A generic approach to homogenization of a diffusion driven by growing incompressible drift

TL;DR

This paper investigates the asymptotic behavior of diffusions with growing incompressible drift, revealing a limit process characterized by a Hunt process and establishing a distributional convergence result.

Contribution

It introduces a general homogenization framework for diffusions driven by increasing incompressible drift, connecting the limit to Hunt processes and distributional limits.

Findings

Limit of the resolvent-family is a selfadjoint pseudo-resolvent.

The associated semi-group corresponds to a Hunt process.

Distributional limit theorem for accelerated diffusion.

Abstract

We study how the resolvent-family of a diffusion behaves, as thedrift grows to infinity. The limit turns out to be a selfadjoint pseudo-resolvent.After reduction of the underlying Hilbert-space, this pseudo-resolvent becomesa resolvent to a strongly continuous semi-group of contractions. We prove thatthis semi-group is associated to some Hunt-process on some suitable state-space which is constructed from equivalence classes of the drifts trajectories.Finally, we show a distributional limit theorem for the accelerated diffusiontoward the associated Hunt process.