisfying the condition The set \mathcal{P}_{\alpha,V}(\mathbb{R}^d) is non-empty for sufficiently large \alpha and is convex, metrizable, and compact in the weak topology. The set of all probability measures with respect to which the function V is integrable is denoted by \mathcal{P}_V(\mathbb{R}^d). We say that a sequence of measures \mu_n\in \mathcal{P}_V(\mathbb{R}^d) converges V-weakly to a measure \mu\in\mathcal{P}_V(\mathbb{R}^d) if
\begin{equation*}
\lim_{n\to\infty}\int_{\mathbb{R}^d}f(x)\, \mu_n(dx)= \int_{\mathbb{R}^d}f(x)\,\mu(dx)
\end{equation*}
\notag
for all continuous functions f such that \lim_{|x|\to\infty} f(x)/V(x)=0. Let us verify that weak convergence of a sequence of measures \mu_n\in \mathcal{P}_{\alpha,V}(\mathbb{R}^d) to a measure \mu\in\mathcal{P}_{\alpha,V}(\mathbb{R}^d) yields V-weak convergence of \mu_n to \mu. Let f be a continuous function, and let \lim_{|x|\to\infty} f(x)/V(x)=0. Set
\begin{equation*}
f_{N}(x)=\min\bigl\{\max\{f(x), -N^{-1}V(x)\}, N^{-1}V(x)\bigr\}.
\end{equation*}
\notag
Since |f_N|\leqslant N^{-1}V and the integrals of V against the given measures do not exceed \alpha, the estimates
\begin{equation*}
\biggl|\int_{\mathbb{R}^d}f\,d\mu_n-\int_{\mathbb{R}^d}(f-f_N)\,d\mu_n\biggr| \leqslant \frac{\alpha}{N}\quad\text{and}\quad \biggl|\int_{\mathbb{R}^d}f\,d\mu-\int_{\mathbb{R}^d}(f-f_N)\,d\mu\biggr| \leqslant \frac{\alpha}{N}
\end{equation*}
\notag
are true, hence the equality
\begin{equation*}
\lim_{n\to\infty}\int_{\mathbb{R}^d}f(x)\,\mu_n(dx)= \int_{\mathbb{R}^d}f(x)\,\mu(dx),
\end{equation*}
\notag
follows since the continuous function f-f_N vanishes outside some ball. Suppose that for every measure \mu\in\mathcal{P}_V(\mathbb{R}^d) and all i,j\leqslant d we are given Borel functions a^{ij}(\,\cdot\,,\mu) and b^i(\,\cdot\,,\mu) such that the matrix A(x,\mu)=(a^{ij}(x,\mu))_{1\leqslant i,j\leqslant d} is symmetric and non-negative definite. Set
\begin{equation*}
L_{\mu}\varphi(x)=\sum_{i,j\leqslant d}a^{ij}(x,\mu) \partial_{x_i}\partial_{x_j}\varphi(x)+ \sum_{i\leqslant d}b^i(x,\mu)\partial_{x_i}\varphi(x).
\end{equation*}
\notag
A measure \mu\in\mathcal{P}_V(\mathbb{R}^d) is a solution to the nonlinear stationary Fokker–Planck–Kolmogorov equation if \mu is a solution to the linear equation with operator L_{\mu}. Theorem 5.1. Suppose that the following conditions are fulfilled: (i) for every ball B and every \alpha>0 the functions a^{ij}(\,\cdot\,,\mu) and b^i(\,\cdot\,,\mu) are bounded on B uniformly in \mu\in \mathcal{P}_{\alpha, V}(\mathbb{R}^d) and are equicontinuous on B uniformly in \mu\in \mathcal{P}_{\alpha, V}(\mathbb{R}^d); (ii) if measures \mu_n\in \mathcal{P}_V(\mathbb{R}^d) converge V-weakly to a measure \mu\in\mathcal{P}_V(\mathbb{R}^d), then for all x we have
\begin{equation*}
\lim_{n\to\infty}b^i(x,\mu_n)=b^i(x,\mu)\quad\textit{and}\quad \lim_{n\to\infty}a^{ij}(x,\mu_n)=a^{ij}(x,\mu);
\end{equation*}
\notag
(iii) there exist numbers \delta\in [0, 1], \Lambda>0, and C>0 such that
\begin{equation*}
L_{\mu}V(x)\leqslant (1-\delta)C+ \Lambda\biggl(\delta\int_{\mathbb{R}^d}V\,d\mu-V(x)\biggr)
\end{equation*}
\notag
for all x\in\mathbb{R}^d and \mu\in\mathcal{P}_{V}(\mathbb{R}^d). Then there exists a solution \mu\in\mathcal{P}_V(\mathbb{R}^d) to the equation L_{\mu}^{*}\mu=0. Proof. Fix \alpha \geqslant C/\Lambda such that the metrizable compact set \mathcal{P}_{\alpha,V}(\mathbb{R}^d) is non-empty. According to [23], Corollary 2.4.4, for every measure \sigma\in\mathcal{P}_{\alpha, V}(\mathbb{R}^d) there exists a solution \mu to the linear equation L_{\sigma}^{*}\mu=0. Since
\begin{equation*}
L_{\sigma}V(x)\leqslant (1-\delta)C+\Lambda\delta\alpha-\Lambda V(x),
\end{equation*}
\notag
according to [23], Theorem 2.3.2, the estimate
\begin{equation*}
\int_{\mathbb{R}^d}V\,d\mu\leqslant (1-\delta)\frac{C}{\Lambda}+\delta\alpha\leqslant \alpha
\end{equation*}
\notag
is true. Denote by \Phi(\sigma) the set of all solutions \mu\in\mathcal{P}_{\alpha, V}(\mathbb{R}^d) to the linear Fokker– Planck– Kolmogorov equation L_{\sigma}^{*}\mu=0. This is a non-empty convex subset of the convex metrizable compact set \mathcal{P}_{\alpha,V}(\mathbb{R}^d). Let us verify that the graph of the multivalued mapping \Phi is closed. Let the measures \sigma_n\in \mathcal{P}_{\alpha, V}(\mathbb{R}^d) converge weakly to a measure \sigma\in\mathcal{P}_{\alpha,V}(\mathbb{R}^d) and the measures \mu_n\in \Phi(\sigma_n) converge weakly to a measure \mu\in \mathcal{P}_{\alpha, V}(\mathbb{R}^d). Then the measures \sigma_n converge V-weakly to \sigma. Let us verify that L_{\sigma}^{*}\mu=0. Let \varphi\in C_0^{\infty}(\mathbb{R}^d). We have the equality Let B be a ball containing the support of \varphi. The sequences a^{ij}(\,\cdot\,,\sigma_n) and b^i(\,\cdot\,,\sigma_n) converge uniformly on B to the continuous functions a^{ij}(\,\cdot\,,\sigma) and b^i(\,\cdot\,, \sigma), respectively. We have
\begin{equation*}
\int_{\mathbb{R}^d}L_{\sigma}\varphi\,d\mu= \int_{\mathbb{R}^d}L_{\sigma}\varphi\,d(\mu-\mu_n)+ \int_{\mathbb{R}^d} (L_{\sigma}\varphi-L_{\sigma_n}\varphi)\,d\mu_n,
\end{equation*}
\notag
where the first term tends to zero as n\to\infty by the continuity of the coefficients and the weak convergence of \mu_n to \mu. The second term tends to zero as n\to\infty due to the uniform convergence on B of the coefficients of the operators L_{\sigma_n} to the coefficients of the operator L_{\sigma}. Letting n\to\infty, we obtain Thus, the graph of the mapping \Phi is closed, hence we can apply the Kakutani–Ky Fan theorem (see [34], Theorem 1.2.12). This theorem, which is a multivalued version of the Schauder–Tychonoff theorem, asserts that if a multivalued mapping \Phi from a convex compact set K in a locally convex space to the set of non-empty convex compact subsets of K has a closed graph, then there exists a point k such that k\in \Phi(k). Thus, there exists \mu\in\Phi(\mu). \Box Note that the factor 1-\delta is needed in condition (iii) to make the term (1-\delta)C vanishing for \delta=1. Let us give a typical example, where the hypotheses of Theorem 5.1 are fulfilled. Example 5.2. Suppose that the functions q^{ij} are continuous and bounded on \mathbb{R}^d\times \mathbb{R}^d and the matrices Q(x,y)=\bigl(q^{ij}(x,y)\bigr)_{i,j\leqslant d} are symmetric and non-negative definite. Suppose also that a vector field \beta=(\beta^i)_{i\leqslant d} is continuous on \mathbb{R}^d, a vector field K=(K^i)_{i\leqslant d} is continuous on \mathbb{R}^d\times\mathbb{R}^d, and there exist positive numbers C_1, C_2, C_3, and C_4 such that C_4<C_2 and for all x and y the estimates
\begin{equation*}
\begin{gathered} \, \langle \beta(x),x\rangle\leqslant C_1-C_2|x|^2, \\ |K(x,y)|\leqslant C_3\bigl(1+|x|^{m-1}+|y|^{m-1}\bigr),\quad\text{and} \quad \langle K(x,y),x\rangle\leqslant C_3+C_4|x|^2 \end{gathered}
\end{equation*}
\notag
hold. For \mu\in \mathcal{P}_V(\mathbb{R}^d) set
\begin{equation*}
a^{ij}(x,\mu)=\int_{\mathbb{R}^d}q^{ij}(x,y)\,\mu(dy) \quad\text{and} \quad b^i(x,\mu)=\beta^i(x)+\int_{\mathbb{R}^d}K^{i}(x,y)\,\mu(dy).
\end{equation*}
\notag
Let us verify that the hypotheses of Theorem 5.1 are fulfilled for the function V(x)=(1+|x|^2)^{m/2}. We have
\begin{equation*}
\begin{aligned} \, L_{\mu}V(x)&\leqslant m(1+|x|^2)^{-1+m/2} \Bigl(\sup_{x,y}\operatorname{trace}Q(x,y) \\ &\qquad+(m-2)|x|^2(1+|x|^2)^{-1}\sup_{x,y}\|Q(x,y)\|- (C_2-C_4)|x|^2+C_3\Bigr). \end{aligned}
\end{equation*}
\notag
Therefore, for some C>0 and \Lambda>0 we have the estimate that is, condition (iii) is fulfilled for \delta=0. Clearly, (ii) is also fulfilled. Let us verify condition (i). Let B be a ball in \mathbb{R}^d, and let For every x\in B we have and
\begin{equation*}
|b^i(x,\mu)|\leqslant \sup_{z\in B}|\beta^i(z)|+C_3+ C_3\sup_{z\in B}|z|^{m-1}+C_3\alpha^{(m-1)/m}.
\end{equation*}
\notag
Let us verify the equicontinuity of the coefficients. It suffices to show that for every continuous function f on \mathbb{R}^d\times\mathbb{R}^d satisfying the condition \sup_{x\in B}|f(x,y)|\leqslant C(B)F(y) on every ball B, where F\geqslant 0 and \lim_{|x|\to\infty} F(x)/V(x)=0, the functions are equicontinuous on every ball uniformly in \mu\in\mathcal{P}_{\alpha,V}(\mathbb{R}^d). For every \varepsilon>0 there exists R>0 such that Then for all x\in B we have
\begin{equation*}
\biggl|\int_{\mathbb{R}^d}f(x, y)\,\mu(dy)- \int_{V\leqslant R}f(x,y)\,\mu(dy)\biggr|\leqslant \varepsilon\alpha C(B).
\end{equation*}
\notag
It remains to observe that the function f is uniformly continuous on the compact set B\times\{y\colon V(y)\leqslant R\}. Thus, all conditions in Theorem 5.1 are fulfilled, and there exists a solution \mu\in\mathcal{P}_V(\mathbb{R}^d) to the nonlinear equation L_{\mu}^{*}\mu=0. Let us consider an example illustrating condition (iii) in Theorem 5.1. Example 5.3. Suppose that d=1 and \varepsilon,\gamma\in\mathbb{R}; let
\begin{equation*}
a(x,\mu)=0 \quad\text{and} \quad b(x,\mu)=-x+\gamma+\varepsilon \int_{\mathbb{R}}y\, \mu(dy).
\end{equation*}
\notag
Our equation has the form (b(x,\mu)\mu)'=0, so that b(x,\mu)\mu=\operatorname{const}. Since |x|\in L^1(\mu), we have b\in L^1(\mu) and b(x,\mu)\mu=0. Integrating the equality b(x,\mu)\mu=0 we obtain Therefore, if \gamma\ne 0 and \varepsilon=1, then there are no solutions. If \varepsilon\ne 1, then
\begin{equation*}
\int_{\mathbb{R}}y\, \mu(dy)=\frac{\gamma}{1-\varepsilon}\quad\text{and} \quad b(x, \mu)=-x+\frac{\gamma}{1-\varepsilon}\,.
\end{equation*}
\notag
The unique solution for \varepsilon\ne 1 is \delta_{\gamma/(1-\varepsilon)}. We show that, whenever |\varepsilon|<1, for the function V(x)=|x|^2/2 one can find numbers \delta\in [0, 1], C>0, and \Lambda>0 for which condition (iii) in Theorem 5.1 is fulfilled. Let us take \beta\in (0,1) such that |\varepsilon|<1-\beta/2. Then
\begin{equation*}
\begin{aligned} \, L_{\mu}V(x)&=-|x|^2+\gamma x+\varepsilon x\int_{\mathbb{R}}y\,\mu(dy) \\ &\leqslant -\biggl(1-\frac{|\varepsilon|+\beta}{2}\biggr)|x|^2+ \frac{\gamma^2}{2\beta}+\frac{|\varepsilon|}{2} \int_{\mathbb{R}^d}|y|^2\, \mu(dy). \end{aligned}
\end{equation*}
\notag
Set
\begin{equation*}
\Lambda=2-|\varepsilon|-\beta, \qquad \delta=\frac{|\varepsilon|}{2-|\varepsilon|-\beta}, \quad\text{and} \quad C=\frac{\gamma^2}{2\beta(1-\delta)}\,.
\end{equation*}
\notag
It is clear that \delta\in(0,1), \Lambda>0, C>0, and
\begin{equation*}
L_{\mu}V(x)\leqslant C(1-\delta)+ \Lambda\biggl(\delta\int_{\mathbb{R}^d}V\,d\mu-V(x)\biggr).
\end{equation*}
\notag
Thus, with the aid of condition (iii) we have managed to indicate precisely one of the intervals of values of \varepsilon for which a solution exists. Recall that for \varepsilon=1 there is no solution. Let us show that for \varepsilon>1 there is no suitable function V. Suppose that there is a function V\in C^2(\mathbb{R}) satisfying the following conditions: V\geqslant 0, \lim_{|x|\to\infty}V(x)=+\infty, and for some numbers \delta\in [0,1], C>0, and \Lambda>0 the inequality
\begin{equation*}
\begin{aligned} \, L_{\mu}V(x)&=-xV'(x)+\gamma V'(x)+\varepsilon V'(x) \int_{\mathbb{R}}y\,\mu(dy) \\ &\leqslant C(1-\delta)+ \Lambda\biggl(\delta\int_{\mathbb{R}}V\, d\mu-V(x)\biggr) \end{aligned}
\end{equation*}
\notag
is fulfilled for all x and all probability measures with finite integrals of the functions |x| and V(x). Substituting \mu=\delta_x into this inequality we obtain
\begin{equation*}
V'(x)\bigl(\gamma+(\varepsilon-1)x\bigr)\leqslant (1-\delta)(C-\Lambda V(x)).
