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

TL;DR

This paper establishes exponential ergodicity and stationary law approximation for Markovian lifts of stochastic Volterra equations, providing a rigorous framework for their long-term behavior and finite-dimensional approximation.

Contribution

It introduces a novel approach to prove ergodicity of Markovian lifts of SVEs and demonstrates finite-dimensional approximation of their stationary solutions.

Findings

Proves exponential ergodicity of the Markovian lift.

Shows weak approximation of stationary laws by finite-dimensional SDEs.

Provides a rigorous foundation for Markovian embedding in statistical physics.

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.