### Jingyu Huang (University of Kansas)

Title: Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency

Abstract:
We study the stochastic heat equation with
multiplicative noises: $\frac {\partial u }{\partial t} =\frac 12 \Delta
u + u \dot{W}$, where $\dot W$ is a mean zero Gaussian noise and
$u \dot{W}$ is interpreted both in the sense of Skorohod and
Stratonovich. The existence and uniqueness of the solution are
studied for noises with general time and spatial covariance
structure. Feynman-Kac formulas for the solutions and for the
moments of the solutions are obtained under general and different
conditions. These formulas are applied to obtain the Hölder
continuity of the solutions. They are also applied to obtain
the intermittency bounds for the moments of the solutions.