\end{equation*}
\notag
Since \varepsilon>1 and V(x)\to +\infty as x\to+\infty, there exists a sequence x_n\to+\infty such that
\begin{equation*}
V'(x_n)\bigl(\gamma+(\varepsilon-1)x_n\bigr)>0, \qquad V(x_n)\to+\infty.
\end{equation*}
\notag
Due to the inequalities 1-\delta\geqslant 0 and \Lambda>0, this leads to a contradiction, because the expression (1-\delta)(C-\Lambda V(x_n)) is negative for sufficiently large n. Recall that for \varepsilon>1 the equation has solutions. Moreover, for \gamma=0 and \varepsilon=1 every measure \delta_a is a solution, but similarly to what has been done above, one can show that also in this case there is no suitable function V. Below we discuss how to modify the hypotheses of Theorem 5.1 for a finer analysis of the existence of solutions. Let W(x)=\sqrt{1+V(x)}. Suppose, in addition to the hypotheses of Theorem 5.1, that for every measure \mu\in\mathcal{P}_V(\mathbb{R}^d) there exists a positive number C_{\mu} such that for all x\in\mathbb{R}^d we have Following [28], let I_0^W denote the set of functions \psi\in C^2(\mathbb{R}^d) such that
\begin{equation*}
\sup_x\bigl(|\psi(x)|+|\nabla\psi(x)|+|D^2\psi(x)|\bigr)W(x)^{-1}<\infty
\end{equation*}
\notag
and for every measure \mu\in\mathcal{P}_V(\mathbb{R}^d) the equality holds. It is clear that the set I_0^W is a linear space and contains 1. Proposition 5.4. Suppose that for some function \psi\in I_0^W there exists a function h\in C(\mathbb{R}^d) such that \sup_x|h(x)|/V(x)<\infty, and for every measure \mu\in \mathcal{P}_V(\mathbb{R}^d) there exist numbers C_1(\mu)\ne 0 and C_2(\mu) satisfying the equality Let \mu\in \mathcal{P}_V(\mathbb{R}^d) be a solution to the linear equation L_{\sigma}^{*}\mu=0 for some measure \sigma\in \mathcal{P}_V(\mathbb{R}^d). Then Proof. We have the chain of equalities
\begin{equation*}
\begin{aligned} \, &C_1(\sigma)\int_{\mathbb{R}^d}h(x)\,\sigma(dx)+C_2(\sigma)= \int_{\mathbb{R}^d}L_{\sigma}\psi(x)\,\sigma(dx)=0 \\ &\qquad=\int_{\mathbb{R}^d}L_{\sigma}\psi(x)\,\mu(dx)= C_1(\sigma)\int_{\mathbb{R}^d}h(x)\,\mu(dx)+C_2(\sigma), \end{aligned}
\end{equation*}
\notag
which, along with the condition C_1\ne 0, implies the required equality. \Box Let \mathcal{H} be the set of functions h\in C(\mathbb{R}^d) such that \displaystyle\lim_{|x|\to\infty}\dfrac{h(x)}{V(x)}=0 and the equality
\begin{equation*}
L_{\mu}\psi(x)=C_1(\mu)h(x)+C_2(\mu), \quad \text{where}\ \ C_1(\mu)\ne 0,
\end{equation*}
\notag
is true for some function \psi\in I_0^W and all measures \mu\in\mathcal{P}_V(\mathbb{R}^d). If
\begin{equation*}
\int_{\mathbb{R}^d}h\,d\mu=\int_{\mathbb{R}^d}h\,d\sigma\quad \forall\, h\in\mathcal{H},
\end{equation*}
\notag
then we write \mu\big|_{\mathcal{H}}=\sigma\big|_{\mathcal{H}}. Theorem 5.5. Let \nu\in\mathcal{P}_V(\mathbb{R}^d). Suppose that the following conditions are fulfilled: (i) for every measure \mu\in\mathcal{P}_V(\mathbb{R}^d) there exists a number C_{\mu}>0 such that
\begin{equation*}
|a^{ij}(x,\mu)|+|b^i(x,\mu)|\leqslant C_{\mu}W(x) \quad \forall\, x\in\mathbb{R}^d{\rm;}
\end{equation*}
\notag
(ii) for every ball B and every \alpha>0, the functions a^{ij}(\,\cdot\,,\mu) and b^i(\,\cdot\,,\mu) on B are uniformly bounded and equicontinuous on B uniformly in \mu\in \mathcal{P}_{\alpha,V}(\mathbb{R}^d) satisfying the equality \mu\big|_{\mathcal{H}}=\nu\big|_{\mathcal{H}}; (iii) if measures \mu_n\in \mathcal{P}_V(\mathbb{R}^d) converge V-weakly to a measure \mu\in\mathcal{P}_V(\mathbb{R}^d), then
\begin{equation*}
\lim_{n\to\infty}b^i(x,\mu_n)=b^i(x,\mu), \quad \lim_{n\to\infty}a^{ij}(x,\mu_n)=a^{ij}(x,\mu) \quad \forall\, x;
\end{equation*}
\notag
(iv) there are numbers \delta\in [0,1], \Lambda>0, and C>0 such that
\begin{equation*}
L_{\mu}V(x)\leqslant (1-\delta)C+ \Lambda\biggl(\delta\int_{\mathbb{R}^d}V\,d\mu-V(x)\biggr)
\end{equation*}
\notag
for all x\in\mathbb{R}^d and all \mu\in\mathcal{P}_{V}(\mathbb{R}^d) such that \mu\big|_{\mathcal{H}}=\nu\big|_{\mathcal{H}}. Then there exists a solution \mu to the equation L_{\mu}^{*}\mu=0 such that \mu\big|_{\mathcal{H}}=\nu\big|_{\mathcal{H}}. Proof. The justification is similar to the proof of Theorem 5.1. We just observe that one has to consider only measures \mu\in \mathcal{P}_{\alpha,V}(\mathbb{R}^d) such that \mu\big|_{\mathcal{H}}=\nu\big|_{\mathcal{H}} and take \alpha greater than C/\Lambda and \|V\|_{L^1(V)}. In addition, if measures \mu_n\in \mathcal{P}_{\alpha,V}(\mathbb{R}^d) converge V-weakly to \mu\in\mathcal{P}_{\alpha, V}(\mathbb{R}^d) and \mu_n\big|_{\mathcal{H}}=\nu\big|_{\mathcal{H}}, then we have \mu\big|_{\mathcal{H}}=\nu\big|_{\mathcal{H}}. \Box Let us consider some examples.
\begin{equation*}
a(x,\mu)=0, \quad\text{and} \quad b(x,\mu)=-x+\int_{\mathbb{R}}y\,\mu(dy).
\end{equation*}
\notag
Set V(x)=|x|^2/2 and W(x)=\sqrt{1+V(x)}. The function \psi(x)=x belongs to I_0^W and hence the function h(x)=x belongs to \mathcal{H}. For every measure \nu=\delta_a the equality \mu\big|_{\mathcal{H}}=\nu\big|_{\mathcal{H}} means that For such measures \mu the operator L_{\mu} has the form and Thus, condition (iv) in Theorem 5.5 is fulfilled for C=a^2/2, \Lambda=1, and \delta=0. This agrees fully with the fact that every measure \mu=\delta_a satisfies the equation L^{*}_{\mu}\mu=0. Example 5.7. Let d\geqslant 1, A={\rm I}, and
\begin{equation*}
b(x,\mu)=-Rx+\int_{\mathbb{R}^d}\langle v,y\rangle\, \mu(dy)\,\theta+ \int_{\mathbb{R}^d} K(x,y)\, \mu(dy),
\end{equation*}
\notag
where R is a constant matrix, K is a vector function, v and \theta are vectors, and
\begin{equation*}
R^{*}v=\lambda v, \quad \langle v, \theta\rangle=\lambda, \quad\text{and} \quad \langle K(x, y), v\rangle =0.
\end{equation*}
\notag
Suppose that
\begin{equation*}
\langle Rx, x\rangle\geqslant q|x|^2, \quad q>0, \quad\text{and} \quad \sup_{x, y}|K(x, y)|<\infty,
\end{equation*}
\notag
and the function K is continuous. Set V(x)=|x|^2/2 and W(x)=\sqrt{1+V(x)}. Observe that the function \psi(x)=\langle v, x\rangle satisfies the equality
\begin{equation*}
L_{\mu}\psi(x)=-\lambda\langle v, x\rangle+ \lambda\int_{\mathbb{R}^d}\langle v,y\rangle\, \mu(dy).
\end{equation*}
\notag
Therefore, \psi\in I_0^W and the function x\mapsto \langle v,x\rangle belongs to \mathcal{H}. Let Q\in \mathbb{R}. We consider measures \mu\in\mathcal{P}_V(\mathbb{R}^d) satisfying the condition Then
\begin{equation*}
|b(x,\mu)| \leqslant \|R\|\,|x|+|Q|\,|\theta|+\sup_{u,v}|K(u,v)|,
\end{equation*}
\notag
and
\begin{equation*}
\begin{aligned} \, L_{\mu}V(x)&\leqslant d-q|x|^2+\bigl(|Q|\,|\theta|+ \sup_{u,v}|K(u,v)|\bigr)|x| \\ &\leqslant d+\frac{1}{2q}\bigl(|Q\theta|+\sup_{u,v}|K(u,v)|\bigr)^2-qV(x). \end{aligned}
\end{equation*}
\notag
Therefore, the inequality
\begin{equation*}
L_{\mu}V\leqslant (1-\delta)C+ \Lambda\biggl(\delta\int_{\mathbb{R}^d} V\,d\mu-V\biggr)
\end{equation*}
\notag
holds for \delta=0, \Lambda=q, and C=d+\bigl(|Q\theta|+\sup_{u, v}|K(u, v)|\bigr)^2/(2q). Thus, the hypotheses of the theorem are fulfilled, hence for every Q there exists a solution \mu to the equation L_{\mu}^{*}\mu=0 that satisfies the condition indicated. In the case where the matrix A is non-degenerate, any solution has a density with respect to Lebesgue measure. Moreover, if the matrix A satisfies the Dini mean oscillation condition, then the solution density has a continuous positive version (see [32], [55], and [57]). These observations enable us in the case of a non-degenerate and sufficiently regular matrix A to construct solutions under weaker restrictions on the drift coefficient and the dependence of coefficients on solutions. Let \mathcal{CL}_V(\mathbb{R}^d) denote the set of continuous functions \varrho such that
\begin{equation*}
\int_{\mathbb{R}^d}\bigl(1+V(x)\bigr)|\varrho(x)|\,dx<\infty.
\end{equation*}
\notag
Let W\in C(\mathbb{R}^d), W>0, and \lim_{|x|\to\infty} W(x)/V(x)=0. We will use the weighted L^1-norm Let \mathcal{CP}_V(\mathbb{R}^d) denote the subset in \mathcal{CL}_V(\mathbb{R}^d) consisting of probability densities. By \mathcal{CP}_{\alpha, V}(\mathbb{R}^d) we denote the subset in \mathcal{CP}_V(\mathbb{R}^d) consisting of densities \varrho satisfying the estimate We say that a measurable function f on a ball B satisfies the Dini mean oscillation condition with modulus of continuity w_B if there exists a continuous increasing function w_B on [0,+\infty) such that w_B(0)=0,
\begin{equation*}
\sup_{x\in B}\frac{1}{|B\cap B(x, r)|}\int_{B\cap B(x, r)}|f(y)-f_{B, r}(x)|\,dy\leqslant w_B(r),
\end{equation*}
\notag
where |B| is the volume of the set B,
\begin{equation*}
\quad f_{B,r}(x)=\frac{1}{|B\cap B(x,r)|}\int_{B\cap B(x,r)}f(y)\,dy,
\end{equation*}
\notag
and It is known that f has a continuous version (see [73]), which we deal with below. Suppose that for every function \varrho\in\mathcal{CP}_V(\mathbb{R}^d) and all i,j\leqslant d we are given Borel functions a^{ij}(\,\cdot\,,\varrho) and b^i(\,\cdot\,,\varrho) on \mathbb{R}^d, where the matrices A(x,\varrho)=\bigl(a^{ij}(x,\varrho)\bigr)_{i,j\leqslant d} are symmetric and non-negative definite. Set
\begin{equation*}
L_{\varrho}\varphi(x)=\sum_{i,j\leqslant d} a^{ij}(x,\varrho)\partial_{x_i}\partial_{x_j}\varphi(x)+ \sum_{i\leqslant d} b^i(x,\varrho)\partial_{x_i}\varphi(x).
\end{equation*}
\notag
A function \varrho\in\mathcal{CP}_V(\mathbb{R}^d) is a solution to the nonlinear stationary Fokker–Planck–Kolmogorov equation if the measure \mu=\varrho\,dx is a solution to the linear equation with operator L_{\varrho}. Theorem 5.8. Suppose that the following conditions are fulfilled: (i) for every ball B and every \alpha>0 one can find a number \lambda_{B,\alpha}>0 and a function w_{B,\alpha} such that for all \varrho\in\mathcal{CP}_{\alpha,V}(\mathbb{R}^d) the functions a^{ij}(\,\cdot\,,\varrho) are continuous on B, satisfy the Dini mean oscillation condition with the modulus of continuity w_{B,\alpha} on B, and for all x\in B the estimate
\begin{equation*}
\lambda_{B,\alpha}\cdot {\rm I}\leqslant A(x,\varrho)\leqslant \lambda_{B,\alpha}\cdot {\rm I}
\end{equation*}
\notag
holds; (ii) for every ball B and every \alpha>0 there exist numbers N(B,\alpha)>0 and p(B,\alpha)>d such that
\begin{equation*}
\|b(\,\cdot\,,\varrho)\|_{L^{p(B,\alpha)}(B)}\leqslant N(B,\alpha)\quad \forall\, \varrho\in\mathcal{CP}_{\alpha,V}(\mathbb{R}^d);
\end{equation*}
\notag
(iii) if some functions \varrho_n\in \mathcal{CP}_V(\mathbb{R}^d) converge to \varrho\in \mathcal{CP}_V(\mathbb{R}^d) locally uniformly and \|\varrho_n-\varrho\|_{W}\to 0, then the functions a^{ij}(\,\cdot\,, \varrho_n) converge pointwise to the function a^{ij}(\,\cdot\,, \varrho_n) and the functions b^i(\,\cdot\,, \varrho_n) converge to the function b^i(\,\cdot\,,\varrho) in L^1(B) for every ball B; (iv) there are numbers \delta\in [0,1], \Lambda>0, and C>0 such that
\begin{equation*}
L_{\varrho}V(x)\leqslant (1-\delta)C+ \Lambda\biggl(\delta\int_{\mathbb{R}^d}V\,d\mu-V(x)\biggr)\quad \forall\, x\in\mathbb{R}^d, \ \varrho\in\mathcal{CP}_{V}(\mathbb{R}^d).
