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Asymptotic Log-Harnack Inequality for Degenerate SPDEs with Reflection

Qi Li, Feng-Yu Wang, Tusheng Zhang

Abstract

By constructing a suitable coupling by change of measures, the asymptotic log- Harnack inequality is established for a class of degenerate SPDEs with reflection. This inequality implies the asymptotic heat kernel estimate, the uniqueness of the invariant probability measure, the asymptotic gradient estimate (hence, asymptotically strong Feller property), and the asymptotic irreducibility. As application, the main result is illustrated by d-dimensional degenerate stochastic Navie-Stokes equations with reflection, where the dissipative operator is the Dirichlet Laplacian with a power θ\geq 1 \vee \frac{d+2}{4}, which includes the Laplacian when d \geq 2.

Asymptotic Log-Harnack Inequality for Degenerate SPDEs with Reflection

Abstract

By constructing a suitable coupling by change of measures, the asymptotic log- Harnack inequality is established for a class of degenerate SPDEs with reflection. This inequality implies the asymptotic heat kernel estimate, the uniqueness of the invariant probability measure, the asymptotic gradient estimate (hence, asymptotically strong Feller property), and the asymptotic irreducibility. As application, the main result is illustrated by d-dimensional degenerate stochastic Navie-Stokes equations with reflection, where the dissipative operator is the Dirichlet Laplacian with a power θ\geq 1 \vee \frac{d+2}{4}, which includes the Laplacian when d \geq 2.
Paper Structure (7 sections, 6 theorems, 79 equations)

This paper contains 7 sections, 6 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Well-posedness and moment estimates
  3. Asymptotic log-Harnack and applications
  4. General results
  5. Main result of the paper
  6. Proof of Theorem \ref{['harnack']}
  7. Application to reflected stochastic Navier-Stokes equations

Key Result

Proposition 2.1

Under assumption (A), for any $x\in \mathbb H$ the equation $(SEE)$ has a unique solution $(X_t^x,L_t^x)$, and

Theorems & Definitions (13)

  • Definition 2.1
  • Proposition 2.1
  • Definition 3.1
  • Definition 3.2: HM
  • Theorem 3.1
  • Theorem 3.2
  • proof : Proof of Theorem \ref{['harnack']}
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • ...and 3 more