Stochastic heat equation with white-noise drift

We study the existence and uniqueness of the solution for a one-dimensional anticipative stochastic evolution equation driven by a two-parameter Wiener process W-t,W-x and based on a stochastic semigroup p (s, t, y, x). The kernel p(s, t, y, x) is supposed to be measurable with respect to the increments of the Wiener process on [s, t] x R. The results are based on L-p-estimates for the Skorohod integral. As a application we deduce the existence of a weak solution for the stochastic partial differential equation partial derivative u/partial derivative t = partial derivative(2)u/partial derivative x(2) + (v) over dot(t, x) partial derivative u/partial derivative x + F(t, x, u) partial derivative(2)W/partial derivative t partial derivative x, where (v) over dot(t, x) is a white-noise in time. (C) 2000 Editions scientifiques et medicales Elsevier SAS.