CS29: Computing the invariant distribution of linear and non linear diffusions by ergodic simulation

date: 7/17/2025, time: 14:00-15:30, room: ICS 118

Organizer: Gilles Pagès (Sorbonne Université)

Chair: Gilles Pagès (Sorbonne Université)

Euler Approximation for the Invariant Measure

Vlad Bally (Université Gustave Eiffel)

We consider a flow of transformations on the Wasserstein space and we ask for sufficient conditions in order that an invariant probability measure exists and is unique. Moreover we are interested in the approximation using Euler Scheme with decreasing step - in the spirit of the approach initiated by D. Lamberton and G. Pages) - for the invariant measure. Our setting covers a large class of non linear stochastic equations depending on the law of the solution (Mc Kean Vlasov or Bolzmann type equations).

Approximation of the invariant distribution for a class of ergodic jump diffusions

Dasha Loukianova (Evry-Paris-Saclay University)

In this talk, we approximate the invariant distribution \(\nu\) of an ergodic Jump Diffusion driven by the sum of a Brownian motion and a Compound Poisson process with sub-Gaussian jumps. We first construct an Euler discretization scheme with decreasing time steps. This scheme is similar to those introduced by Lamberton and Pagès in \([1]\) for a Brownian diffusion and extended by Panloup in \([2]\) to a diffusion with Lévy jumps. We obtain a non-asymptotic qiasi-Gaussian concentration bound for the difference between the invariant distribution and the empirical distribution computed with the scheme of decreasing time step along appropriate test functions \(f\) such that \(f-\nu(f)\) is is a coboundary of the infinitesimal generator.

Bibliography

\([1]\) Damien Lamberton and Gilles Pagés. "Recursive computation of the invariant distribution of a diffusion." Bernoulli, vol. 8, no. 3, 2002, pp. 367–405.

\([2]\) Fabien Panloup. "Recursive computation of the invariant measure of a stochastic differential equation driven by a Lévy process." Ann. Appl. Probab., vol. 18, no. 2, 2008, pp. 379–426.

Computing the invariant distribution of McKean-Vlasov SDEs by ergodic simulation with rates in Wasserstein distance

Gilles Pagès (Sorbonne Université)

We design a fully implementable scheme to compute the invariant distribution of an ergodic McKean-Vlasov SDE satisfying a uniform confluence property, see \([3]\). Under natural conditions, we prove various convergence results notably we obtain rates for the Wasserstein distance in quadratic mean and almost sure sense of the empirical measure of this scheme toward the (unique) invariant distribution.

These rates depend on their counterparts in the simpler setting where the SDE is a standard Brownian diffusion. We will first revisit in \([2]\) and \([4]\) this problem, inspired by the paper \([1]\) by Fournier & Guillin on convergence rate of the empirical measure of i.i.d. sequences.

Bibliography

\([1]\) Jean-François Chassagneux & Gilles Pagès. Computing the invariant distribution of McKean-Vlasov SDEs by ergodic simulation, arXiv:2503.20411 , 2025

\([2]\) Jean-François Chassagneux & Gilles Pagès. A note on the \({\cal W}_2\)-convergence rate of the empirical measure of an ergodic \(R^d\)-valued diffusion, arXiv:2502.07704, 2025.

\([3]\) Nicolas Fournier F. & Arnaud Gullin . On the rate of convergence in Wasserstein distance of the empirical measure. Probab. Theory Relat. Fields vol. 162, no. 3-4, 2015, pp. 707-738.

\([4]\) Gilles Pagès & Fabien Panloup. Mean convergence rate in Wasserstein distance of the empirical measure of a Markov process toward its invariant distribution, in progress, 2025.