\end{equation*}
\notag
Then there exists a unique solution to the equation L_{\varrho}^{*}\varrho=0 and this solution is a continuous positive function. Proof. Let \alpha>0 and \sigma\in\mathcal{CP}_{V}(\mathbb{R}^d). According to [29], Theorems 4.2 and 4.7, there exists a unique solution \varrho to the equation L_{\sigma}^{*}\varrho=0. In addition, by [29], Theorem 3.5, for every ball B one can find a number C(B,\alpha) and a modulus of continuity \omega_{B,\alpha} (a strictly increasing function on [0,+\infty) vanishing at zero), which do not depend on \sigma, such that
\begin{equation*}
|\varrho(x)-\varrho(y)|\leqslant\omega_{B, \alpha}(|x-y|) \quad\text{and}\quad \sup_{B}\varrho\leqslant C(B,\alpha)\inf_{B}\varrho.
\end{equation*}
\notag
Observe that
\begin{equation*}
\sup_{B}\varrho\leqslant C(B,\alpha)|B|^{-1}\int_B\varrho(x)\,dx\leqslant C(B,\alpha)|B|^{-1}.
\end{equation*}
\notag
Let us consider the set X_{\alpha} of functions \varrho\in\mathcal{CP}_{\alpha, V}(\mathbb{R}^d) such that for every ball B, whenever x,y\in B, we have
\begin{equation*}
|\varrho(x)-\varrho(y)|\leqslant\omega_{B,\alpha}(|x-y|) \quad\text{and}\quad \sup_{B}\varrho\leqslant C(B,\alpha)|B|^{-1}.
\end{equation*}
\notag
The set X_{\alpha} is non-empty for sufficiently large \alpha. Every sequence \varrho_n\in X_{\alpha} contains a subsequence \varrho_{n_j} such that it converges uniformly on any ball to some non-negative continuous function \varrho, \varrho\,dx is a probability measure, and the measures \varrho_{n_j}\,dx converge V-weakly to \varrho\,dx. Therefore, \|\varrho_{n_j}-\varrho\|_W\to 0. Thus, the set X_{\alpha} is convex and compact in the space \mathcal{CL}_V(\mathbb{R}^d) with the metric generated by the norm \|\cdot\|_W.
Fix a number \alpha>0 such that the set X_{\alpha} is non-empty and \alpha>C/\Lambda. For every function \sigma\in X_{\alpha} we denote by \Phi(\sigma) the unique solution \varrho to the equation L^{*}_{\sigma}\varrho=0. Let us verify the continuity of the mapping \Phi. Suppose that \sigma_n\in X_{\alpha}, \varrho_n=\Phi(\sigma_n), \sigma\in X_{\alpha}, and \|\sigma-\sigma_n\|_W\to 0. Every subsequence \{\varrho_{n_k}\} contains a further subsequence \{\varrho_{n_{k_j}}\} that converges locally uniformly and in the norm \|\cdot\|_W to \varrho\in X_{\alpha}. Moreover, we can assume that \{\sigma_{n_{j_k}}\} converges locally uniformly to \sigma. Let us verify that \varrho=\Phi(\sigma). It will follow from this that the whole original sequence \{\varrho_n\} converges to \varrho. For every function \varphi\in C_0^{\infty}(\mathbb{R}^d) we have
\begin{equation*}
\int_{\mathbb{R}^d}\bigl[\operatorname{trace}\bigl(A(x,\sigma_{n_{j_k}}) D^2\varphi(x)\bigr)+\langle b(x,\sigma_{n_{j_k}}), \nabla\varphi(x)\rangle\bigr]\,\varrho_{n_{j_k}}(x)\, dx=0.
\end{equation*}
\notag
Let B be a ball containing the support of \varphi. We observe that the sequence of functions a^{ij}(\,\cdot\,,\sigma_{n_{j_k}}) converges to a^{ij}(\,\cdot\,, \sigma) uniformly on B, and the functions \varrho_{n_{j_k}} converge to \varrho in L^1(B). Therefore,
\begin{equation*}
\lim_{k\to\infty}\int_{\mathbb{R}^d}\operatorname{trace} \bigl(A(x, \sigma_{n_{j_k}})D^2\varphi(x)\bigr)\varrho_{n_{j_k}}(x)= \int_{\mathbb{R}^d}\operatorname{trace}\bigl(A(x,\sigma)D^2\varphi(x)\bigr) \varrho(x)\,dx.
\end{equation*}
\notag
Since the functions b^{i}(\,\cdot\,,\sigma_{n_{j_k}}) converge to b^i(\,\cdot\,,\sigma) in L^1(B) and the sequence \{\varrho_{n_{j_k}}\} is uniformly bounded on B and converges uniformly on B to \varrho, we have
\begin{equation*}
\lim_{k\to\infty}\int_{\mathbb{R}^d}\bigl\langle b(x,\sigma_{n_{j_k}}), \nabla\varphi(x)\bigr\rangle\varrho_{n_{j_k}}(x)=\int_{\mathbb{R}^d} \langle b(x,\sigma),\nabla\varphi(x)\rangle\varrho(x)\,dx.
\end{equation*}
\notag
Thus, passing to the limit, we obtain
\begin{equation*}
\int_{\mathbb{R}^d}\bigl[\operatorname{trace} \bigl(A(x,\sigma)D^2\varphi(x)\bigr)+\langle b(x,\sigma), \nabla\varphi(x)\rangle\bigr]\,\varrho(x)\, dx=0.
\end{equation*}
\notag
Therefore, \varrho=\Phi(\sigma). Since the mapping \Phi from X_{\alpha} to X_{\alpha} is continuous and X_{\alpha} is convex and compact with respect to the norm \|\cdot\|_W, by Schauder’s theorem (see [34], Theorem 1.12.8) there exists \varrho\in X_{\alpha} such that \varrho=\Phi(\varrho). Consider an example showing that one cannot expect the uniqueness of solutions under the assumptions of Theorem 5.8.
\begin{equation*}
\varrho_1(x)=\frac{1}{\sqrt{2\pi}}\, e^{-x^2/2} \quad\text{and}\quad \varrho_2(x)=\frac{2}{\sqrt{2\pi}}\, e^{-2x^2}.
\end{equation*}
\notag
Observe that c=\|\varrho_1-\varrho_2\|_{L^1(\mathbb{R})}>0 and for every \varrho\in L^1(\mathbb{R}) one has
\begin{equation*}
\|\varrho-\varrho_2\|_{L^1(\mathbb{R})}+ \|\varrho_1-\varrho\|_{L^1(\mathbb{R})}\geqslant c.
\end{equation*}
\notag
Set a(x,\varrho)=1, b(x,\varrho)=-c^{-1}\bigl(\|\varrho- \varrho_2\|_{L^1(\mathbb{R})}+ 4\|\varrho_1-\varrho\|_{L^1(\mathbb{R})}\bigr)x, W(x)=1 and V(x)=|x|^2/2. Since
\begin{equation*}
|b(x,\varrho)|\leqslant 10c^{-1}|x|, \qquad |b(x,\varrho)-b(x, \sigma)|\leqslant c^{-1}|x|\,\|\varrho-\sigma\|_{W},
\end{equation*}
\notag
and
\begin{equation*}
L_{\varrho}V(x)=1-c^{-1}|x|^2\bigl(\|\varrho-\varrho_2\|_{L^1(\mathbb{R})}+ 4\|\varrho_1-\varrho\|_{L^1(\mathbb{R})}\bigr)\leqslant 1-|x|^2,
\end{equation*}
\notag
the hypotheses of the theorem are fulfilled for
\begin{equation*}
W=1, \ \ \ V(x)=\frac{|x|^2}{2}\,, \ \ \ N(B,\alpha)=10c^{-1}\sup_B|x|, \ \ \ \text{and}\ \ \ \delta=0, \ \ \ C=1, \ \ \ \Lambda=1.
\end{equation*}
\notag
However, the equation L_{\varrho}^{*}\varrho=0 has two solutions \varrho_1 and \varrho_2. We draw attention to the fact that in this example the diffusion coefficient is one and the drift coefficient is smooth in x and Lipschitz in \varrho with respect to the weighted norm \|\cdot\|_W. Therefore, uniqueness is connected not only with the regularity of coefficients. The key role in obtaining sufficient conditions for uniqueness is played by estimates for the distances between solutions. Let us give an example of sufficient conditions for uniqueness obtained in this way. Theorem 5.10. Let A(x,\mu)=\mathrm{I} and V(x)=|x|^2. Suppose that for every measure \mu\in\mathcal{P}_V(\mathbb{R}^d) we are given a function b(\,\cdot\,,\mu) bounded on every ball and satisfying the condition
\begin{equation*}
\langle b(x,\mu)-b(x,\mu),x-y\rangle\leqslant -\kappa(\mu)|x-y|^2
\end{equation*}
\notag
for all x,y\in\mathbb{R}^d and some number \kappa(\mu). Suppose also that
\begin{equation*}
|b(x,\mu)-b(x,\sigma)|\leqslant Q(x)W_2(\mu,\sigma)\quad \forall\, x\in\mathbb{R}^d, \ \ \mu,\sigma\in\mathcal{P}_V(\mathbb{R}^d)
\end{equation*}
\notag
for some Borel non-negative function Q. Let
\begin{equation*}
L_{\mu}^{*}\mu=0, \quad L_{\sigma}^{*}\sigma=0, \quad\textit{and}\quad \|Q\|_{L^2(\sigma)}<\kappa(\mu),
\end{equation*}
\notag
where
\begin{equation*}
L_{\mu}\varphi(x)=\Delta\varphi(x)+\langle b(x,\mu),\nabla\varphi(x)\rangle
\end{equation*}
\notag
and
\begin{equation*}
L_{\sigma}\varphi(x) =\Delta\varphi(x)+ \langle b(x,\sigma),\nabla\varphi(x)\rangle.
\end{equation*}
\notag
Then \mu=\sigma. In particular, if \kappa(\mu)=\kappa does not depend on \mu and Q(x)=Q does not depend on x, then for Q<\kappa the equation L_{\mu}^{*}\mu=0 has at most one solution. Proof. From Example 4.3 we obtain the estimate
\begin{equation*}
W_2(\mu,\sigma)\leqslant \frac{1}{\kappa(\mu)}\|b(\,\cdot\,,\mu)- b(\,\cdot\,,\sigma)\|_{L^2(\sigma)}.
\end{equation*}
\notag
Therefore, W_2(\mu,\sigma)\leqslant \kappa(\mu)^{-1}\|Q\|_{L^2(\sigma)}W_2(\mu,\sigma). If \|Q\|_{L^2(\sigma)}<\kappa(\mu), then we obtain W_2(\mu,\sigma)=0. \Box Consider an example showing that the conditions in Theorem 5.10 are sharp. Example 5.11. Let d=1, \varepsilon>0, and Observe that It is known (see [15], § 3.2) that there exists a probability measure \pi for which
\begin{equation*}
W_2(\mu,\sigma)^2=\int_{\mathbb{R}\times\mathbb{R}}|x-y|^2\, \pi(dx\,dy).
\end{equation*}
\notag
Then
\begin{equation*}
|b(x,\mu)-b(x,\sigma)|\leqslant \varepsilon\int_{\mathbb{R}\times\mathbb{R}}|x-y|\, \pi(dx\,dy)\leqslant \varepsilon W_2(\mu,\sigma).
\end{equation*}
\notag
Thus, the assumptions of the theorem are fulfilled for \kappa=1 and Q=\varepsilon. If \varepsilon<1, then the solution is unique. It is readily verified that for \varepsilon=1 every measure \mu given by a density (2\pi)^{-1/2}e^{-(x-a)^2/2}, a\in\mathbb{R}, is a solution. We draw the reader’s attention to the fact that the condition Q<\kappa in Theorem 5.10 and the condition
\begin{equation*}
L_\mu V(x)\leqslant C(1-\delta)+ \Lambda\biggl(\delta\int_{\mathbb{R}}V\,d\mu-V(x)\biggr)
\end{equation*}
\notag
in the existence theorem mean a certain smalleness of the nonlinear (in \mu) part of the coefficients. The same can be seen from the examples presented above. If we consider a nonlinear equation as a perturbation of a linear equation, then it is natural to expect that under small perturbations existence and uniqueness theorems valid for linear equations remain in forth. Let us mention the following result from [28]. Let \mathcal{P}_k^a(\mathbb{R}^d) denote the subset of L^1(\mathbb{R}^d) consisting of almost everywhere non-negative functions \varrho such that
\begin{equation*}
\int_{\mathbb{R}^d}\varrho(x)\,dx=1\quad\text{and} \quad \int_{\mathbb{R}^d}|x|^k\varrho(x)\,dx<\infty.
