Evolution equation of a stochastic semigroup with white-noise drift

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 measu rable 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 conti nuous solution with values in L-2(R-d). As an application we prove the exis tence of a mild solution of the stochastic parabolic equation du(t) = Delta(x)udt + v(dt, x) . del 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) . de l(x).