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
$\Delta_x + v(dt, x) \cdot \nabla_x$ .