Homogenization of Lévy-type Operators with Oscillating Coefficients

Abstract

The paper deals with homogenization of Lévy-type operators with rapidly oscillating coefficients. We consider cases of periodic and random statistically homogeneous micro-structures and show that in the limit we obtain a Lévy-operator. In the periodic case we study both symmetric and non-symmetric kernels whereas in the random case we only investigate symmetric kernels. We also address a nonlinear version of this homogenization problem.MSC : 45E10, 60J75, 35B27, 45M05

Homogenization of symmetric stable-like processes in stationary ergodic medium

“…They showed that the limiting process is a symmetric α-stable Lévy process. Kassmann, Piatnitski and Zhizhina in [33] investigated homogenization of a class of symmetric stable-like processes in ergodic environment whose jumping kernels are of product form. Homogenization problem of symmetric stable-like processes in two-parameter ergodic environment was also studied in [33].…”

Periodic homogenization for weakly elliptic Hamilton-Jacobi-Bellman equations with critical fractional diffusion

“…If on one hand it is natural that H inherits the nonlocal nature in its third variable, on the other hand no explicit formula can be obtained in general. Some examples of explicit nonlocal effective equations can be found in [6] and [28], but we stress that these methods cannot be applied in the setting and/or the generality presented here. In particular, we establish a non-trivial ellipticity-growth condition for H that further allows to manipulate the effective problem in spite of not knowing its explicit form.…”

The fractional p-Laplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unbounded-range jumps

“…The homogenization of the fractional Laplace operator seems to be only recent and rather unexplored. However, there are a few results in the literature: Most of them are focused on the periodic homogenization of the continuous fractional Laplace operator (−∆) s , starting from a work by Piatnitskii and Zhizhina [20] and Kassmann, Piatnitskii and Zhizhina [14]. A first result on the stochastic homogenization of the (continuum) fractional Laplace operator with uniformly bounded c is given in [21].…”