| Contents |
| In this talk we discuss the stochastic heat equation on $\mathbb{R}^d$
\[
\frac {\partial u }{\partial t} =\frac 12 \Delta
u + u \dot{W},
\]
where $\dot W$ is a mean zero Gaussian noise with covariance $
E\left[ \dot{W}_{t,x} \dot{W}_{s,y}\right] =\gamma(s-t) \, \Lambda(x-y)$,
and $\gamma$ and $\Lambda$ are general nonnegative and nonnegative
definite (generalized) functions satisfying some integrability
conditions. The product
$u \dot{W}$ can be interpreted in both the Skorohod and
Stratonovich sense. We will present recent results on the existence and uniqueness of a solution and its H\"older continuity.
Moreover we will establish Feynman-Kac formulas for the solution and for its
moments, which allow us to derive moment estimates and intermittency properties. |
|