Quenched large deviations for multiscale diffusion processes in random environments

Konstantinos Spiliopoulos

doi:10.1214/ejp.v20-3729

Abstract

We consider multiple time scales systems of stochastic differential equations with small noise in random environments. We prove a quenched large deviations principle with explicit characterization of the action functional. The random medium is assumed to be stationary and ergodic. In the course of the proof we also prove related quenched ergodic theorems for controlled diffusion processes in random environments that are of independent interest. The proof relies entirely on probabilistic arguments, allowing to obtain detailed information on how the rare event occurs. We derive a control, equivalently a change of measure, that leads to the large deviations lower bound. This information on the change of measure can motivate the design of asymptotically efficient Monte Carlo importance sampling schemes for multiscale systems in random environments

Highlights

Let 0 < ε, δ 1 and consider the process (X, Y ) = {(Xt, Yt ), t ∈ [0, T ]} taking values in the space Rm × Rd−m that satisfies the system of stochastic differential equation (SDE’s)
In [33], it was assumed that the coefficients are periodic with respect to the y−variable and based on the derived large deviations principle, asymptotically efficient importance sampling Monte Carlo methods for estimating rare event probabilities were obtained
We focus on quenched large deviations for the case /δ ↑ ∞ and the situation is more complex when compared to the periodic case since the coefficients are random fields themselves and the fast motion does not take values in a compact space

Summary

Introduction

For a measurable function φ : Rm × Γ → Rm we consider the (locally) stationary random field (x, y) → φ(x, τyγ) = φ(x, y, γ) We follow this procedure to define the random fields b, c, σ, f, g, τ1, τ2 that play the role of the coefficients of (1.1), which guarantees that they are ergodic and stationary random fields. Following [24], we assume the following condition on the structure of the operator defined in Definition 2.3 This condition allows to have a closed form for the unique ergodic invariant measure for the environment process {γt}t≥0, Proposition 2.6. As in Proposition 3.2. of [24], we have that for almost all γ ∈ Γ δχ0(y/δ, γ) → 0, as δ ↓ 0, a.s. y ∈ Y

Main results

Compactness of level sets and quenched lower and upper bounds

A Quenched ergodic theorems

Time shifts and uniformity

Ergodic theorems with perturbation by small drift-Uncontrolled case