12/03/2025, 16:00 — 17:00 — Online
Le Chen, Auburn University
Long-term behavior of the nonlinear stochastic heat equation on $\R^d$ without a drift term
In this talk, we will present recent studies on 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, spatially homogeneous Gaussian noise that is white in time. We identify a set of conditions on the initial data, the correlation measure, and the weight function $\rho$, which together guarantee the existence of an invariant measure or a limiting random field in the weighted space $L^2_\rho(\R^d)$. In particular, our results include the parabolic Anderson model (i.e., the case when $b(u) = \lambda u$) starting from unbounded initial data such as the Dirac delta measure. This study has implications for the study of continuous random polymers. This talk is based on joint work with Nicholas Eisenberg and another collaboration with Ouyang Cheng, Samy Tindel, and Panqiu Xia.
