Invariant measures for the nonlinear stochastic heat equation on $\mathbb{R}^d$ with no drift term
数学专题报告
报告题目(Title)：Invariant measures for the nonlinear stochastic heat equation on $\mathbb{R}^d$ with no drift term
报告人(Speaker)：陈乐 (Auburn University)
地点(Place)：后主楼1220
时间(Time)： 2024年6月14日上午10:00-11:00
邀请人(Inviter)：蒲飞
报告摘要
In this talk, we will present some recent study 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 and spatially homogeneous Gaussian noise that is white in time. Using moment formulas obtained in Chen and Kim 2019 and Chen and Huang 2019, 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 unbounded initial data such as the Dirac delta measure and harmonic polynomials. This talk is based on the joint work with Nicholas Eisenberg and an on-going project with Alexandra Dunlap and Panqiu Xia.
