Averaging principle for diffusion processes via Dirichlet forms

Abstract

Abstract. We study diffusion processes driven by a Brownian motion with regular drift in a finite dimension setting. The drift has two components on different time scales, a fast conservative component and a slow dissipative component. Using the theory of Dirichlet form and Mosco-convergence we obtain simpler proofs, interpretations and new results of the averaging principle for such processes when we speed up the conservative component. As a result, one obtains an effective process with values in the space of connected level sets of the conserved quantities. The use of Dirichlet forms provides a simple and nice way to characterize this process and its properties.

Variational approach to coarse-graining of generalized gradient flows

“…Recently, Barret and Von Renesse [15] provided an alternative proof using Dirichlet forms and their convergence. The latter approach is closer to ours in the sense that it is mainly PDE-based method and of variational type.…”

“…The latter approach is closer to ours in the sense that it is mainly PDE-based method and of variational type. However, in [15] the authors consider a perturbation of the Hamiltonian by a friction term and a non-degenerate noise, i.e. the noise is present in both space and momentum variables; this non-degeneracy appears to be essential in their method.…”

“…In [38] the authors consider the case of degenerate noise by using probabilistic and analytic techniques based on hypoelliptic operators. More recently this problem has been handled using PDE techniques [44] (the elliptic case) and Dirichlet forms [15]. In Sect.…”

“…The rigorous construction of the graph and the topology on it has been done several times [15, 36, 37]; for our purposes it suffices to note that (a) inside each edge, the usual topology and geometry of apply, and (b) across the whole graph there is a natural concept of distance, and therefore of continuity. It will be practical to think of functions as defined on the disjoint union .…”

“…The results of the current and the next sections were proved by Freidlin and co-authors in a series of papers [36–40], using probabilistic techniques. Recently, Barret and Von Renesse [15] provided an alternative proof using Dirichlet forms and their convergence. The latter approach is closer to ours in the sense that it is mainly PDE-based method and of variational type.…”