\end{equation*}
\notag
Suppose that for every function \varrho\in\mathcal{P}_k^a(\mathbb{R}^d) (as usual, we do not mean a single function, but rather the whole equivalence class with respect to equality almost everywhere) and every \varepsilon\in[0,1] we are given Borel functions a^{ij}_{\varepsilon}(\,\cdot\,,\varrho) and b^i_{\varepsilon}(\,\cdot\,,\varrho). The corresponding operator is denoted by L_{\varrho,\varepsilon}. For all functions f such that (1+|x|^k)|f|\in L^1(\mathbb{R}^d) we set Theorem 5.12. Suppose that the following conditions are fulfilled: (i) the matrices A_{\varepsilon}(x, \varrho)= \bigl(a^{ij}_{\varepsilon}(x, \varrho)\bigr)_{i,j\leqslant d} are symmetric, non-negative definite, and there exist positive numbers \lambda and \theta such that for all \varrho\in\mathcal{P}_k^a(\mathbb{R}^d) and \varepsilon\in[0,1] one has
\begin{equation*}
|A_{\varepsilon}(x, \varrho)-A(y, \varrho)|\leqslant\lambda|x-y| \quad\textit{and}\quad \theta\cdot {\rm I}\leqslant A_{\varepsilon}(x, \varrho)\leqslant \theta^{-1}\cdot {\rm I};
\end{equation*}
\notag
(ii) there exist numbers \beta_0>0, \beta_1>0, \beta_2>0, \beta_3>0, and m\geqslant 0 such that for all \varrho\in\mathcal{P}_k^a(\mathbb{R}^d) and \varepsilon\in[0,1] one has
\begin{equation*}
|b_{\varepsilon}(x, \varrho)|\leqslant\beta_0(1+|x|)^m\|\varrho\|_k \quad\textit{and}\quad \langle b_{\varepsilon}(x, \varrho), x\rangle\leqslant \beta_1-\beta_2|x|^{2k}+\varepsilon\beta_3\|\varrho\|_k^2;
\end{equation*}
\notag
(iii) there exists a number N>0 such that for all \varrho,\sigma\in \mathcal{P}_k^a(\mathbb{R}^d) and \varepsilon\in[0,1] the estimates
\begin{equation*}
\|A_{\varepsilon}(x,\varrho)-A_{\varepsilon}(x,\sigma)\|\leqslant \varepsilon N\|\varrho-\sigma\|_k,
\end{equation*}
\notag
and
\begin{equation*}
|\partial_{x_j}a_{\varepsilon}^{ij}(x, \varrho)- \partial_{x_j}a_{\varepsilon}^{ij}(x, \sigma)|+ |b_{\varepsilon}(x,\varrho)-b_{\varepsilon}(x,\sigma)|\leqslant \varepsilon N(1+|x|)^m\|\varrho-\sigma\|_k
\end{equation*}
\notag
hold for almost all x. Then there exists \varepsilon_0\in (0, 1] such that for every \varepsilon\in[0, \varepsilon_0] a solution \varrho_{\varepsilon}\in\mathcal{P}_k^a(\mathbb{R}^d) to the equation L_{\varrho, \varepsilon}^{*}\varrho=0 exists and is unique. The proof makes use of estimates for the distances between solutions and a priori estimates for the logarithmic gradient of the solution. Let consider an example where the hypotheses of Theorem 5.12 are fulfilled.
\begin{equation*}
A_{\varepsilon}(x,\varrho)={\rm I}+ \varepsilon\int_{\mathbb{R}^d}Q(x,y)\varrho(y)\,dy
\end{equation*}
\notag
and
\begin{equation*}
b_{\varepsilon}(x,\varrho) =-x+ \varepsilon\int_{\mathbb{R}^d}K(x,y)\varrho(y)\,dy,
\end{equation*}
\notag
where the entries of the matrix Q and the vector field K are Borel functions. Suppose that Q is a symmetric matrix such that
\begin{equation*}
0\leqslant Q(x, y)\leqslant C_1\cdot {\rm I} \quad\text{and}\quad |Q(x, y)-Q(z, y)|\leqslant C_2|x-z|.
\end{equation*}
\notag
Suppose also that
\begin{equation*}
\langle K(x, y), x\rangle\leqslant C_3(1+|x|)(1+|y|) \quad\text{and}\quad |K(x, y)|\leqslant C_4(1+|x|)^m(1+|y|).
\end{equation*}
\notag
Then the hypotheses of Theorem are fulfilled for k=1. The parameter \varepsilon in the hypotheses of Theorem 5.12 is responsible for the size of the nonlinear part. For \varepsilon=0 condition (iii) yields that the coefficients do not depend on \varrho and the equation is linear. Correspondingly, for the values of the parameter \varepsilon close to zero the nonlinear equation does not differ much from the linear equation and inherits the existence and uniqueness theorems valid for it. Consider a frequently encountered form of the dependence of coefficients on the parameter, when
\begin{equation*}
a^{ij}_{\varepsilon}(x,\varrho)=a^{ij}(x,\varepsilon\varrho) \quad\text{and}\quad b^i_{\varepsilon}(x,\varrho)=b^i(x,\varepsilon\varrho).
\end{equation*}
\notag
Suppose that for some \varepsilon>0 there exists a probability solution \varrho_{\varepsilon} to the nonlinear equation with coefficients a^{ij}_{\varepsilon}(x,\varrho) and b^i_{\varepsilon}(x,\varrho). Then \sigma=\varepsilon\varrho_{\varepsilon} is a solution to the equation with the coefficients a^{ij}(x,\varrho) and b^i(x,\varrho) independent of \varepsilon. However, \sigma is not a probability density and the integral of \sigma equals \varepsilon. Thus, in place of the problem of the construction of a probability solution for small values of the parameter \varepsilon one can solve the problem of constructing a non-negative solution with integral equal to \varepsilon. In some physical problems exactly this setting is natural; for example, this is the case for solutions of the Bose–Einstein equation. We have discussed above conditions on the coefficients admitting only a non-local dependence on solutions, a typical example of which is the expression of the form However, many popular physical, biological, and economic models (see [48], [65], and [109]) lead to Fokker–Planck–Kolmogorov equations with local nonlinearity, when the coefficients depend on the values of the solution at points.Let us mention a result from [33] on the existence and uniqueness of non-negative solutions with a prescribed value of the integral for a nonlinear equation with local and non-local nonlinearities. For a fixed number k\geqslant 1 we denote by \mathcal{CL}_k^{+}(\mathbb{R}^d) the set of non-negative continuous functions \varrho satisfying the condition
\begin{equation*}
\int_{\mathbb{R}^d}\bigl(1+|x|^k\bigr)\varrho(x)\,dx<\infty.
\end{equation*}
\notag
Suppose that the matrices A(x)=(a^{ij}(x))_{i,j\leqslant d} are symmetric, and there exists a positive constant \theta such that
\begin{equation*}
\theta^{-1}\cdot {\rm I}\leqslant A(x)\leqslant \theta\cdot {\rm I} \quad\text{and}\quad \|A(x)-A(y)\|\leqslant \theta|x-y| \quad \forall\,x, y\in\mathbb{R}^d.
\end{equation*}
\notag
Suppose we are given a mapping
\begin{equation*}
b\colon \mathbb{R}^d \times [0, +\infty)\times \mathcal{CL}_{k}^{+}(\mathbb{R}^d) \to \mathbb{R}^d
\end{equation*}
\notag
such that for every g\in \mathcal{CL}_k^{+}(\mathbb{R}^d) the mapping (x,u)\mapsto b(x,u,g) is Borel measurable and bounded on the sets
\begin{equation*}
Q_R=\{(x,u)\colon |x|\leqslant R, \, 0\leqslant u\leqslant R\}, \qquad R>0.
\end{equation*}
\notag
Below we also require the Lipschitz property in g, which implies joint Borel measurability. Set
\begin{equation*}
L_{g}\varphi(x)=\sum_{i,j\leqslant d}a^{ij}(x)\partial_{x_i}\partial_{x_j} \varphi(x)+\sum_{i\leqslant d}b^i(x,g(x),g)\partial_{x_i}\varphi(x).
\end{equation*}
\notag
As above, a function \varrho \in \mathcal{CL}_k^{+}(\mathbb{R}^d) is called a solution to the nonlinear equation L^{*}_{\varrho}\varrho=0 if \varrho is a solution to the linear equation with the operator L_{\varrho}. For any function g in the linear span of \mathcal{CL}_k^{+}(\mathbb{R}^d) we set and for a bounded function we set \|g\|_{\infty}=\sup_{x\in\mathbb{R}^d}|g(x)|. Let m\geqslant 1.Theorem 5.14. Suppose that the following conditions are fulfilled: (i) there exists a positive number \beta_0 such that (ii) there exist positive numbers \beta_1, \beta_2, and r_0 such that for every function g\in \mathcal{CL}_k^{+}(\mathbb{R}^d) such that \|g\|_k\leqslant r_0 and all x\in\mathbb{R}^d and u\in [0,r_0] one has
\begin{equation*}
\langle x, b(x, u, g)\rangle \leqslant \beta_1-\beta_2|x|^2;
\end{equation*}
\notag
(iii) for every r>0 there exists a number C_r>0 such that the inequality
\begin{equation*}
|b(x, u, g)-b(x, v, h)|\leqslant C_r(1+|x|^m)\bigl(|u-v|+\|g-h\|_{k}\bigr)
\end{equation*}
\notag
holds for all x\in\mathbb{R}^d, u,v\in[0,r], and all g, h\in \mathcal{CL}_k^{+}(\mathbb{R}^d) such that \|g\|_k\leqslant r and \|h\|_k\leqslant r. Then there exists M_0>0 such that for every M \in (0, M_0) the stationary equation L^{*}_{\varrho}\varrho=0 has a unique solution \varrho\in \mathcal{CL}_k^{+}(\mathbb{R}^d) satisfying the conditions
\begin{equation*}
\int_{\mathbb{R}^d}\varrho(x)\,dx=M, \quad \|\varrho\|_{\infty}\leqslant r_0, \quad\textit{and}\quad \|\varrho\|_{k}\leqslant r_0.
\end{equation*}
\notag
Moreover, \varrho is Hölder continuous on every compact set. Let us consider an example where the hypotheses of Theorem 5.14 are fulfilled. Example 5.15. Conditions (i)–(iii) of Theorem 5.14 for k=m=1 are fulfilled for
\begin{equation*}
b(x, u, g)=-x+u^{\nu}b(x)+u^{\kappa}\int_{\mathbb{R}^d} K(x, y)g(y)\,dy,
\end{equation*}
\notag
where \nu>0, \kappa>0, and |b(x)|+|K(x, y)|\leqslant c_1+c_2|x|+c_3|y|. Our next example shows that in the general case one cannot expect the existence of solutions for all M>0. Example 5.16. Consider the stationary Bose–Einstein equation
\begin{equation*}
\Delta\varrho+\operatorname{div}\bigl(x\varrho(1+\varrho)\bigr)=0.
\end{equation*}
\notag
The drift b has the form b(x,u,\varrho)=-x-xu, that is, this equation is a particular case of the equation from the previous example. The hypotheses of Theorem 5.14 are fulfilled, hence for sufficiently small positive M there exists a positive continuous solution the integral of which equals M. However, in this case we can fully investigate the problem of the existence of solutions. It is readily verified that the functions
\begin{equation*}
\varrho_{\beta}(x)=\frac{1}{\beta e^{|x|^2/2}-1}\,, \qquad \beta>1,
\end{equation*}
\notag
are solutions. We show that there are no other solutions integrable over \mathbb{R}^d. Let \varrho be a positive continuous integrable solution. Since for k\geqslant 2 we have
\begin{equation*}
\Delta |x|^k-\langle x(1+\varrho),\nabla |x|^k\rangle= \bigl(k(k+d-2)-|x|^2(1+\varrho)\bigr)|x|^{k-2},
\end{equation*}
\notag
by [23], Theorem 2.3.2, the function |x|^k\varrho(x)(1+\varrho(x)) is integrable over \mathbb{R}^d. Set Let us verify that \varrho(1+\varrho)U^2\in L^1(\mathbb{R}^d). We have
\begin{equation*}
U(x)^2\leqslant 2\log^2\frac{\varrho(x)}{1+\varrho(x)}+\frac{|x|^4}{2}\,.
\end{equation*}
\notag
Therefore, it suffices to justify the integrability of the function
\begin{equation*}
\varrho(x)(1+\varrho(x))\log^2\frac{\varrho(x)}{1+\varrho(x)}\,.
\end{equation*}
\notag
Since for \varrho(x)>1 the estimate \varrho(x)/(1+\varrho(x))\geqslant 1/2 holds, it suffices to establish integrability on the set \{x\colon \varrho(x)\leqslant 1\}. Moreover, it suffices to prove integrability on this set for the function \varrho(x)\log^2\varrho(x). Observe that on this set we have the estimate for some positive constant C, and the integrability of the function \sqrt{\varrho(x)} follows from the estimate
\begin{equation*}
\sqrt{\varrho(x)}\leqslant \frac{1}{2(1+|x|)^{2d}}+ \frac{1}{2}(1+|x|)^{2d}\varrho(x).
\end{equation*}
\notag
With the aid of the function U the original equation is written in the form
\begin{equation*}
\operatorname{div}\bigl(\varrho(1+\varrho)\nabla U\bigr)=0.
\end{equation*}
\notag
Multiplying this equation by U\psi^2, where \psi\in C_0^{\infty}(\mathbb{R}^d), and integrating by parts we obtain
\begin{equation*}
\int_{\mathbb{R}^d} \varrho(1+\varrho)|\nabla U|^2\psi^2\,dx= -2\int_{\mathbb{R}^d}\varrho(1+\varrho) \langle\nabla U,\nabla\psi\rangle\psi U\,dx.
\end{equation*}
\notag
Applying the Cauchy–Bunyakowskii inequality to the right-hand side, we arrive at the inequality
\begin{equation*}
\int_{\mathbb{R}^d}\varrho(1+\varrho)|\nabla U|^2\psi^2\,dx\leqslant 4\int_{\mathbb{R}^d}\varrho(1+\varrho)|\nabla\psi|^2U^2\,dx.
\end{equation*}
\notag
Substituting in place of \psi the function \psi_N(x)=\zeta(x/N), where \zeta\in C_0^{\infty}(\mathbb{R}^d) and \zeta(x)=1 for |x|\leqslant 1, we obtain
\begin{equation*}
\int_{\mathbb{R}^d} \varrho(1+\varrho)|\nabla U|^2\psi_N^2\,dx\leqslant \frac{4\max|\nabla\zeta|^2}{N^2}\int_{\mathbb{R}^d}\varrho(1+\varrho)U^2\,dx.
\end{equation*}
\notag
Letting N\to\infty, we obtain the equality Therefore, the function U is constant, that is, we have the equality from which the formula for \varrho_{\beta}(x) follows immediately. In any dimension d\geqslant 3 the integral is finite, and therefore there is no stationary solution with integral over \mathbb{R}^d greater than M_1. In the one-dimensional and two-dimensional cases this integral diverges, hence for every M>0 there is a positive solution with integral equal to M.It was proved in [39], [49], and [47] that in the one-dimensional and two-dimensional cases, for every M>0 there exists a global in time solution \varrho to the Cauchy problem for the equation
\begin{equation}
\partial_t\varrho=\Delta\varrho+ \operatorname{div}\bigl(x\varrho(1+\varrho)\bigr)
\end{equation}
\tag{5.1}
such that the integral of the function x\mapsto \varrho(x,t) over \mathbb{R}^d equals M for every t, and it was also shown that, as t\to+\infty, this solution converges to the stationary solution with integral equal to M. In [111] sufficient conditions were obtained for the non-existence of a global in time solution in the three-dimensional case. Finally, in [46] and [69] approximation of equation (5.1) by more regular equations was considered in the multidimensional case, and it was shown that the solutions converge to the curve in the space of measures that is the sum of the Dirac measure at zero and the measure given by the density with respect to Lebesgue measure such that this density satisfies equation (5.1) away from the origin. 6. Nonlinear parabolic equationsNonlinear parabolic Fokker–Planck–Kolmogorov equations, like nonlinear elliptic equations, differ from linear ones by the dependence of their coefficients on solutions. The nature of the dependence of coefficients is determined by the applied problem that leads to the equation. An important particular case of nonlinear Fokker–Planck–Kolmogorov equations are the famous Vlasov equations, arising, for example, in the study of the behaviour of a large number of interrelated particles. Let us consider the system of differential equations
\begin{equation*}
\dot{x}^i_t=\frac{1}{N}\sum_{j=1}^NK(x^i_t, x^j_t), \qquad x^i_t\in\mathbb{R}^d,\quad 1\leqslant i\leqslant N.
