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Mixing times for the stochastic $p$-Laplace equation

Gerardo Barrera, Jonas M. Tölle

TL;DR

The paper provides a systematic survey of long-time behavior and mixing times for the stochastic $p$-Laplace equation with additive noise ($p>1$). By leveraging coupling methods, Girsanov's theorem, and disintegration arguments, it derives explicit upper and lower bounds for the $\ au_{\textsf{mix}}(\varepsilon;x)$ in Wasserstein distances and organizes results across the regimes determined by $p$ and noise degeneracy, including both deterministic and stochastic settings. It highlights exponential mixing for non-degenerate noise, polynomial or extinction-type decay in the singular and degenerate cases, and presents comprehensive tables of known rates and asymptotics, while pointing out open problems, notably exponential ergodicity in the singular, degenerate-noise regime. The findings clarify how noise structure and dimensionality influence ergodic rates in nonlinear diffusion models and inform analyses of power-law diffusion phenomena in applied contexts.

Abstract

We give an overview on existing quantitative results on long-time behavior of the stochastic $p$-Laplace equation with additive Wiener noise, $p>1$. We summarize the existing results in a table. We give explicit quantitative upper and lower estimates for the $\varepsilon$-mixing times of the stochastic $p$-Laplace equations for $p>1$. We summarize the mixing time asymptotics in a table.

Mixing times for the stochastic $p$-Laplace equation

TL;DR

The paper provides a systematic survey of long-time behavior and mixing times for the stochastic -Laplace equation with additive noise (). By leveraging coupling methods, Girsanov's theorem, and disintegration arguments, it derives explicit upper and lower bounds for the in Wasserstein distances and organizes results across the regimes determined by and noise degeneracy, including both deterministic and stochastic settings. It highlights exponential mixing for non-degenerate noise, polynomial or extinction-type decay in the singular and degenerate cases, and presents comprehensive tables of known rates and asymptotics, while pointing out open problems, notably exponential ergodicity in the singular, degenerate-noise regime. The findings clarify how noise structure and dimensionality influence ergodic rates in nonlinear diffusion models and inform analyses of power-law diffusion phenomena in applied contexts.

Abstract

We give an overview on existing quantitative results on long-time behavior of the stochastic -Laplace equation with additive Wiener noise, . We summarize the existing results in a table. We give explicit quantitative upper and lower estimates for the -mixing times of the stochastic -Laplace equations for . We summarize the mixing time asymptotics in a table.
Paper Structure (13 sections, 1 theorem, 76 equations)

This paper contains 13 sections, 1 theorem, 76 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. The stochastic $p$-Laplace equation
  4. Known results and open problems
  5. Overview on existing results
  6. The deterministic case
  7. The case of degenerate noise
  8. Exponential convergence
  9. Summary of the results
  10. Mixing times asymptotics
  11. Example
  12. Mixing times asymptotics
  13. A disintegration result for the Wasserstein distance

Key Result

Lemma A.1

Let $r\geq 1$ be fixed. Assume that there exists a unique invariant probability measure $\mu_\ast$ on $(H,\mathcal{B}(H))$ with finite absolute moment of order $r$ for the dual semigroup $(P^\ast_t)_{t\ge 0}$ on the space of probability measures on $(H,{\mathcal{B}}(H))$. Then for any $x\in H$ and $ In particular, it follows that

Theorems & Definitions (5)

  • Definition 2.1
  • Definition 2.2
  • Remark 3.1
  • Remark 3.2
  • Lemma A.1: Disintegration