Homogenization with fractional random fields

Abstract

We consider a system of differential equations in a fast long range dependent random environment and prove a homogenization theorem involving multiple scaling constants. The effective dynamics solves a rough differential equation, which is 'equivalent' to a stochastic equation driven by mixed Itô integrals and Young integrals with respect to Wiener processes and Hermite processes. Lacking other tools we use the rough path theory for proving the convergence, our main technical endeavour is on obtaining an enhanced scaling limit theorem for path integrals (Functional CLT and non-CLT's) in a strong topology, the rough path topology, which is given by a Hölder distance for stochastic processes and their lifts. In dimension one we also include the negatively correlated case, for the second order / kinetic fractional BM model we also bound the error.

Generating Diffusions with Fractional Brownian Motion

“…See also [1,7,12,14] for more recent results with a similar flavour. In the case when the fast dynamics is non-Markovian and solves an equation driven by a fractional Brownian motion, a collection of homogenisation results were obtained in [23][24][25][26], while stochastic averaging results with non-Markovian fast motions are obtained in [51,52] for the case H > 1 2 . The former group of results are proved using rough path techniques, but there is of course an extensive literature on functional limit theorems based on either central or non-central limit theorems, see for example [3,5,6,10,54,62,63].…”

Rough invariance principle for delayed regenerative processes

“…In the context of semimartingales and rough paths with jumps [3,8,4], CLT on nilpotent covering graphs and crystal lattices [15,24,16], additive functional of Markov process and random walks in random environment [6]. For homogenization in the continuous settings [5,18,19], and for additive functionals of fractional random fields [11,12,13].…”