\end{equation*}
\notag
The distribution of the points x^1_t,\dots,x^N_t in \mathbb{R}^d can be described with the aid of the measures Observe that for every function \varphi\in C_0^{\infty}(\mathbb{R}^d) we have
\begin{equation*}
\begin{aligned} \, \frac{d}{dt}\int_{\mathbb{R}^d}\varphi\, d\mu_t^N&=\frac{1}{N}\sum_{i=1}^N \biggl\langle\nabla\varphi(x^i_t),\frac{1}{N} \sum_{j=1}^NK(x^i_t, x^j_t)\biggr\rangle \\ &=\int_{\mathbb{R}^d}\langle\nabla\varphi(x),b(x,\mu_t^N)\rangle\,\mu_t^N(dx), \end{aligned}
\end{equation*}
\notag
where Thus, the measure \mu^N=\mu_t^N(dx)\,dt is a solution to Vlasov’s equation
\begin{equation*}
\partial_t\mu_t=-\operatorname{div}\bigl(b(x,\mu_t)\mu_t\bigr).
\end{equation*}
\notag
Under broad conditions on the function K one can show (see Dobrushin’s celebrated paper [54]) that if the initial distributions \mu_0^N converge weakly to some measure \nu, then the solutions \mu^N converge weakly to the solution \mu of the Cauchy problem for the same Vlasov equation with initial condition \nu. Note that in [54] the Kantorovich–Rubinshtein metric was used for the first time to prove the solvability of the nonlinear Vlasov equation with the aid of the contracting mapping theorem. In that paper the drift coefficient has the form where a and B have bounded continuous first order derivatives and B is also bounded. Now consider the system of stochastic equations
\begin{equation*}
dx^i_t=\frac{1}{N}\sum_{j=1}^NK(x^i_t,x^j_t)\,dt+\sqrt{2}\,dw_t^i, \qquad 1\leqslant i\leqslant N,
\end{equation*}
\notag
where the w_t^i are independent d-dimensional Wiener processes. Suppose that random variables x^1_0,\dots,x^N_0 are independent and have distribution \nu. Under broad conditions on K, for every t the sequence of empiric measures \mu_t^N=\dfrac{1}{N}\displaystyle\sum_{i=1}^N\delta_{x^i_t(\omega)} converges in probability, as N\to\infty, to a (non-random) probability measure \mu_t, where the measure \mu=\mu_t\,dt is a solution to the Cauchy problem for the nonlinear Fokker–Planck–Kolmogorov equation
\begin{equation*}
\partial_t\mu_t=\Delta\mu_t- \operatorname{div}\bigl(b(x,\mu_t)\mu_t\bigr),\qquad b(x,\mu_t)=\int_{\mathbb{R}^d}K(x,y)\, \mu_t(dy),
\end{equation*}
\notag
with the initial condition \mu_0=\nu. Moreover, as N\to\infty, the processes x_t^1,\dots,x_t^N converge to N independent processes y_t^1,\dots,y_t^N satisfying the stochastic Mckean–Vlasov equations where the w_t^i are independent d-dimensional Wiener processes, \mu_t is the distribution of the random variable y_t^i, and the random variables y_0^i are independent and have distribution \nu. This phenomenon is called ‘propagation of chaos’ (see [77], [93], and [94]). Thus, a typical example of the dependence of coefficients on solutions, as already noted in the elliptic case, is delivered by the expression The most important questions in the theory of nonlinear Fokker–Plank–Kolmogorov equations are the existence and uniqueness of solutions to the Cauchy problem and their convergence to the stationary solution as t\to+\infty. In what follows we discuss solutions in the classes of probability measures or non-negative measures; however, equations for signed measures are also of great interest.Let us proceed to precise formulations and examples. Let T>0 and let V\in C^2(\mathbb{R}^d) be a function such that
\begin{equation*}
V\geqslant 0 \quad\text{and}\quad \lim_{|x|\to\infty}V(x)=+\infty.
\end{equation*}
\notag
For every \tau\in(0,T] let \mathcal{M}_{\tau}(V) denote the set of finite non-negative Borel measures \mu on \mathbb{R}^d\times [0,\tau] of the form \mu=\mu_t\, dt, where \mu_t\in \mathcal{P}(\mathbb{R}^d), for which the mapping t\mapsto \mu_t is continuous in the weak topology and We say that a sequence of measures \mu^n=\mu^n_t\,dt in \mathcal{M}_{\tau}(V) converges V-weakly to a measure \mu=\mu_t\,dt in \mathcal{M}_{\tau}(V) if for all t\in [0,\tau] the sequence of measures \mu_t^n converges V-weakly to \mu. Let C^{+}[0,\tau] be the set of non-negative continuous functions on the interval [0,\tau]. For \tau\in(0,T] and a function \alpha\in C^{+}[0,\tau] we denote by M_{\tau,\alpha}(V) the set of measures \mu=\mu_t\,dt\in \mathcal{M}_{\tau}(V) such that for all t\in[0,\tau] we have As we noted in the elliptic case, if \mu_n=\mu^n_t\, dt\in M_{\tau,\alpha}(V) are measures such that for every t the measures \mu_t^n converge weakly to measures \mu_t and \mu=\mu_t\, dt, then one has the V-convergence of the measures \mu_n to \mu.Below we assume that each measure \mu=\mu_t\,dt\in \mathcal{M}_{\tau}(V) is extended to [0,T]\times\mathbb{R}^d by the equality \mu_t=\mu_{\tau} if t>\tau. If \mu=\mu_t\,dt\in \mathcal{M}_{\tau,\alpha}(V), then after this extension the new measure \widetilde{\mu} belongs to \mathcal{M}_{\tau,\widetilde{\alpha}}(V), where \widetilde{\alpha}(t)=\alpha(t) for t\in[0,\tau] and \widetilde{\alpha}(t)=\alpha(\tau) for t>\tau. If a sequence of measures \mu^n in \mathcal{M}_{\tau,\alpha}(V) converges V-weakly to a measure \mu\in\mathcal{M}_{\tau,\alpha}(V), then after this extension the sequence of measures \widetilde{\mu}^n converges V-weakly to the measure \widetilde{\mu}. We define similarly the classes \mathcal{SM}_{\tau}(V) and \mathcal{SM}_{\tau,\alpha}(V) consisting of the measures \mu=\mu_t\,dt, where each \mu_t is subprobability measure on \mathbb{R}^d. Suppose that for all \mu\in\mathcal{M}_{T}(V) and all i, j\leqslant d we are given Borel functions a^{ij}(\,\cdot\,{,}\,\cdot\, ,\mu) and b^{i}(\,\cdot\, {,} \,\cdot\,, \mu) on \mathbb{R}^d\times[0,T] such that the matrix A(x,t,\mu)=(a^{ij}(x,t,\mu))_{i,j\leqslant d} is symmetric and non-negative definite. Set
\begin{equation*}
\begin{gathered} \, L_{\mu}\varphi(x,t)=\operatorname{trace}(A(x,t,\mu)D^2\varphi(x))+ \langle b(x,t,\mu),\nabla\varphi(x)\rangle, \\ b(x,t,\mu)=(b^i(x,t,\mu))_{i\leqslant d}. \end{gathered}
\end{equation*}
\notag
Let \nu\in \mathcal{P}(\mathbb{R}^d). A measure \mu=\mu_t\, dt\in\mathcal{M}_{\tau}(V) on \mathbb{R}^d\times[0,\tau] is called a solution to the Cauchy problem for the nonlinear Fokker–Planck–Kolmogorov equation if \mu is a solution to the Cauchy problem for the linear equation with the operator L_{\mu}. As in the linear case, for the measure \mu=\mu_t\, dt this expression is identical to the shortened version The next theorem generalizes some results from [92] (see also [23], Chap. 6). An analogous assertion for the transport equation was obtained earlier in [24]. Theorem 6.1. Suppose that the following conditions are fulfilled: (i) for each ball B\subset\mathbb{R}^d and each function \alpha\in C^{+}[0, T] the functions a^{ij}(\,\cdot\,, t, \mu) and b^i(\,\cdot\,, t, \mu) are bounded on B uniformly in \mu\in \mathcal{M}_{T,\alpha}(V) and t\in[0,T], and for almost all t\in[0,T] they are equicontinuous on B uniformly in \mu\in \mathcal{M}_{T,\alpha}(V); (ii) if a sequence of measures \mu^n\in \mathcal{M}_{T,\alpha}(V) converges V-weakly to some measure \mu\in \mathcal{M}_{T,\alpha}(V), then for almost every t\in[0,T], for all x\in B we have
\begin{equation*}
\lim_{n\to\infty}a^{ij}(x, t, \mu^n)=a^{ij}(x, t, \mu \quad\textit{and}\quad \lim_{n\to\infty}b^{i}(x, t, \mu^n)=b^{i}(x, t, \mu);
\end{equation*}
\notag
(iii) there are mappings \Lambda_1 and \Lambda_2 associating to any function \alpha\in C^{+}[0,T] functions \Lambda_1[\alpha] and \Lambda_2[\alpha] in C^{+}[0,T] such that for all \alpha\in C^{+}[0,T] and all \mu\in\mathcal{M}_{T,\alpha}(V) we have
\begin{equation*}
L_{\mu}V(x, t)\leqslant \Lambda_1[\alpha](t)+\Lambda_2[\alpha](t)V(x) \quad \forall \, x\in\mathbb{R}^d,\ t\in[0, T].
\end{equation*}
\notag
Then for every probability measure \nu such that V\in L^1(\nu) there exists \tau in (0,T] such that there is a measure \mu=\mu_t\,dt\in\mathcal{M}_{\tau}(V) on \mathbb{R}^d\times[0,\tau] that is a solution to the Cauchy problem \partial_t\mu_t=L_{\mu}^{*}\mu_t, \mu_0=\nu. Moreover, if
\begin{equation*}
\Lambda_1[\alpha](t)=g_1(\alpha(t))\quad\textit{and}\quad \Lambda_2[\alpha](t)=g_2(\alpha(t)),
\end{equation*}
\notag
where g_1 and g_2 are positive non-decreasing continuous functions on [0,+\infty), then as \tau one can take an arbitrary number not greater than T and smaller than \tau^{*}, where Proof. Let \sigma=\sigma_t\,dt\in\mathcal{M}_{\tau,\alpha}(V). According to [23], Theorems 6.7.3 and 7.1.1, the Cauchy problem \partial_t\mu_t=L_{\sigma}^{*}\mu_t, \mu_0=\nu has a solution \mu=\mu_t\,dt\in\mathcal{M}_{\tau}(V) for which
\begin{equation*}
\int_{\mathbb{R}^d} V\,d\mu_t\leqslant Q(t)+R(t)\int_{\mathbb{R}^d} V\, d\nu,
\end{equation*}
\notag
where
\begin{equation*}
R(t)=\exp\biggl(\int_0^t\Lambda_2[\alpha](s)\,ds\biggr)\quad\text{and}\quad Q(t)=R(t)\int_0^t\frac{\Lambda_1[\alpha](s)}{R(s)}\,ds.
\end{equation*}
\notag
When the mappings \Lambda_1 and \Lambda_2 are arbitrary, we set and take \tau\in(0,T] sufficiently small so that Q(t)\leqslant 1 and R(t)\leqslant 2 for t\in[0,\tau]. When \Lambda_1[\alpha](t)=g_1(\alpha(t)) and \Lambda_2[\alpha](t)=g_2(\alpha(t)), we fix a number \tau satisfying the conditions
\begin{equation*}
0<\tau\leqslant T \quad\text{and}\quad \tau<\int_{u_0}^{+\infty}\frac{du}{g_1(u)+ug_2(u)}\,,
\end{equation*}
\notag
and find the function \alpha by solving on [0,\tau] the differential equation
\begin{equation*}
\alpha'=g_1(\alpha)+\alpha g_2(\alpha), \quad \alpha(0)=u_0.
\end{equation*}
\notag
It is clear that the integral of V against the measure \mu_t does not exceed \alpha(t). Choosing \tau and \alpha in this way, we obtain that the measure \sigma\in\mathcal{M}_{\tau, \alpha}(V) corresponds to the solution \mu in \mathcal{M}_{\tau, \alpha}(V). Let \varphi\in C_0^{\infty}(\mathbb{R}^d). Since for all t,s\in[0,\tau] we have
\begin{equation*}
\int_{\mathbb{R}^d}\varphi\,d\mu_t-\int_{\mathbb{R}^d}\varphi\,d\mu_s= \int_s^t\int_{\mathbb{R}^d} L_{\sigma}\varphi(x,\theta)\,d\mu_{\theta}\,d\theta
\end{equation*}
\notag
and by condition (i) the coefficients of operator L_{\sigma} on the support of \varphi are bounded uniformly in \sigma\in\mathcal{M}_{\tau,\alpha}(V) and t\in[0,\tau], we obtain
\begin{equation*}
\biggl|\int_{\mathbb{R}^d}\varphi\,d\mu_t- \int_{\mathbb{R}^d}\varphi\,d\mu_s\biggr|\leqslant C(\varphi)|t-s|,
\end{equation*}
\notag
where
\begin{equation*}
\begin{aligned} \, C(\varphi)&=\sup\bigl\{|\operatorname{trace}A(x, t, \sigma)D^2\varphi(x)|+ |\langle b(x, t, \sigma), \nabla\varphi(x)\rangle|\colon \\ &\qquad x\in\mathbb{R}^d,\ t\in[0,\tau],\ \sigma\in\mathcal{M}_{\tau,\alpha}(V)\bigr\}. \end{aligned}
\end{equation*}
\notag
Consider the set \mathcal{K} of measures \sigma=\sigma_t\,dt\in\mathcal{M}_{\tau,\alpha}(V) for which
\begin{equation*}
\biggl|\int_{\mathbb{R}^d}\varphi\,d\sigma_t- \int_{\mathbb{R}^d}\varphi\,d\sigma_s\biggr|\leqslant C(\varphi)|t-s|\quad \forall\, \varphi\in C_0^{\infty}(\mathbb{R}^d), \ \ t, s\in[0,\tau].
\end{equation*}
\notag
The set \mathcal{K} is convex and non-empty. We show that it is compact in the space of finite non-negative measures on \mathbb{R}^d\times[0,\tau] (which is metrizable by the Kantorovich–Rubinshtein metric). Let \sigma^n\in\mathcal{K}. Since \|V\|_{L^1(\sigma_t^n)}\leqslant\alpha(t), for each fixed t we can find a weakly convergent subsequence of \{\sigma_t^n\}. Using the diagonal procedure we extract a subsequence \{n_k\} such that the measures \sigma_r^{n_k} converge weakly to some probability measures \sigma_r for all rational numbers r\in [0,\tau]. Let t\in [0,T] and \varepsilon>0. For every function \varphi\in C_0^{\infty}(\mathbb{R}^d) there exists a rational number r\in[0,T] such that C(\varphi)|t-r|<\varepsilon. Let N be such that
\begin{equation*}
\biggl|\int_{\mathbb{R}^d}\varphi\,d\sigma_r^{n_k}- \int_{\mathbb{R}^d}\varphi\,d\sigma_r^{n_m}\biggr|\leqslant \varepsilon \quad \forall\, k, m>N.
