Analysis and Applied Mathematics Seminar
Le Chen
Auburn University
Invariant measures for the nonlinear stochastic heat equation on $\mathbb{R}^d$ with no drift term
Abstract: In this talk, we will present a recent joint work with
Dr. Nicholas Eisenberg (arXiv:2209.04771). This paper deals with the long term
behavior of the solution to the nonlinear stochastic heat equation $\partial u
/\partial t - \frac{1}{2}\Delta u = b(u)\dot{W}$, where $b$ is assumed to be a
globally Lipschitz continuous function and the noise $\dot{W}$ is a centered and
spatially homogeneous Gaussian noise that is white in time. Using the moment
formulas obtained in Chen & Kim [10] and Chen & Huang [9], we
identify a set of conditions on the initial data, the correlation measure and
the weight function $\rho$, which will together guarantee the existence of an
invariant measure in the weighted space $L^2_\rho(\mathbb{R}^d)$. In particular, our
result includes the parabolic Anderson model (i.e., the case when $b(u)
= \lambda u$) starting from the Dirac delta measure.
Monday October 3, 2022 at 4:00 PM in 1227 SEO