Damping perturbations of dynamic systems and convergence conditions for recursive stochastic procedures

Let dynamic system $\dot x_t=b(x_t)$ in $R^d$ has stable equilibrium state at the point 0. Random perturbations of this system are considered as $dX_t=b(X_t)\,dt+d\zeta(t,X_t)$, where $\zeta(t,x)$ for any $x$ is the process with independent increments which damps when $t\to\infty$. Following [9] we show that $X_t$-paths leave an arbitrary domain $D_0$ containing point 0 during time $T$ after moment $t_0$ with probability the main term of which for $t_0\to\infty$ has the form
$$\exp\{-g_T(t_0)V_T(D_0)\},\quad g_t(t_0)\to\infty,\quad V_T(D_0)>0.$$
In many cases this probability may be estimated from above and from below by $\exp\{-g(t_0)(V(D_0)\pm h)\}$ with arbitrary small $h>0$. In such a case either $X_t$-paths leave the domain $D_0$ with probability 1 after any moment $t_0$ or stay in $D_0$ with probability which tends to 1 when $t_0\to\infty$. These two possibilities depend on the divergence or convergence of the integral
$$\int_0^{\infty}\exp\{-g(t_0)V(D_0)\}\,dt_0.$$

The results are applied to the investigation of convergence conditions for some stochastic recursive procedures. In a number of cases for Robbins–Monro and Kiefer–Wolfowitz procedures the necessary and sufficient conditions are obtained.