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

Abstract

We study a general class of discrete p-Laplace operators in the random conductance model with long-range jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional p-Laplace operator.Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Laplace operator to the continuous fractional Laplace operator.

Homogenization of symmetric stable-like processes in stationary ergodic medium

“…As far as we know, for all the results in literature, the limiting process is always non-degenerate with Γ = R d , even if the coefficients of scaled processes are degenerate -see for instance [9,15,16,18,33,28]. Our paper provides examples for symmetric jump processes that both coefficients of scaled processes and the limiting process are degenerate.…”

“…We shall mention that in [33] random coefficients of the jumping kernel are assumed to be uniformly elliptic and bounded. Recently, Flegel and Heida [28] considered the corresponding problem in the discrete setting under some moment conditions on coefficients in two-parameter ergodic environment. The stochastic homogenization of a class of fully non-linear integral-differential equations in ergodic environment was studied by Schwab [45].…”

“…The first one is [33], where the infinitesimal generator is given by (1.5) with Γ = R d and κ(x, y, ω) = λ 1 (τ x ω)λ 2 (τ y ω) in one-parameter ergodic random environment. The second one is [28], which is under two-parameters ergodic environment. The setting of our paper is more general, and it also includes the symmetrization of those in [24,30,45]; see (A1) introduced above.…”

Markov chain approximations for nonsymmetric processes

“…This opens the door to the consideration of homogenization problems for irreversible Markov chains on random media. Quenched invariance principles for symmetric random conductance models with bounded, respectively long range and limiting generators of the form (1.2), respectively (1.3) can be found in [Bis11], [ABDH13], [ADS15], respectively [BKU10], [CKK13], [CKW20], [CKW21], [FH20], [Bos20], [BCKW21]. Moreover, let us point to a related direction of research, namely homogenization problems for local, respectively nonlocal operators with random coefficients.…”

Stochastic homogenization of random walks on point processes

“…Our homogenized equation is of diffusive type. We refer to [5,18] and references therein for homogenization on non-local random operators with non diffusive homogenized equation.…”