Evolution equation of a stochastic semigroup with white-noise drift

Abstract

We study the existence and uniqueness of the solution of a function-valued stochastic evolution equation based on a stochastic semigroup whose kernel $p(s,t,y,x)$ is Brownian in $s$ and $t$.The kernel $p$ is supposed to be measurable with respect to the increments of an underlying Wiener process in the interval $[s, t]$. The evolution equation is then anticipative and, choosing the Skorohod formulation,we establish existence and uniqueness of a continuous solution with values in $L^2(\mathbb{R}^d)$.

As an application we prove the existence of a mild solution of the stochastic parabolic equation

du_t = \Delta_x u dt + v(dt, x) \cdot \nabla u + F(t, x, u) W(dt, x),

where $v$ and $W$ are Brownian in time with respect to a common filtration. In this case, p is the formal backward heat kernel of $\Delta_x + v(dt, x) \cdot \nabla_x$ .