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Law of the Iterated Logarithm for Markov Semigroups with Exponential Mixing in the Wasserstein Distance

Dawid Czapla, Sander C. Hille, Katarzyna Horbacz, Hanna Wojewódka-Ściążko

Abstract

In this paper, we establish the law of the iterated logarithm for a wide class of non-stationary, continuous-time Markov processes evolving on Polish spaces. Specifically, our result applies to certain additive functionals of processes governed by stochastically continuous Markov-Feller semigroups that exhibit exponential mixing and non-expansiveness in the Wasserstein distance, provided that a suitable moment condition involving the initial distribution is satisfied. Furthermore, we outline the application of this result to a Markov process arising as the solution of an infinite-dimensional stochastic differential equation with dissipative drift and additive noise.

Law of the Iterated Logarithm for Markov Semigroups with Exponential Mixing in the Wasserstein Distance

Abstract

In this paper, we establish the law of the iterated logarithm for a wide class of non-stationary, continuous-time Markov processes evolving on Polish spaces. Specifically, our result applies to certain additive functionals of processes governed by stochastically continuous Markov-Feller semigroups that exhibit exponential mixing and non-expansiveness in the Wasserstein distance, provided that a suitable moment condition involving the initial distribution is satisfied. Furthermore, we outline the application of this result to a Markov process arising as the solution of an infinite-dimensional stochastic differential equation with dissipative drift and additive noise.

Paper Structure

This paper contains 10 sections, 20 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Assumptions and Formulation of the Main Result
  4. Proof of the Main Result
  5. Discretization and a Martingale-Based Decomposition
  6. Proof of the LIL for the Associated Martingale
  7. Summary
  8. Proof of Lemma \ref{['lem:5']}
  9. On representations of $\sigma_g^2$
  10. An Application to an SDE with Dissipative Drift and Additive Noise

Key Result

Proposition 3.1

Suppose that conditions cnd:a2-cnd:a4 hold. Then $\{P_t\}_{t\geq 0}$ possesses a unique invariant probability measure $\mu_*$, which belongs to $\mathcal{M}_{1,\zeta}(X)$. Moreover, $\{P_t\}_{t\geq 0}$ is then asymptotically stable, i.e., $\nu P_t\stackrel{w}{\to}\mu_*$ for every $\nu\in\mathcal{M}_

Theorems & Definitions (51)

  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Example 3.1
  • Proposition 3.1
  • proof
  • Remark 3.5
  • Theorem 3.1
  • Lemma 4.1
  • ...and 41 more