\end{equation*}
\notag
Then
\begin{equation*}
\biggl|\int_{\mathbb{R}^d}\varphi\,d\sigma_t^{n_k}- \int_{\mathbb{R}^d}\varphi\,d\sigma_t^{n_m}\biggr|\leqslant 3\varepsilon.
\end{equation*}
\notag
Thus, for every function \varphi\in C_0^{\infty}(\mathbb{R}^d) the sequence of integrals is Cauchy. This yields the weak convergence of the sequence \{\sigma_t^{n_k}\} to some probability measure \sigma_t. Observe that \|V\|_{L^1(\sigma_t)}\leqslant\alpha(t) and the mapping t\mapsto\sigma_t is continuous, because
\begin{equation*}
\biggl|\int_{\mathbb{R}^d}\varphi\,d\sigma_t- \int_{\mathbb{R}^d}\varphi\,d\sigma_s\biggr|\leqslant C(\varphi)|t-s|.
\end{equation*}
\notag
Thus, we have obtained a measure \sigma=\sigma_t\,dt\in\mathcal{M}_{\tau, \alpha}(V) to which the subsequence \{\sigma^{n_k}\} converges weakly.
Consider the multivalued mapping \Phi that with every measure \sigma\in\mathcal{K} associates the subset \Phi(\sigma)\subset\mathcal{K} consisting of the solutions to the Cauchy problem \partial_t\mu=L_{\sigma}^{*}\mu, \mu_0=\nu. It is clear that \Phi(\sigma) is non-empty and convex. Let us verify that the graph of \Phi is closed. Suppose that measures \sigma^n\in\mathcal{K} converge weakly to \sigma\in\mathcal{K} and measures \mu^n\in\mathcal{K} converge weakly to \mu. We show that \mu\in\Phi(\sigma). Passing to a subsequence \{n_k\} we can assume that for every t the measures \sigma^{n_k}_t and \mu^{n_k}_t converge V-weakly to \sigma_t and \mu_t, respectively. Then by conditions (i) and (ii), on every ball, for almost all t the sequences of functions a^{ij}(\,\cdot\,, t, \sigma^{n_k}), b^i(\,\cdot\, , t, \sigma^{n_k}) converge uniformly to the continuous functions a^{ij}(\,\cdot\,, t, \sigma) and b^i(\,\cdot\,, t, \sigma). This observation and the weak convergence of \{\mu^{n_k}_t\} to \mu_t enable us to justify the equality
\begin{equation*}
\lim_{k\to\infty}\int_{\mathbb{R}^d}L_{\sigma^{n_k}} \varphi(x,t)\,\mu^{n_k}_t(dx)= \int_{\mathbb{R}^d}L_{\sigma}\varphi(x,t)\,\mu_t(dx)
\end{equation*}
\notag
for every fixed function \varphi\in C_0^{\infty}(\mathbb{R}^d) and almost all t\in[0,\tau]. From Lebesgue’s dominated convergence theorem we obtain
\begin{equation*}
\lim_{k\to\infty}\int_0^t\int_{\mathbb{R}^d} L_{\sigma^{n_k}}\varphi(x, s)\,\mu^{n_k}_s(dx)\,ds= \int_0^t\int_{\mathbb{R}^d}L_{\sigma}\varphi(x, s)\,\mu_s(dx)\,ds.
\end{equation*}
\notag
Therefore, we can pass to the limit in the equality
\begin{equation*}
\int_{\mathbb{R}^d}\varphi\,d\mu_t^{n_k}= \int_{\mathbb{R}^d}\varphi\,d\nu^{n_k}+\int_0^t\int_{\mathbb{R}^d} L_{\sigma^{n_k}}\varphi\,d\mu_s^{n_k}\,ds
\end{equation*}
\notag
as k\to\infty and obtain the equality
\begin{equation*}
\int_{\mathbb{R}^d}\varphi\,d\mu_t=\int_{\mathbb{R}^d}\varphi\,d\nu+ \int_0^t\int_{\mathbb{R}^d}L_{\sigma}\varphi\,d\mu_s\,ds.
\end{equation*}
\notag
Thus, the measure \mu=\mu_t\,dt belongs to \Phi(\sigma). It also follows that \Phi(\sigma) is closed. By the Kakutani–Ky Fan theorem there exists a measure \mu\in\mathcal{K} belonging to \Phi(\mu), that is, \mu is a solution to the Cauchy problem \partial_t\mu=L^{*}_{\mu}\mu, \mu_0=\nu. By the definition of the set \mathcal{K} the mapping t\mapsto\mu_t is continuous. \Box Let us give a typical example when Theorem 6.1 is applicable.
\begin{equation*}
a^{ij}(x,t,\mu)=\int_{\mathbb{R}^d}q^{ij}(x, y)\,\mu_t(dy) \quad\text{and}\quad b^i(x,t,\mu)=\int_{\mathbb{R}^d}h^i(x,y)\,\mu_t(dy),
\end{equation*}
\notag
where q^{ij} and h^i are continuous functions, the matrix Q(x,y)=(q^{ij}(x,y))_{i,j\leqslant d} is symmetric, and 0\leqslant Q(x,y)\leqslant \theta\cdot {\rm I} for some number \theta and all x and y. Suppose that the vector field h=(h^i)_{i\leqslant d} satisfies the estimates
\begin{equation*}
\langle h(x,y),x\rangle\leqslant C_1+C_2|x|^2\quad\text{and} \quad |h(x,y)|\leqslant C_3(1+|x|^{m-1}+|y|^{m-1}),
\end{equation*}
\notag
where C_1, C_2, and C_3 are positive numbers, m\geqslant 2. Then for the function V(x)=|x|^m there exists C>0 such that
\begin{equation*}
\begin{aligned} \, L_{\mu}V(x,t) &\leqslant m|x|^{m-2}\sup\operatorname{trace}Q+m(m-2) |x|^{m-2}\sup\|Q\| \\ &\quad+mC_1|x|^{m-2}+mC_2|x|^m \leqslant C+CV(x)\quad \forall\, (x, t). \end{aligned}
\end{equation*}
\notag
Thus, the hypotheses of Theorem 6.1 are fulfilled for the constant functions g_1=g_2=C, hence a solution to the Cauchy problem exists on the interval [0,\tau] for every \tau>0 since the integral of the function 1/(C+Cu) diverges. With the aid of the superposition principle (see, for example, [30]), from the existence of a solution to the Cauchy problem for the Fokker–Planck–Kolmogorov equation one can derive the existence of a solution to the nonlinear martingale problem. Recall that the superposition principle for a solution (\mu_t)_{t\in [0,T]} to the Cauchy problem for the Fokker–Planck–Kolmogorov equation with the operator L gives (under broad assumptions on L) a Borel probability measure P on the path space C([0,T],\mathbb{R}^d) such that \mu_t is the image P\circ e_t^{-1} of the measure P under the mapping e_t(\omega)=\omega(t), t\in [0,T], and for all \varphi\in C_0^{\infty}(\mathbb{R}^d) the process
\begin{equation*}
\xi_t(\omega)=\varphi(\omega(t))-\varphi(\omega(0))- \int_0^tL \varphi(\omega(s),s)\,ds
\end{equation*}
\notag
is a martingale with respect to P and the filtration \mathcal{F}_t=\sigma\bigl(\omega(s), s\leqslant t\bigr). The measure P is called a solution to the martingale problem with operator L. Thus, the superposition principle means a representation of the solution to the Cauchy problem for the Fokker–Planck–Kolmogorov equation in the form of the one-point distributions of a solution to the martingale problem. Theorem 6.3. Suppose that for every measure \mu\in\mathcal{M}_T(V) there exists C(\mu)>0 such that Let \tau\in(0,T], and let \nu be a probability measure such that V\in L^1(\nu). Suppose that a measure \mu=\mu_t\,dt\in\mathcal{M}_{\tau}(V) is a solution to the Cauchy problem \partial_t\mu_t=L_{\mu}^{*}\mu_t, \mu_0=\nu.Then there exists a probability measure P on C([0,T],\mathbb{R}^d) that is a solution to the martingale problem for the operator L_{\mu} such that \mu_t=P\circ e_t^{-1}. Proof. We observe that the functions \|A(\,\cdot\,{,}\,\cdot\,,\mu)\| and |b(\,\cdot\,{,}\,\cdot\,,\mu)| are integrable against the measure \mu on \mathbb{R}^d\times[0,\tau] and apply the superposition principle for the linear equation with the operator L_{\mu}, which is possible under the conditions indicated in the theorem (see [30]). \Box The existence and uniqueness of solutions to the nonlinear martingale problem and the corresponding stochastic equation were studied in [64], [95], and [96]. Now consider an example where a solution does not exist on every interval [0,\tau]. Example 6.4. Let d=1, a(x,t,\mu)=0, and The hypotheses of Theorem 6.1 are fulfilled for V(x)=|x|^2 because
\begin{equation*}
L_{\mu}V(x)=2|x|^2 \int_{\mathbb{R}} y\,\mu_t(dy) \leqslant 2V(x)\biggl(\int_{\mathbb{R}}V(y)\, \mu_t(dy)\biggr)^{1/2}.
\end{equation*}
\notag
Thus, here we have g_1(u)=0 and g_2(u)=2u^{1/2}, and a solution exists on [0,\tau] whenever
\begin{equation*}
\tau<\int_{u_0}^{+\infty}\frac{du}{2u^{3/2}}=\frac{1}{\sqrt{u_0}}\,,\qquad u_0=\int_{\mathbb{R}}|y|^2\, \nu(dy).
\end{equation*}
\notag
Set Then from the equation for \mu one can derive the equalities
\begin{equation*}
f'(t)=f(t)^2 \quad\text{and}\quad f(0)=\int_{\mathbb{R}} y\, \nu(dy).
\end{equation*}
\notag
Let \nu=\delta_a, where a>0. Then f(t) is defined on [0,a^{-1}), and the Cauchy problem for the equation \partial_t\mu_t=L_{\mu}^{*}\mu_t with initial condition \nu has no solutions on [0,\tau] for \tau\geqslant a^{-1}, which agrees with the estimate \tau<u_0^{-1/2} obtained above since here we have u_0=a^2. Let us give a sufficient condition for the absence of a global solution. The problem of the ‘blow-up’ of solutions to equations of the form under consideration has a vast literature; see [97], [98], [14], [43], and [85]. The following result was obtained in [92]. Theorem 6.5. Suppose that conditions (i) and (ii) in Theorem 6.1 are fulfilled and there is a continuous increasing positive function G on [0,+\infty) such that
\begin{equation*}
{\rm (a)}\ \ L_{\mu}V(x,t)\geqslant G\biggl(\int_{\mathbb{R}^d}V\, d\mu_{t}\biggr)V(x)\quad \textit{or}\quad {\rm (b)}\ \ L_{\mu }V(x,t)\geqslant G\biggl(\int_{\mathbb{R}^d}V\, d\mu_{t}\biggr).
\end{equation*}
\notag
Also let |\sqrt{A(x,t,\mu)}\,\nabla V(x)|^2\leqslant C_1+C_2V(x) for some positive numbers C_1 and C_2. Suppose that u_{0}=\|V\|_{L^1(\nu)}>0 and in case (a) the function 1/(uG(u)) is integrable on [u_0,+\infty) and its integral equals \tau_1<T, while in case (b) the function 1/G(u) is integrable on [u_0,+\infty) and its integral equals \tau_1<T. Then the Cauchy problem \partial_t\mu_t=L_{\mu}^{*}\mu_t, \mu_0=\nu has no solution in \mathcal{M}_T(V) on the interval [0,\tau] for \tau\geqslant \tau_1. In [68] some integral conditions involving Lyapunov functions were considered. Let us state for example the following assertion, which is analogous to a result in [68] for the stochastic equation, proved under some additional restrictions on the growth of coefficients, but for more general Lyapunov functions. The existence result for the stochastic equation implies the solvability of the Cauchy problem for the nonlinear Fokker–Planck–Kolmogorov equation (but the converse result can be obtained from the superposition principle). Let a^{ij}(x,t,\mu) and b^i(x,t,\mu) be defined for all measures \mu\in\mathcal{SM}_T(V). Theorem 6.6. Let conditions (i) and (ii) in Theorem 6.1 be fulfilled, but for the classes \mathcal{SM}_{T,\alpha}(V) in place of \mathcal{M}_{T,\alpha}(V). Suppose that there exist positive numbers C_1 and C_2 such that for all measures \mu\in\mathcal{SM}_{T,\alpha}(V) with compact support in \mathbb{R}^d\times[0,T] we have
\begin{equation*}
\int_{\mathbb{R}^d}L_{\mu}V(x,t)\,\mu_t(dx)\,dt\leqslant C_1+C_2\int_{\mathbb{R}^d}V(x)\,\mu_t(dx).
\end{equation*}
\notag
Then, for every probability measure \nu such that V\in L^1(\nu) there exists a solution \mu=\mu_t\,dt\in\mathcal{M}_T(V) to the Cauchy problem \partial_t\mu_t=L_{\mu}^{*}\mu_t, \mu_0=\nu. Proof. Suppose that \varphi\in C_0^{\infty}(\mathbb{R}^d), 0\leqslant\varphi\leqslant 1, and \varphi(x)=1 if |x|<1, \nu^n is a probability measure with support in the ball |x|<n, the measures \nu^n converge weakly to \nu, and the norms \|V\|_{L^1(\nu^n)} are uniformly bounded by some number C_{\nu}. Set \varphi_n(x)=\varphi(x/n) and L_{\mu,n}=\varphi_nL_{\varphi_n\mu}. By Theorem 6.1 there exists a solution \mu^n\in\mathcal{M}_T(V) to the Cauchy problem \partial_t\mu_t=L_{\mu,n}^{*}\mu_t, \mu_0=\nu^n. Observe that \varphi_n\mu^n \in \mathcal{SM}_{T,\alpha}(V), and so
\begin{equation*}
\begin{aligned} \, \int_{\mathbb{R}^d}L_{\mu^n,n}V(x,t)\,\mu^n_t(dx)&= \int_{\mathbb{R}^d}[L_{\varphi_n\mu^n}V(x,t)]\varphi_n(x)\,\mu^n_t(dx) \\ &\leqslant C_1+C_2\int_{\mathbb{R}^d}V(x)\varphi_n(x)\,\mu^n_t(dx) \\ &\leqslant C_1+C_2\int_{\mathbb{R}^d}V(x)\,\mu^n_t(dx). \end{aligned}
\end{equation*}
\notag
It follows from our conditions on coefficients that in this case equality (3.7) remains valid for all functions \varphi\in C^2(\mathbb{R}^d) with compact support, but since the function \varphi_n vanishes outside a ball, for \mu=\mu^n this equality is also true for \varphi=V. Therefore,
\begin{equation*}
\int_{\mathbb{R}^d}V(x)\,\mu_t^n(dx)\leqslant \int_{\mathbb{R}^d}V(x)\,\nu^n(dx)+ \int_0^t\biggl[C_1+C_2\int_{\mathbb{R}^d}V(x) \varphi_n(x)\,\mu^n_s(dx)\biggr]\,ds.
