Functional Limit Theorems for Volterra Processes and Applications to Homogenization

TL;DR

This paper establishes functional limit theorems for multi-dimensional Volterra processes in rough path topology and applies these results to analyze the homogenization of solutions to random ODEs, showing convergence to rough differential equations and stochastic flows.

Contribution

It provides an enhanced limit theorem for additive functionals of Volterra processes and applies it to prove convergence of solutions of random ODEs to rough differential equations and stochastic flows.

Findings

Weak convergence of solutions to a rough differential equation driven by Gaussian fields.

Convergence of stochastic flows to Kunita type Itô SDEs.

Enhanced limit theorem for additive functionals of Volterra processes.

Abstract

We prove an enhanced limit theorem for additive functionals of a multi-dimensional Volterra process $(y_t)_{t\geq 0}$ in the rough path topology. As an application, we establish weak convergence as $\varepsilon\to 0$ of the solution of the random ordinary differential equation (ODE) $\frac{d}{dt}x^\varepsilon_t=\frac{1}{\sqrt \varepsilon} f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}})$ and show that its limit solves a rough differential equation driven by a Gaussian field with a drift coming from the L\'evy area correction of the limiting rough driver. Furthermore, we prove that the stochastic flows of the random ODE converge to those of the Kunita type It\^o SDE $dx_t=G(x_t,dt)$, where $G(x,t)$ is a semi-martingale with spatial parameters.