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 L2(Rd).

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 Δx+v(dt,x)⋅∇x .