\end{equation*}
\notag
Applying Gronwall’s inequality we obtain
\begin{equation*}
\int_{\mathbb{R}^d}V(x)\,\mu_t^n(dx)\leqslant \frac{C_1}{C_2}\,e^{C_2t}+C_{\nu}e^{C_2t}.
\end{equation*}
\notag
Set \alpha(t)=C_1C_2^{-1}e^{C_2t}+C_{\nu}e^{C_2t}. Then
\begin{equation*}
\mu^n\in\mathcal{M}_{T,\alpha}(V)\quad\text{and}\quad \varphi_n\mu^n\in\mathcal{SM}_{T,\alpha}(V).
\end{equation*}
\notag
Since \varphi_n(x)a^{ij}(x,t,\varphi_n\mu^n) and \varphi_n(x)b^{i}(x,t,\varphi_n\mu^n) are bounded on every ball B\subset \mathbb{R}^d uniformly in t\in[0,T] and n, for every function \psi\in C_0^{\infty}(\mathbb{R}^d) we obtain the estimate
\begin{equation*}
\biggl|\int_{\mathbb{R}^d}\psi(x)\,\mu_t^n(dx)- \int_{\mathbb{R}^d}\psi(x)\, \mu_s^n(dx)\biggr|\leqslant C(\psi)|t-s|
\end{equation*}
\notag
for some constant C(\psi) independent of t, s, and n. Repeating the reasoning from the proof of Theorem 6.1, we find a subsequence \{n_k\} such that the sequence \{\mu_t^{n_k}\} converges weakly to some probability measure \mu_t; moreover, \|V\|_{L^1(\mu_t)}\leqslant\alpha(t) and the mapping t\mapsto\mu_t is continuous. Thus, we have obtained the measure \mu=\mu_t\,dt\in\mathcal{M}_{T,\alpha}(V). It is clear that the measures \mu^{n_k} converge V-weakly to \mu, and the analogous assertion is true for the sequence \{\varphi_{n_k}\mu^{n_k}\}. For every function \psi\in C_0^{\infty}(\mathbb{R}^d) we have
\begin{equation*}
\int_{\mathbb{R}^d}\psi(x)\, \mu_t^{n_k}(dx)= \int_{\mathbb{R}^d}\psi(x)\,\nu^{n_k}(dx)+ \int_0^t\int_{\mathbb{R}^d} \varphi_{n_k}(x)L_{\varphi_{n_k} \mu^{n_k}}\psi(x,s)\,\mu_s^{n_k}(dx)\,ds.
\end{equation*}
\notag
By the hypotheses of the theorem the sequences a^{ij}(x,t,\varphi_n\mu^n) and b^{i}(x,t,\varphi_n\mu^n) converge uniformly on every ball to functions a^{ij}(x,t,\mu) and b^{i}(x,t,\mu) continuous in x. This enables us to pass to the limit as n\to\infty and obtain the equality
\begin{equation*}
\int_{\mathbb{R}^d}\psi(x)\,\mu_t(dx)=\int_{\mathbb{R}^d}\psi(x)\,\nu(dx)+ \int_0^t\int_{\mathbb{R}^d} L_{\mu}\psi(x,s)\,\mu_s(dx)\,ds.
\end{equation*}
\notag
Thus, the measure \mu=\mu_t\,dt is a solution to the Cauchy problem on \mathbb{R}^d\times[0,T]. \Box Let us consider an example illustrating Theorem 6.6.
\begin{equation*}
a(x,t,\mu)=x^2\biggl(\int_{\mathbb{R}}y\, \mu_t(dy)\biggr)^4 \quad\text{and}\quad b(x,t,\mu)=-x^3\biggl(\int_{\mathbb{R}}y\, \mu_t(dy)\biggr)^2.
\end{equation*}
\notag
If we take V(x)=x^2/2, then
\begin{equation*}
L_{\mu}V(x,t)=x^2\biggl(\int_{\mathbb{R}}y\, \mu_t(dy)\biggr)^4- x^4\biggl(\int_{\mathbb{R}}y\, \mu_t(dy)\biggr)^2.
\end{equation*}
\notag
Let a\ne 0. Consider the function h(x)=x^2a^4-x^4a^2. Its maximum a^6/4 is attained at x^2=a^2/2. Thus, the best upper estimate has the form
\begin{equation*}
L_{\mu}V(x, t)\leqslant \frac{1}{4}\biggl(\int_{\mathbb{R}}y\, \mu(dy)\biggr)^6\leqslant 2\biggl(\int_{\mathbb{R}}V(y)\, \mu_t(dy)\biggr)^3.
\end{equation*}
\notag
In this situation Theorem 6.1 enables us to construct a solution only for \tau<u_0^{-2}, where u_0=\|V\|_{L^1(\nu)}. The condition from the theorem has the form
\begin{equation*}
\int_{\mathbb{R}^d}L_{\mu}V(x, t)\, \mu_t(dx)\leqslant \biggl(\int_{\mathbb{R}}y\,d\mu_t\biggr)^2 \biggl[\biggl(\int_{\mathbb{R}}y^2\,\mu_t(dy)\biggr)^2- \int_{\mathbb{R}}y^4\,\mu_t(dy)\biggr]\leqslant 0.
\end{equation*}
\notag
However, by Theorem 6.6 a solution exists for all \tau>0. One also considers Lyapunov functions depending on the measure \mu. In this case an important role is played by differential calculus on the space of probability measures and generalizations of Itôs formula to the case of functions depending on the distributions of random process (see [41], Chap. 5, and [68]). Applications of differential calculus on the space of probability measures to the justification of the existence and uniqueness of solutions to nonlinear martingale problems and the corresponding nonlinear Fokker–Planck–Kolmogorov equations were discussed in [50]. The main novelty of that work consists in conditions on the dependence of the diffusion coefficient on the solution, because in place of the classical Lipschitz condition the existence and regularity of a linear derivative of A with respect to the measure are assumed. In the case where the matrix A is non-degenerate and sufficiently regular, a solution possesses a continuous density with respect to Lebesgue measure, and it is more natural to consider nonlinear equations in the space of densities rather than measures. Let us give an example of such an assertion. We denote by \mathcal{L}_{\tau}(V) and \mathcal{L}_{\tau,\alpha}(V) the subsets of \mathcal{M}_{\tau}(V) and \mathcal{M}_{\tau,\alpha}(V), respectively, consisting of the measures \mu=\mu_t\,dt such that \mu_t(dx)=\varrho(x,t)\,dx for every t\in(0,\tau]. Note that we do not assume the absolute continuity of the measure \mu_0. Let B be an open ball in \mathbb{R}^d. Let B_r(z) denote the intersection B\cap B(z,r), where B(z,r) is the open ball of radius r with centre z. Set We say that a measurable function f on B\times(0, T) satisfies the Dini mean oscillation condition in x with modulus of continuity w_B if there exists a continuous increasing function w_B on [0,+\infty) such that w_B(0)=0 and
\begin{equation*}
\sup_{Q_r(z,s)\subset \overline{B}\times[0,T]}\frac{1}{|Q_r(z,s)|} \int_{Q_r(z,s)}|f(y,t)-f_{r}(z,t)|\,dy\,dt\leqslant w_B(r),
\end{equation*}
\notag
where and, moreover, Now assume that the coefficients of the operator L_{\mu} are defined only on the set \mathcal{L}_T(V). By a solution to the Cauchy problem \partial_t\mu=L^{*}_{\mu}\mu, \mu_0=\nu we mean a measure \mu\in\mathcal{L}_T(V) that is a solution to the Cauchy problem for the linear equation with operator L_{\mu} and initial condition \nu. Theorem 6.8. Suppose that the following conditions are fulfilled: (i) for every ball B and every function \alpha\in C^{+}[0,T] one can find a positive number \lambda_{B,\alpha} and function w_{B,\alpha} such that for all \mu\in\mathcal{L}_{T,\alpha}(V) and all t\in [0,T] the functions a^{ij}(\,\cdot\,,t,\mu) are continuous on B, satisfy the Dini mean oscillation condition with modulus of continuity w_{B,\alpha} on B\times(0,T), and for all x\in B and t\in[0,T] one has
\begin{equation*}
\frac{1}{\lambda_{B,\alpha}}\cdot{\rm I}\leqslant A(x,t,\mu)\leqslant \lambda_{B,\alpha}\cdot {\rm I};
\end{equation*}
\notag
(ii) for every ball B\subset\mathbb{R}^d and every function \alpha\in C^{+}[0, T] the functions b^i(\,\cdot\,,t,\mu) are bounded on B uniformly in \mu\in \mathcal{L}_{T,\alpha}(V) and t\in[0,T], and for almost all t\in[0,T] they are equicontinuous on B uniformly in \mu\in \mathcal{L}_{T,\alpha}(V); (iii) if measures \mu^n=\varrho^n(x,t)\,dx\,dt and \mu=\varrho(x,t)\,dx\,dt belong to \mathcal{L}_{T,\alpha}(V) and for almost every t\in (0,T] the sequence \varrho^n(\,\cdot\,,t) converges to \varrho(\,\cdot\,,t) in L^1(\mathbb{R}^d), then for almost every t\in[0,T], for all x\in\mathbb{R}^d we have
\begin{equation*}
\lim_{n\to\infty}a^{ij}(x,t,\mu^n)=a^{ij}(x,t,\mu)\quad\textit{and}\quad \lim_{n\to\infty}b^{i}(x,t,\mu^n)=b^{i}(x,t,\mu);
\end{equation*}
\notag
(iv) there exist positive numbers C_1 and C_2 such that and
\begin{equation*}
L_{\mu}V(x, t)\leqslant C_2+C_2V(x) \quad \forall \, \mu\in\mathcal{L}_{T}(V), \ x\in\mathbb{R}^d, \ t\in[0,T].
\end{equation*}
\notag
Then for every probability measure \nu satisfying the condition V\!\in L^1(\nu) there exists a solution \mu\in\mathcal{L}_T(V) to the Cauchy problem \partial_t\mu_t=L_{\mu}^{*}\mu_t, \mu_0=\nu. Proof. Let \omega\in C_0^{\infty}(\mathbb{R}^d) satisfy
\begin{equation*}
\omega\geqslant 0, \qquad \omega(x)=0 \quad \text{for } \ |x|>1 \quad\text{and}\quad \|\omega\|_{L^1(\mathbb{R}^d)}=1.
\end{equation*}
\notag
Set \omega_n(x)=n^d\omega(nx). Let
\begin{equation*}
(\omega_n*\mu)(x,t)=\int_{\mathbb{R}^d}\omega_n(x-y)\, \mu_t(dy).
\end{equation*}
\notag
Observe that for \mu\in\mathcal{M}_{T, \alpha}(V) each measure (\omega_n*\mu)(x, t)\,dx is a probability measure and
\begin{equation*}
\int_{\mathbb{R}^d}V(x)(\omega_n*\mu)(x, t)\,dx\leqslant C_1+C_1\alpha(t)=\widetilde{\alpha}(t),
\end{equation*}
\notag
that is, (\omega_n*\mu)(x,t)\,dx\,dt\in\mathcal{L}_{T,\widetilde{\alpha}}. Moreover, if some measures \mu^k\in\mathcal{M}_{T,\alpha}(V) converge V-weakly to \mu\in\mathcal{M}_{T, \alpha}(V), then for every t\in[0,T] the probability densities x\mapsto (\omega_n*\mu^k)(x,t) converge pointwise to the probability density x\mapsto (\omega_n*\mu)(x,t), which implies their convergence in L^1(\mathbb{R}^d).
Set L_{\mu, n}=L_{(\omega_n*\mu)\,dx\,dt}. According to the observations made above, the coefficients of the operator L_{(\omega_n*\mu)\,dx\,dt} satisfy all conditions in Theorem 6.1 for \Lambda_1=\Lambda_2= C_2. Therefore, on \mathbb{R}^d\times[0,T] there exists a solution \mu^n\in\mathcal{M}_T(V) to the Cauchy problem \partial_t\mu_t=L_{\mu, n}^{*}\mu_t, \mu_0=\nu. Observe that \mu^n satisfies the inequality
\begin{equation*}
\int_{\mathbb{R}^d}V(x)\,\mu^n_t(dx)\leqslant C_2e^{C_2t}+ e^{C_2t}\int_{\mathbb{R}^d}V(x)\,\nu(dx)=\alpha(t),
\end{equation*}
\notag
that is, \mu^n\in\mathcal{M}_{T, \alpha}(V) for all n. Hence \omega_n*\mu^n\in\mathcal{M}_{T,\widetilde{\alpha}}(V), where \widetilde{\alpha}=C_1+C_1\alpha. Moreover, the solution \mu^n has a continuous density \varrho^n with respect to Lebesgue measure, and according to [56], Theorem 1.4, for every ball B and every interval J\subset(0,T) one can find a number C(B,J,\widetilde{\alpha}) and a continuous increasing function \omega_{B,J,\widetilde{\alpha}} independent of n such that for all x,y\in B and t,s\in J one has
\begin{equation*}
\varrho^n(x,t)\leqslant C(B,J,\widetilde{\alpha}) \quad\text{and}\quad |\varrho^n(x,t)-\varrho^n(y,s)|\leqslant \omega_{B,J,\widetilde{\alpha}}(|x-y|+|t-s|).
\end{equation*}
\notag
Passing to a subsequence, we can assume that \{\varrho^n\} converges to some continuous function \varrho on \mathbb{R}^d\times(0,T) uniformly on every cylinder B\times J. Since for t>0 the functions \varrho^n(\,\cdot\,,t) are probability densities and the integrals of V(x)\varrho^n(x,t) over \mathbb{R}^d are uniformly bounded, the function \varrho(\,\cdot\,,t) is a probability density and the sequence \varrho^n(\,\cdot\,,t) converges to it in L^1(\mathbb{R}^d). In addition, for every t>0 the sequence (\omega_n*\mu^n)(\,\cdot\,, t) converges to \varrho(\,\cdot\,, t) in L^1(\mathbb{R}^d) because
\begin{equation*}
\begin{aligned} \, \|(\omega_n*\mu^n)(\,\cdot\,,t)-\varrho(\,\cdot\,,t)\|_{L^1(\mathbb{R}^d)} &\leqslant\|\varrho^n(\,\cdot\,,t)- \varrho(\,\cdot\,,t)\|_{L^1(\mathbb{R}^d)} \\ &\qquad+\|(\omega_n*\varrho)(\,\cdot\,,t)- \varrho(\,\cdot\,,t)\|_{L^1(\mathbb{R}^d)}. \end{aligned}
\end{equation*}
\notag
These observations enable us to pass to the limit in the equality
\begin{equation*}
\int_{\mathbb{R}^d}\varphi(x)\varrho^n(x, t)\,dx= \int_{\mathbb{R}^d}\varphi(x)\,\nu(dx)+\int_0^t\int_{\mathbb{R}^d} L_{(\omega_n*\mu^n)\,dx\,dt}\varphi(x, s)\varrho^n(x, s)\,dx\,ds
\end{equation*}
\notag
as n\to\infty for every function \varphi\in C_0^{\infty}(\mathbb{R}^d) and obtain the equality
\begin{equation*}
\int_{\mathbb{R}^d}\varphi(x)\varrho(x, t)\,dx= \int_{\mathbb{R}^d}\varphi(x)\,\nu(dx)+\int_0^t\int_{\mathbb{R}^d} L_{\varrho\,dx\,dt}\varphi(x, s)\varrho(x, s)\,dx\,ds.
