Homogenization for Deterministic Maps and Multiplicative Noise

TL;DR

This paper extends the homogenization results for deterministic maps with chaotic fast dynamics to discrete time and multiplicative noise, clarifying stochastic integral interpretations and deriving superdiffusive limits driven by Levy processes.

Contribution

It generalizes previous continuous-time results to discrete time and addresses multiplicative noise, including the interpretation of stochastic integrals and superdiffusive limits.

Findings

Homogenization from deterministic to stochastic systems with multiplicative noise.

Stochastic integrals are of Marcus type for multiplicative noise.

Derived superdiffusive limits driven by stable Levy processes.

Abstract

A recent paper of Melbourne & Stuart, A note on diffusion limits of chaotic skew product flows, Nonlinearity 24 (2011) 1361-1367, gives a rigorous proof of convergence of a fast-slow deterministic system to a stochastic differential equation with additive noise. In contrast to other approaches, the assumptions on the fast flow are very mild. In this paper, we extend this result from continuous time to discrete time. Moreover we show how to deal with one-dimensional multiplicative noise. This raises the issue of how to interpret certain stochastic integrals; it is proved that the integrals are of Stratonovich type for continuous time and neither Stratonovich nor Ito for discrete time. We also provide a rigorous derivation of superdiffusive limit where the stochastic differential equation is driven by a stable Levy process. In the case of one-dimensional multiplicative noise, the stochastic integrals are of Marcus type both in the discrete and continuous time contexts.