Invariant measure of duplicated diffusions and application to Richardson–Romberg extrapolation

Lemaire, Vincent; Pagès, Gilles; Panloup, Fabien

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1562-1596.

Initialement motivés par des problèmes numériques d’analyse en temps long d’une diffusion ergodique, nous abordons ici la question suivante : si une diffusion Brownienne ergodique a une unique probabilité invariante, quelles sont les probabilités invariantes associées à sa dupliquée, i.e. au système formé par deux copies de la diffusion initiale. Nous nous focalisons notamment sur le cas où ces deux copies sont dirigées par le même mouvement Brownien ($2$-point motion). Sous cette hypothèse, nous montrons que l’unicité de la probabilité invariante relative à la diffusion dupliquée (confluence faible) est essentiellement toujours vraie en dimension $1$. En dimension supérieure, après avoir exhibé un contre-exemple, nous proposons une série de critères de confluence faible (de type intégral) mais aussi de confluence trajectorielle presque sûre, lisibles sur les coefficients de la diffusion au travers d’un exposant de Lyapounov non-infinitésimal. Ces critères permettent en particulier de traiter des cas non triviaux comme certaines classes de systèmes-gradients à potentiel (sur-quadratique) non convexe ou, à l’inverse, des systèmes pour lesquels la confluence est induite par le coefficient de diffusion. Nous montrons enfin que la propriété de confluence faible est associée à un problème de transport optimal. Dans un second temps, nous appliquons nos résultats pour optimiser la mise en œuvre de l’extrapolation Richardson–Romberg pour l’approximation par simulation de mesures invariantes.

With a view to numerical applications we address the following question: given an ergodic Brownian diffusion with a unique invariant distribution, what are the invariant distributions of the duplicated system consisting of two trajectories? We mainly focus on the interesting case where the two trajectories are driven by the same Brownian path. Under this assumption, we first show that uniqueness of the invariant distribution (weak confluence) of the duplicated system is essentially always true in the one-dimensional case. In the multidimensional case, we begin by exhibiting explicit counter-examples. Then, we provide a series of weak confluence criterions (of integral type) and also of a.s. pathwise confluence, depending on the drift and diffusion coefficients through a non-infinitesimal Lyapunov exponent. As examples, we apply our criterions to some non-trivially confluent settings such as classes of gradient systems with non-convex potentials or diffusions where the confluence is generated by the diffusive component. We finally establish that the weak confluence property is connected with an optimal transport problem. As a main application, we apply our results to the optimization of the Richardson–Romberg extrapolation for the numerical approximation of the invariant measure of the initial ergodic Brownian diffusion.

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