\end{equation*}
\notag
Thus, the measure \mu=\varrho(x,t)\,dx\,dt is the required solution. \Box Note that in the proof of Theorem 6.8 we actually approximated the nonlinear equation for the density \varrho by a nonlinear equation for the measure \mu, by using the substitution of the expression \omega_n*\mu in place of \varrho into the coefficients of the equation. This approach has successfully been applied to the approximation of nonlinear Fokker–Planck–Kolmogorov equations with local nonlinearity by equations with a nonlinearity of nonlocal type (see [52]). One cannot expect the uniqueness of solutions in Theorem 6.8. Example 6.9. Let d=1 and \nu=\delta_0. Set
\begin{equation*}
\varrho_1(x,t)=\frac{1}{\sqrt{4\pi t}}e^{-x^2/(4t)} \quad\text{and}\quad \varrho_2(x,t)=\frac{1}{\sqrt{8\pi t}}e^{-x^2/(8t)}.
\end{equation*}
\notag
Observe that
\begin{equation*}
\|\varrho_1(\,\cdot\,,t)-\varrho_2(\,\cdot\,,t)\|_{L^1(\mathbb{R})}= \|\varrho_1(\,\cdot\,,1)-\varrho_2(\,\cdot\,,1)\|_{L^1(\mathbb{R})}=c>0
\end{equation*}
\notag
for t>0. Let V(x)=|x|^2, b=0,
\begin{equation*}
a(x,t,\mu)=1+c^{-1}\|\mu_t-\varrho_1(\,\cdot\,, t)\,dx\|,\quad \mu=\mu_t\,dt\in\mathcal{L}_T(V)\quad\text{for}\ \ t>0
\end{equation*}
\notag
and a(x,0,\mu)=1+c^{-1}\|\mu_0-\delta_0\|. Since a(x,t,\varrho_1\,dx\,dt)=1 and a(x,t,\varrho_2\,dx\,dt)=2 for t>0, the measures \varrho_1\,dx\,dt and \varrho_2\,dx\,dt are solutions to the Cauchy problem
\begin{equation*}
\partial_t\mu_t=\partial_x^2\bigl(a(x,t,\mu)\mu_t\bigr), \quad \mu_0=\delta_0.
\end{equation*}
\notag
Note that a(x,t,\mu) does not depend on x, the inequality 1\leqslant a(x,t,\mu)\leqslant 1+2c^{-1} holds, and
\begin{equation*}
|a(x,t,\mu)-a(x,t,\sigma)|\leqslant c^{-1}\|\mu_t-\sigma_t\|,
\end{equation*}
\notag
that is, in this example the coefficient a is non-degenerate, constant in x, and Lipschitz with respect to \mu. Other examples of non-uniqueness can be found in [90]. One method for deriving sufficient conditions for uniqueness consists in applying a suitable estimate for distances between solutions to linear equations. Let us mention an assertion from [108]. Theorem 6.10. Let V(x)=1+|x|^{2m}, m\geqslant 1. Suppose that the following conditions are fulfilled: (i) for every measure \mu\in\mathcal{L}_T(V) there exist positive numbers \lambda(\mu) and \theta(\mu) such that for all x, y\in\mathbb{R}^d and all t\in[0, T] one has
\begin{equation*}
\frac{1}{\theta(\mu)}\cdot {\rm I}\leqslant A(x,t,\mu)\leqslant \theta(\mu)\cdot {\rm I} \quad\textit{and}\quad |A(x,t,\mu)-A(y,t,\mu)|\leqslant\lambda(\mu)|x-y|;
\end{equation*}
\notag
(ii) for every measure \mu\in\mathcal{L}_T(V) there exists \gamma(\mu)>0 such that for all x\in\mathbb{R}^d and all t\in[0, T] one has
\begin{equation*}
\langle b(x,t,\mu),x\rangle\leqslant\gamma(\mu)+\gamma(\mu)|x|^2 \quad\textit{and}\quad |b(x, t, \mu)|\leqslant \gamma(\mu)+\gamma(\mu)|x|^m;
\end{equation*}
\notag
(iii) there are positive numbers C_1 and C_2 such that for any fixed \mu, \sigma\in\mathcal{L}_T(V) the inequalities
\begin{equation*}
|a^{ij}(x,t,\mu)-a^{ij}(x,t,\sigma)|+|\nabla_xa^{ij}(x,t,\mu)- \nabla_xa^{ij}(x,t,\sigma)|\leqslant C_1\|\mu_t-\sigma_t\|
\end{equation*}
\notag
and
\begin{equation*}
|b^i(x,t,\mu)-b^i(x,t,\sigma)|\leqslant C_2(1+|x|^m)\|\mu_t-\sigma_t\|
\end{equation*}
\notag
hold for almost all (x,t)\in\mathbb{R}^d\times[0,T]. Then for every probability measure \nu=\varrho_0\,dx satisfying the conditions \varrho_0\log\varrho_0\in L^1(\mathbb{R}^d) and V\varrho_0\in L^1(\mathbb{R}^d), the Cauchy problem \partial_t\mu_t=L_{\mu}^{*}\mu_t, \mu_0=\nu can have at most one solution \mu\in \mathcal{L}_T(V). The key role in justification is played by the estimate from Theorem 4.7. Various sufficient conditions for uniqueness are presented in [90]. An important problem is the study of the behaviour of solutions \mu=\mu_t\,dt to the parabolic equation as t\to\infty and, in particular, obtaining sufficient conditions for convergence to the stationary solution. A considerable number of papers are devoted to this question, among which we single out [36], [44], [45], [67], [82], [86], and [107]. For example, in [36] solutions \mu_t\in \mathcal{P}_2(\mathbb{R}^n) to the nonlinear equation
\begin{equation*}
\partial_t \mu_t=\Delta \mu_t+\nabla \cdot (\mu_t(\nabla V+\nabla W*\mu_t))
\end{equation*}
\notag
were considered that are gradient flows for the energy functional
\begin{equation*}
F(\mu)=\int_{\mathbb{R}^n} \mu\log \mu\, dx+ \int_{\mathbb{R}^n} V\, d\mu+ \frac{1}{2} \int\!\!\!\int_{\mathbb{R}^{2n}} W(x-y)\, \mu(dx)\, \mu(dy),
\end{equation*}
\notag
and conditions for their convergence to the stationary solution \mu_\infty, along with the estimate
\begin{equation*}
W_2(\mu_t,\mu_\infty)\leqslant e^{-ct}W_2(\mu_0,\mu_\infty),
\end{equation*}
\notag
were obtained. Convergence to the stationary solution in the total variation distance in place of the Kantorovich metric was investigated in [38], [28], [29], and [12]. Let us present a result from [28]. Let A(x)=(a^{ij}(x))_{i,j\leqslant d} be a symmetric positive definite matrix such that there exist positive numbers \theta and \lambda for which
\begin{equation*}
\theta^{-1}\cdot {\rm I}\leqslant A(x)\leqslant \theta\cdot {\rm I},\qquad \|A(x)-A(y)\|\leqslant \lambda|x-y|.
\end{equation*}
\notag
Let V\geqslant 1. Recall that \mathcal{P}_V(\mathbb{R}^d) is the set of probability measures \mu on \mathbb{R}^d for which V\in L^1(\mu), and \mathcal{P}_{\alpha, V}(\mathbb{R}^d) is the subset of \mathcal{P}_V(\mathbb{R}^d) consisting of measures \mu for which \|V\|_{L^1(\mu)}\leqslant\alpha. Suppose that for every \varepsilon\in[0, 1] and every measure \mu\in\mathcal{P}_V(\mathbb{R}^d) we are given a Borel vector field b_{\varepsilon}(\,\cdot\,,\mu) on \mathbb{R}^d. Set
\begin{equation*}
L_{\mu,\varepsilon}\varphi(x)= \operatorname{trace}\bigl(A(x)D^2\varphi(x)\bigr)+ \langle b_{\varepsilon}(x,\mu),\nabla\varphi(x)\rangle.
\end{equation*}
\notag
Let \gamma\in(0,1/2] and W(x)=V(x)^{\gamma}. Theorem 6.11. Suppose that the following conditions are fulfilled: (i) for every \alpha>0 there exist positive numbers N_1(\alpha) and N_2(\alpha) such that for every \varepsilon\in[0, 1], all \mu\in\mathcal{P}_{\alpha,V}(\mathbb{R}^d), and all x\in\mathbb{R}^d one has
\begin{equation*}
|b_{\varepsilon}(x,\mu)| \leqslant N_1(\alpha)V(x)^{1/2-\gamma}
\end{equation*}
\notag
and
\begin{equation*}
|b_{\varepsilon}(x,\mu)-b_{\varepsilon}(x, \sigma)| \leqslant \varepsilon N_2(\alpha)V(x)^{1/2-\gamma}\|W(\mu-\sigma)\|;
\end{equation*}
\notag
(ii) there are positive numbers C_1 and C_2 such that
\begin{equation*}
L_{\mu, \varepsilon}V(x)\leqslant C_1-C_2V(x) \quad \forall\, \varepsilon\in[0, 1], \ x\in\mathbb{R}^d, \ \mu\in\mathcal{P}_V(\mathbb{R}^d).
\end{equation*}
\notag
Then one can find \varepsilon_0\in (0, 1] such that for every \varepsilon\in[0, \varepsilon_0] there exists a solution \mu^{\varepsilon}\in\mathcal{P}_V(\mathbb{R}^d) to the stationary equation L^{*}_{\mu,\varepsilon}\mu=0, and for every probability measure \nu\in\mathcal{P}_V(\mathbb{R}^d) on \mathbb{R}^d\times[0,+\infty) there exists a unique solution \mu_t^{\varepsilon}\,dt with \mu_t^{\varepsilon}\in\mathcal{P}_V(\mathbb{R}^d) to the Cauchy problem \partial_t\mu_t=L_{\mu_t,\varepsilon}^{*}\mu_t, \mu_0=\nu, moreover,
\begin{equation*}
\|W(\mu_t^{\varepsilon}-\mu)\|\leqslant q_1e^{-q_2 t} \quad \forall\,t>0,
\end{equation*}
\notag
where q_1 and q_2 are positive numbers and q_2 does not depend on \nu. Example 6.12. Let A={\rm I} and
\begin{equation*}
b_{\varepsilon}(x,\mu)= \beta(x)+\varepsilon\int_{\mathbb{R}^d}K(x,y)\,\mu(dy),
\end{equation*}
\notag
where the Borel functions \beta and K satisfy the following conditions:
\begin{equation*}
\begin{alignedat}{2} \langle \beta(x),x\rangle&\leqslant c_1-c_2|x|^2, &\quad \langle K(x, y),x\rangle&\leqslant c_3+c_4|x|^2, \qquad c_2>c_4, \\ |\beta(x)|&\leqslant c_5+c_5|x|^m, &\quad\text{and}\quad |K(x,y)|&\leqslant c_6(1+|x|^m)(1+|y|^m), \end{alignedat}
\end{equation*}
\notag
where c_1, c_2, c_3, c_4, c_5, and c_6 are positive numbers and m\geqslant 1. The hypotheses of the theorem are fulfilled for the functions A similar result on convergence to the stationary solution in the case where the diffusion matrix depends on the solution was obtained in [29]. The existence of a global solution to a parabolic equation and its convergence to the stationary solution in the case where the equation contains nonlinear terms of local and non-local type were studied in [33]. In [70]–[72], [74], and [116], nonlinear Fokker–Plank–Kolmogorov equations on infinite-dimensional spaces were considered. Let E=C([-1,0],\mathbb{R}^d) be the space of continuous mappings with the standard \sup-norm, and let the functions b^i and a^{ij} on E\times \mathcal{P}(\mathbb{R}^d) be Borel measurable. For a fixed measure \mu\in\mathcal{P}(\mathbb{R}^d) the operator L_{\mu} on C_0^\infty(\mathbb{R}^d) with values in the space of Borel functions on E is defined by the formula
\begin{equation*}
L_{\mu}f(v)=\sum_{i,j\leqslant d}a^{ij}(v,\mu)\partial_{x_i}\partial_{x_j}f(v(0))+ \sum_{i\leqslant d} b^{i}(v,\mu)\partial_{x_i} f(v(0)).
\end{equation*}
\notag
Then a Borel mapping t\mapsto \mu_t\in \mathcal{P}(E) with values in the space of probability measures on E is called a solution to the nonlinear Cauchy problem \partial_{t}\mu(t)=L_{\mu_t}^*\mu_t with initial condition \mu_0, where \mu(t) is the image of the measure \mu_t under the projection v\mapsto v(0) of E onto \mathbb{R}^d, if for all f\in C_0^\infty(\mathbb{R}^d) the function L_{\mu_t}f is integrable against the measure \mu_t\, dt on the sets E\times [0,T] and
\begin{equation*}
\int_{\mathbb{R}^d} f(x)\, \mu(t)(dx)-\int_{\mathbb{R}^d}f(x)\, \mu(0)(dx)= \int_0^t\int_E L_{\mu_s}f(x,s)\, \mu_s(dx)\, ds.
\end{equation*}
\notag
In the papers cited above one can find conditions for the solvability of such equations and some properties of solutions. In [113] the author studied the differentiability of nonlinear functionals on the space of measures with respect to a solution to a parabolic Fokker–Planck–Kolmogorov equation. For the selection of solutions with the flow property in the case of the non-unique solvability of a nonlinear equation, see [103]. We do not discuss equations on domains. In this connection see, for example, [102], where the authors considered the equation with nonlinearity in the right-hand side, of the form where b is a vector field independent of time and the solution and h is a function of the solution u(\,\cdot\,,\cdot).In [101] a stochastic approximation of the nonlinear equation
\begin{equation*}
\partial_t u(t,x)=\Delta u(t,x)-\nabla \cdot (u(t,x) K * u(t,x))
\end{equation*}
\notag
with a singular convolution kernel K was considered. The stochastic approximation of the Navier–Stokes–Vlasov–Fokker–Planck system was studied in [61]. A linearization of nonlinear parabolic Fokker–Planck–Kolmogorov equations was discussed in [105] and [104]. The authors thank the referee for useful comments. 
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\Bibitem{BogSha24} |
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