Invariant measures for the nonlinear stochastic heat equation with no drift term

2022/09/11 by Le Chen, Chen, Le, Nicholas Eisenberg +1
Economics, Econometrics and Finance · Mathematics · #60F05 #60H07 #60H15 #FOS: Mathematics #Mathematical Biology Tumor Growth #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2209.04771

openalex publication_date 2022/09/11 · openalex created_date 2022/09/14 · openalex updated_date 2026/07/28

Abstract

This paper deals with the long term behavior of the solution to the nonlinear stochastic heat equation ∂ u /∂ t - (1)/(2)Δu = b(u)W, where b is assumed to be a globally Lipschitz continuous function and the noise W is a centered and spatially homogeneous Gaussian noise that is white in time. Using the moment formulas obtained in [9, 10], we identify a set of conditions on the initial data, the correlation measure and the weight function ρ, which will together guarantee the existence of an invariant measure in the weighted space L²_ρ(ℝ^d). In particular, our result includes the parabolic Anderson model (i.e., the case when b(u) = λu) starting from the Dirac delta measure.

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