Functional Limit Theorems for Volterra Processes and Applications to Homogenization

Abstract

We prove an enhanced limit theorem for additive functionals of a multi-dimensional Volterra process $(y_t{)}_{t\ge 0}$ in the rough path topology. As an application, we establish weak convergence as $\epsilon \to 0$ of the solution of the random ordinary differential equation (ODE) $\frac{d}{dt}{x}_t^{\epsilon }=\frac{1}{\sqrt{\epsilon }}f(x_t^{\epsilon },y_{\frac{t}{\epsilon }})$ and show that its limit solves a rough differential equation driven by a Gaussian field with a drift coming from the Lévy 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 Itô SDE $d{x}_t=G(x_t,dt)$, where $G(x,t)$ is a semi-martingale with spatial parameters.

Published version with minor typos corrected; 32 pages

References in corpus (5)

• Central limit theorems for sequences of multiple stochastic integrals
• Limit theorems for weighted nonlinear transformations of Gaussian stationary processes with singular spectra
• Homogenization with fractional random fields
• Diffusive and rough homogenisation in fractional noise field
• Functional Limit Theorems of moving averages of Hermite processes and an application to homogenization

Cited by in corpus (3)

• Generating diffusions with fractional Brownian motion
• Mild Stochastic Sewing Lemma, SPDE in Random Environment, and Fractional Averaging
• Lévy areas, Wong Zakai anomalies in diffusive limits of Deterministic Lagrangian Multi-Time Dynamics