SPD

Abstract

We are here concerned with a Fokker--Planck equation in a separable Hilbert space $H$ of the form \begin{equation} \label{e1} \int_{0}^T\int_H \mathcal K_0^F\,u(t,x)\,\mu_t(dx)dt=-\int_H u(0,x)\,\zeta(dx),\quad\forall\;u\in\mathcal E \end{equation} The unknown is a probability kernel $(\mu_t)_{t\in [0,T]}$. Here $K_0^F$ is the Kolmogorov operator $$ K_0^Fu(t,x)=D_tu(t,x)+\frac12\mbox{Tr}\;[BB^*D^2_xu(t,x)]+\langle Ax+F(t,x),D_xu(t,x)\rangle $$ where $A:D(A)\subset H\to H$ is self-adjoint, $F:[0,T]\times D(F)\to H$ is nonlinear and $\mathcal E$ is a space of suitable test functions. $K_0^F$ is related to the stochastic PDE \begin{equation} \label{e2} dX=(AX+F(t,X))dt+BdW(t) X(0)=x. \end{equation} We present some existence and uniqueness results for equation (1) both when problem (2) is well posed and when it is not.