Large Deviations for Multiscale Diffusions via Weak Convergence Methods

TL;DR

This paper establishes large deviations principles for multiscale stochastic differential equations with small noise, using weak convergence methods to analyze different regimes and facilitate efficient importance sampling.

Contribution

It introduces a unified weak convergence framework for large deviations in multiscale diffusions, covering all regimes of noise and oscillation scales.

Findings

Derived action functionals for all regimes.

Identified controls nearly achieving large deviations bounds.

Provided insights for importance sampling in multiscale systems.

Abstract

We study the large deviations principle for locally periodic stochastic differential equations with small noise and fast oscillating coefficients. There are three possible regimes depending on how fast the intensity of the noise goes to zero relative to the homogenization parameter. We use weak convergence methods which provide convenient representations for the action functional for all three regimes. Along the way we study weak limits of related controlled SDEs with fast oscillating coefficients and derive, in some cases, a control that nearly achieves the large deviations lower bound at the prelimit level. This control is useful for designing efficient importance sampling schemes for multiscale diffusions driven by small noise.