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arXiv · 2402.02907

Additive-multiplicative stochastic heat equations, stationary solutions, and Cauchy statistics

Alexander Dunlap·Chiranjib Mukherjee
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Abstract

We study long-term behavior and stationary distributions for stochastic heat equations forced simultaneously by a multiplicative noise and an independent additive noise with the same distribution. We prove that nontrivial space-time translation-invariant measures exist for all values of the parameters. We also show that if the multiplicative noise is sufficiently strong, the invariant measure has Cauchy-distributed marginals. Using the same techniques, we prove a similar result on Cauchy-distributed marginals for a logarithmically attenuated version of the problem in two spatial dimensions. The proofs rely on stochastic analysis and elementary potential theory.

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Alexander Dunlap, Chiranjib Mukherjee. 2025-04-23. Additive-multiplicative stochastic heat equations, stationary solutions, and Cauchy statistics. https://arxiv.org/abs/2402.02907

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