On boundary conditions for stochastic evolution equations with an extremally chaotic source

Stochastic differential equation of the form
$$d\xi_t=A\xi_t\,dt+Bd\eta_t^0, \qquad t\in I=(t_0,t_1),$$
are considered for a generalized random field
$$\xi_t\equiv(\varphi,\xi_t), \quad \varphi\in C_0^\infty(G),$$
in the domain $G\subseteq\mathbb R^d$ with stochastic boundary conditions on the boundary corresponding to an operator $A\leqslant0$ and an extremal operator coefficient $B$ (strengthening the chaotic source $d\eta^0_t$ of ‘white noise’ type).