Poisson Equations with locally-Lipschitz coefficients and Uniform in Time Averaging for Stochastic Differential Equations via Strong Exponential Stability

Dan Crisan, Paul Dobson, Ben Goddard, Michela Ottobre, Iain Souttar

Research output: Contribution to journalArticlepeer-review

7 Downloads (Pure)

Abstract

We study averaging for Stochastic Differential Equations (SDEs) and Poisson equations. We succeed in obtaining a uniform in time (UiT) averaging result, with a rate, for fully coupled SDE models with super-linearly growing coefficients. This is the main result of this paper and it is, to the best of our knowledge, the first UiT multiscale result with a rate. More precisely, the main feature of our averaging theorem is that it holds uniformly in time; the technique of proof we use gives, as a biproduct, a rate of convergence as well. Very few UiT averaging results exist in the literature, and they almost exclusively apply to multiscale systems of Ordinary Differential Equations. Among these few, none of those we are aware of comes with a rate of convergence. The UiT nature of this result and the fact that the main theorem comes with a rate of convergence as well, make it important as theoretical underpinning for a range of applications, such as applications to statistical methodology, molecular dynamics etc. Key to obtaining both our UiT averaging result and to enable dealing with the super-linear growth of the coefficients (of the slow-fast system and of the associated Poisson equation)is conquering exponential decay in time of the space-derivatives of appropriate Markov semigroups. We refer to semigroups which enjoy this property as being Strongly Exponentially Stable.

There are various approaches in the literature to proving averaging results. The analytic approach we take here requires studying a family of Poisson problems associated with the generator of the (fast component of the) SDE dynamics. The study of Poisson equations in non-compact state space is notoriously difficult, with current literature mostly covering the case when the coefficients of the Partial Differential Equation (PDE) are either bounded or satisfy linear growth assumptions (with the latter case having been achieved only recently). In this paper we treat Poisson equations on non-compact state spaces for coefficients that can grow superlinearly. In particular, we demonstrate how Strong Exponential Stability can be employed not only to prove the UiT result for the slow-fast system but also to overcome some of the technical hurdles in the analysis of Poisson problems. Poisson equations are essential tools in both probability theory and PDE theory. Their vast range of applications includes the study of the asymptotic behaviour of solutions of parabolic PDEs, the treatment of multi-scale and homogenization problems as well as the theoretical analysis of approximations of solutions of Stochastic Differential Equations (SDEs). So our result on Poisson equations is of independent interest as well.
Original languageEnglish
JournalAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
Publication statusAccepted/In press - 20 Aug 2024

Keywords

  • math.PR
  • 60J60, 60H10, 35B30, 34K33, 34D20, 47D07, 65M75

Fingerprint

Dive into the research topics of 'Poisson Equations with locally-Lipschitz coefficients and Uniform in Time Averaging for Stochastic Differential Equations via Strong Exponential Stability'. Together they form a unique fingerprint.

Cite this