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Exponential ergodicity and finite-dimensional approximation for Markovian lifts of stochastic Volterra equations

Yushi Hamaguchi

Abstract

This paper investigates the long-time asymptotics and the existence of stationary solutions for a class of stochastic Volterra equations (SVEs). To address the non-Markovian nature of SVEs, we employ a Markovian lifting technique, formulating a Markovian lift as the solution to a stochastic evolution equation (SEE) on a Gelfand triplet. Our main objective is to establish the ergodicity of this Markovian lift via the generalized Harris' theorem, which in turn yields the asymptotic results for the original SVE. Despite the challenges posed by the highly degenerate, infinite-dimensional nature of the SEE, we achieve this by constructing a generalized coupling and a distance function that exploit the structural properties arising from the non-local operators in its coefficients. Furthermore, we prove that the invariant probability measure and, more generally, the stationary law on the path space of the SEE can be weakly approximated by those of finite-dimensional SDEs. This yields a novel approximation result for the stationary solution of the original SVE, while offering a rigorous mathematical framework that supports the validity of the Markovian embedding concept widely utilized in statistical physics.

Exponential ergodicity and finite-dimensional approximation for Markovian lifts of stochastic Volterra equations

Abstract

This paper investigates the long-time asymptotics and the existence of stationary solutions for a class of stochastic Volterra equations (SVEs). To address the non-Markovian nature of SVEs, we employ a Markovian lifting technique, formulating a Markovian lift as the solution to a stochastic evolution equation (SEE) on a Gelfand triplet. Our main objective is to establish the ergodicity of this Markovian lift via the generalized Harris' theorem, which in turn yields the asymptotic results for the original SVE. Despite the challenges posed by the highly degenerate, infinite-dimensional nature of the SEE, we achieve this by constructing a generalized coupling and a distance function that exploit the structural properties arising from the non-local operators in its coefficients. Furthermore, we prove that the invariant probability measure and, more generally, the stationary law on the path space of the SEE can be weakly approximated by those of finite-dimensional SDEs. This yields a novel approximation result for the stationary solution of the original SVE, while offering a rigorous mathematical framework that supports the validity of the Markovian embedding concept widely utilized in statistical physics.
Paper Structure (10 sections, 18 theorems, 240 equations)

This paper contains 10 sections, 18 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized Harris' theorem
  4. Markovian lifts of SVEs
  5. Main results
  6. Proofs of the main results
  7. Proof of \ref{['theo_coupling']}: Contractivity, $d$-smallness, and the asymptotic log-Harnack inequality
  8. Proofs of \ref{['theo_Lyapunov']} and \ref{['prop_Lyapunov-sufficient']}: Construction of a Lyapunov function
  9. Finite-dimensional approximation of the invariant probability measure
  10. Appendix

Key Result

Theorem 2.1

Let $\{P_t\}_{t\geq0}$ be a measurable Markov semigroup on a Polish space $E$ satisfying the Feller propertyIn HaMaSc11, the measurability of $\{P_t\}_{t\geq0}$ is implicitly assumed.. Assume that $\{P_t\}_{t\geq0}$ admits a continuous Lyapunov function $V:E\to[0,\infty)$. Suppose further that there Then, $\{P_t\}_{t\geq0}$ possesses a unique invariant probability measure $\pi\in\mathcal{P}(E)$. F

Theorems & Definitions (60)

  • Definition 2.1
  • Remark 2.1
  • Theorem 2.1: The generalized Harris' theorem; HaMaSc11
  • Remark 2.2
  • Proposition 2.1
  • Remark 2.3
  • Definition 2.2
  • Remark 2.4
  • Example 2.1
  • Lemma 2.1
  • ...and 50 more