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

Alexander Dunlap, Chiranjib Mukherjee

TL;DR

This work analyzes an additive-multiplicative stochastic heat equation driven by independent Gaussian noises with a shared spatial covariance. Using a backward-in-time Duhamel framework and martingale techniques, it proves the existence of space-time stationary solutions for all α≥0, β>0, obtained as long-time limits from zero initial data. In the strong-noise regime, linear combinations of point values of the stationary solution have exact Cauchy marginals with scale α/β, while in the weak regime marginals possess finite moments; a 2D logarithmically attenuated variant yields the same Cauchy-type limits under a critical scaling. The results reveal a robust mechanism for non-Gaussian invariant measures in stochastic PDEs through a martingale-exit structure, with implications for nonlinear SPDEs and higher-dimensional behavior.

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.

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

TL;DR

This work analyzes an additive-multiplicative stochastic heat equation driven by independent Gaussian noises with a shared spatial covariance. Using a backward-in-time Duhamel framework and martingale techniques, it proves the existence of space-time stationary solutions for all α≥0, β>0, obtained as long-time limits from zero initial data. In the strong-noise regime, linear combinations of point values of the stationary solution have exact Cauchy marginals with scale α/β, while in the weak regime marginals possess finite moments; a 2D logarithmically attenuated variant yields the same Cauchy-type limits under a critical scaling. The results reveal a robust mechanism for non-Gaussian invariant measures in stochastic PDEs through a martingale-exit structure, with implications for nonlinear SPDEs and higher-dimensional behavior.

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.
Paper Structure (9 sections, 14 theorems, 84 equations)

This paper contains 9 sections, 14 theorems, 84 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Main results
  4. Source of the Cauchy distribution
  5. The multiplicative problem and the Duhamel formula
  6. Convergence of marginal distributions
  7. Cauchy-distributed limits
  8. Construction of space-time stationary solutions
  9. The phase transition

Key Result

Theorem 1.2

For any $\alpha\ge0$ and any $\beta>0$, there is a space-time stationary solution $(\tilde{v}_{t}^{{{({\alpha,\beta}})}})_t$ to eq:theSPDE, with $\tilde{v}_{t}^{{{({\alpha,\beta}})}}\in\bigcap_{p\in [1,\infty),\xi>0} L^p_{w_{\mathcal{X};\xi}}(\mathcal{X})$ a.s. for each $t$, such that for each $t\in

Theorems & Definitions (31)

  • Theorem 1.2
  • Proposition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Proposition 1.9
  • proof : Proof of \ref{['prop:howtogetCauchy']}
  • Remark 1.10
  • ...and 21 more