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Euler-type approximation for the invariant measure: An abstract framework

Aurélien Alfonsi, Vlad Bally, Arturo Kohatsu-Higa

Abstract

We establish a general framework to study the rate of convergence of a Euler type approximation scheme with decreasing time steps to the invariant measure, for a general class of stochastic systems. The error is measured in general Wasserstein distances, which enables to encompass cases with non global contractivity conditions. Our main assumption is a coupling property which is expressed in terms of the one-step approximation. We show that the proposed set-up can be applied to a wide range of equations that may be law dependent, such as Langevin equations, reflected equations, Boltzmann type equations and for a recent McKean Vlasov type model for neuronal activity.

Euler-type approximation for the invariant measure: An abstract framework

Abstract

We establish a general framework to study the rate of convergence of a Euler type approximation scheme with decreasing time steps to the invariant measure, for a general class of stochastic systems. The error is measured in general Wasserstein distances, which enables to encompass cases with non global contractivity conditions. Our main assumption is a coupling property which is expressed in terms of the one-step approximation. We show that the proposed set-up can be applied to a wide range of equations that may be law dependent, such as Langevin equations, reflected equations, Boltzmann type equations and for a recent McKean Vlasov type model for neuronal activity.

Paper Structure

This paper contains 18 sections, 29 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Framework and main results
  3. Coupling on a time grid
  4. Invariant measure
  5. Probabilistic representation
  6. Some useful tools
  7. Using the sewing lemma to prove coupling
  8. A Foster-Lyapunov type criterion to prove $p$-boundedness
  9. Examples with non global contraction conditions (Langevin type equations)
  10. Examples with global contraction conditions
  11. A stochastic equation with reflection in $W_2$
  12. A neuronal model in $W_1$
  13. Boltzmann type equations
  14. Proof of Lemma \ref{['lem_tech']}
  15. Rates of convergence for the Ornstein-Uhlenbeck process
  16. ...and 3 more sections

Key Result

Theorem 2.4

Let $(\theta_{s,t})_{0\le s\le t}$ be a family of operators on $\mathcal{P}_p(B)$ with the flow property ($\theta_{s,t}\circ \theta_{r,s}= \theta_{r,t}$ for $r\le s\le t$). We assume that $\theta$ is time homogeneous (i.e. $\theta_{s,t}=\theta_{0,t-s}$) and such that for all $s\le t$, $\mu \mapsto \ Let $(\Theta_{s,t})_{0\le s\le t}$ be a family of operators on $\mathcal{P}_p(B)$ such that $\theta

Theorems & Definitions (68)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • Remark 2.8
  • Corollary 2.9
  • ...and 58 more