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Mathematics Colloquium

Le Chen
Auburn University

Title: Invariant measures for the nonlinear stochastic heat equation on $\R^d$ with no drift term
Date: Monday, October 24, 2022
Place and Time: LOV 101, 3:05-3:55 pm


In this talk, we will first give a short introduction to stochastic partial differential equations, and in particular, the stochastic heat equation. Then we will present a recent joint work with \textit{Dr. Nicholas Eisenberg} (arXiv:2209.04771). In this paper, we study 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 sufficient 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 $L^2_\rho(\R^d)$. In particular, our result includes the \textit{parabolic Anderson model} (i.e., the case when $b(u) = \lambda u$) starting from the Dirac delta measure.