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Invariant Measures for the Nonlinear Stochastic Heat Equation with No Drift Term

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Abstract

This paper deals with the long-term behavior of the solution to the nonlinear stochastic heat equation \(\frac{\partial u}{\partial t} - \frac{1}{2}\Delta u = b(u){\dot{W}}\), 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. We identify a set of nearly optimal conditions on the initial data, the correlation measure of the noise, and the weight function \(\rho \), which together guarantee the existence of an invariant measure in the weighted space \(L^2_\rho ({\mathbb {R}}^d)\). In particular, our result covers the parabolic Anderson model (i.e., the case when \(b(u) = \lambda u\)) starting from the Dirac delta measure.

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Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Notes

  1. In [17], the authors constructed a spectral measure \({\widehat{f}}\) such that \(\Upsilon (0)<\infty \) and \(\Upsilon _\alpha (\beta ) = \infty \) for all \(\alpha \in (0,1)\).

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Acknowledgements

L.C. thanks Sandra Cerrai for some helpful discussions and for pointing out the reference [30] during the conference Frontier Probability Days 2018. L.C. also thanks Yu Gu for some helpful discussions regarding the stationary limit obtained in [12] during the Workshop: Stochastic PDEs & Related Topics, Nov. 14–16, 2022 at Brin Mathematics Research Center, University of Maryland, and for pointing out the reference [13]. Both authors would like to thank the anonymous referee for helpful comments and suggestions.

Funding

Le Chen is partially supported by NSF Grant DMS-2246850 and a collaboration grant from the Simons Foundation (# 959981).

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  1. Department of Mathematics and Statistics, Auburn University, 221 Parker Hall, Auburn, AL, 36849, USA

    Le Chen & Nicholas Eisenberg

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  1. Le Chen
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Chen, L., Eisenberg, N. Invariant Measures for the Nonlinear Stochastic Heat Equation with No Drift Term. J Theor Probab (2024). https://doi.org/10.1007/s10959-023-01302-4

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