32. chen.eisenberg:22:invariant

32. chen.eisenberg:22:invariant#

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

Le Chen and Nicholas Eisenberg

  • To appeare in Journal of Theoretical Proabibility

Abstract: This paper deals with the long term behavior of the solution to the nonlinear stochastic heat equation

\[\frac{\partial u(t,x)}{\partial t} - \frac{1}{2}\Delta u(t,x) = b\left(u(t,x)\right)\dot{W}, \quad t>0, x\in\mathbb{R}^d,\]

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 textit{parabolic Anderson model} (i.e., the case when \(b(u) = \lambda u\)) starting from the Dirac delta measure.

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[CE22b] Le Chen & Nicholas Eisenberg (2022) ‘Invariant measures for the nonlinear stochastic heat equation with no drift term’, J. Theoret. Probab. (pending revision, preprint arXiv:2209.04771)

MSC 2010 subject classifications: Primary 60H15. Secondary 60H07, 60F05.

keyworkds: stochastic heat equation, parabolic Anderson model, invariant measure, Dirac delta initial condition, weighted \(L^2\) space, Matérn class of correlation functions, Bessel kernel

@article{chen.eisenberg:22:invariant,
  title         = {Invariant measures for the nonlinear stochastic heat equation with no drift term},
  author        = {Le Chen and Nicholas Eisenberg},
  year          = {2022},
  month         = {September},
  journal       = {J. Theoret. Probab. Feb. 2024},
  url           = {http://arXiv.org/abs/2209.04771}
}

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