Functional limit theorems for Volterra processes and applications to homogenization* *Johann Gehringer is supported by an EPSRC studentship, Xue-Mei Li by the EPRSC grants EP/V026100/1 and EP/S023925/1, and Julian Sieber by the EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Simulation (EP/S023925/1).

Abstract

We prove an enhanced limit theorem for additive functionals of a multi-dimensional Volterra process in the rough path topology. As an application, we establish weak convergence as ɛ → 0 of the solution of the random ordinary differential equation (ODE) 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 dx t = G(x t , dt), where G(x, t) is a semi-martingale with spatial parameters.