Scaling limits for symmetric Ito-Levy processes in random medium

TL;DR

This paper investigates the scaling limits of solutions to stochastic differential equations driven by Poisson measures and Brownian motions in random media, revealing diffusive or superdiffusive behaviors based on measure properties.

Contribution

It establishes an annealed convergence theorem for these processes, characterizing their limiting behavior under different integrability conditions.

Findings

Limit processes can be diffusive or superdiffusive.

Scaling limits depend on the integrability of the Poisson measure.

Provides a rigorous framework for understanding stochastic dynamics in random media.

Abstract

We are concerned with scaling limits of the solutions to stochastic differential equations with stationary coefficients driven by Poisson random measures and Brownian motions. We state an annealed convergence theorem, in which the limit exhibits a diffusive or superdiffusive behavior, depending on the integrability properties of the Poisson random measure