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Asymptotic Log-Harnack Inequality for Monotone SPDE with Multiplicative Noise

Zhihui Liu

Abstract

We derive an asymptotic log-Harnack inequality for nonlinear monotone SPDE driven by possibly degenerate multiplicative noise. Our main tool is the asymptotic coupling by the change of measure. As an application, we show that, under certain monotone and coercive conditions on the coefficients, the corresponding Markov semigroup is asymptotically strong Feller, asymptotic irreducibility, and possesses a unique and thus ergodic invariant measure. The results are applied to highly degenerate finite-dimensional or infinite-dimensional diffusion processes.

Asymptotic Log-Harnack Inequality for Monotone SPDE with Multiplicative Noise

Abstract

We derive an asymptotic log-Harnack inequality for nonlinear monotone SPDE driven by possibly degenerate multiplicative noise. Our main tool is the asymptotic coupling by the change of measure. As an application, we show that, under certain monotone and coercive conditions on the coefficients, the corresponding Markov semigroup is asymptotically strong Feller, asymptotic irreducibility, and possesses a unique and thus ergodic invariant measure. The results are applied to highly degenerate finite-dimensional or infinite-dimensional diffusion processes.

Paper Structure

This paper contains 10 sections, 6 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotic Log-Harnack Inequality
  4. Asymptotic Log-Harnack Inequality
  5. Applications to Degenerate Diffusions
  6. Examples
  7. Asymptotic Log-Harnack Inequality for Degenerate SODEs
  8. Asymptotic Log-Harnack Inequality for SPDEs
  9. Nondegenerate case
  10. Degenerate case

Key Result

Lemma 2.1

Let $T>0$ and Assumption ap hold. For any $\mathscr F_0$-measurable $z \in L^2(\Omega, H)$, Eq. eq-z with initial datum $Z_0=z$ exists a unique solution $\{Z_t:\ t\in [0,T]\}$ in $L^2(\Omega; \mathcal{C}([0,T]; H)) \cap L^\alpha(\Omega \times (0, T); V)$ which is a Markov process such that holds a.s. for all $v \in V$ and $t \in [0, T]$.

Theorems & Definitions (15)

  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Corollary 3.1
  • proof
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.1
  • ...and 5 more