From averaging to homogenization in cellular flows – An exact description of the transition

Abstract

We consider a two-parameter averaging-homogenization type elliptic problem together with the stochastic representation of the solution. A limit theorem is derived for the corresponding diffusion process and a precise description of the two-parameter limit behavior for the solution of the PDE is obtained.

A fractional kinetic process describing the intermediate time behaviour of cellular flows

“…The vast majority of results concerning scaling limits of diffusions obtain a limiting behaviour that is again a diffusion, if not a rescaled Brownian motion. A time changed Brownian motion was first obtained in [HKP14] on time scales of order 1/ε, and here we extend this result to much shorter time scales.…”

“…As discussed in the previous section,X homogenizes on time scales larger than O(1/ε). Under a compactness assumption (e.g., if the periodic flow is replaced by a flow on a torus) classical results of Freidlin (discussed below) show thatX averages along the flow lines of v. In the non-compact setting that we consider, a recent result [HKP14] shows thatX transitions between the homogenized and averaged behaviour in a very natural way, and we describe this behaviour here. To study the behaviour on time scales of order t ≈ 1/ε, consider the time rescaled process X defined by…”

“…2 We emphasise that this only determines the effective behaviour of X projected onto the compact Reeb graph of H, when H is viewed as a function on the torus. A recent paper [HKP14] showed how this can be used to obtain the effective behaviour of X on the whole plane R 2 . The main theorem in [HKP14] shows that…”

“…A recent paper [HKP14] showed how this can be used to obtain the effective behaviour of X on the whole plane R 2 . The main theorem in [HKP14] shows that…”

“…To relate this to the classical homogenization results, note that equation (2.5) provides information on the effective behaviour ofX on time scales of order 1/ε; the borderline time scale, beyond which homogenization results are valid. In fact, for the processZ ε,δ defined by (2.1), the result of [HKP14] states that for δ = ε, the limiting process in (2.3) is now a subordinated Brownian motion. In contrast, for δ ε (as we had in (2.1)), the limiting process is simply an effective Brownian motion without any subordination.…”