Homogenization of symmetric stable-like processes in stationary ergodic medium

Chen, Xin; Chen, Zhen-Qing; Kumagai, Takashi; Wang, Jian

doi:10.48550/arxiv.2003.08581

Cited by 2 publications

(3 citation statements)

References 33 publications

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“…(2.2) See e.g. [11,Lemma 2.1]. Since the coefficients of the generator L ε are multivariate ε-periodic, the process X ε can be also viewed as an T d -valued process if 1/ε is an integer.…”

Section: Preliminariesmentioning

confidence: 99%

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Periodic homogenization of non-symmetric Lévy-type processes

Chen¹,

Chen²,

Kumagai³

et al. 2020

Preprint

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In this paper, we study homogenization problem for strong Markov processes on R d having infinitesimal generatorsin periodic media, where Π is a non-negative measure on R d that does not charge the origin 0, satisfiesand can be singular with respect to the Lebesgue measure on R d . Under a proper scaling, we show the scaled processes converge weakly to Lévy processes on R d . The results are a counterpart of the celebrated work [6,7] in the jump-diffusion setting. In particular, we completely characterize the homogenized limiting processes when b(x) is a bounded continuous multivariate 1-periodic R d -valued function, k(x, z) is a non-negative bounded continuous function that is multivariate 1-periodic in both x and z variables, and, in spherical coordinate z = (r, θ) ∈ R+ × S d−1 ,with α ∈ (0, ∞) and ̺0 being any finite measure on the unit sphere S d−1 in R d . Different phenomena occur depending on the values of α; there are five cases: α ∈ (0, 1), α = 1, α ∈ (1, 2), α = 2 and α ∈ (2, ∞).

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Section: Preliminariesmentioning

confidence: 99%

“…A closely related topic is homogenization of non-local operators or jump diffusions in random media, which typically requires a different approach than the periodic media case, see [36,40] for example. Recently we have studied homogenization of symmetric stable-like processes in stationary ergodic random media in [11].…”

Section: Introductionmentioning

confidence: 99%

Periodic homogenization of non-symmetric Lévy-type processes

Chen¹,

Chen²,

Kumagai³

et al. 2020

Preprint

Self Cite

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Stochastic homogenization of random walks on point processes

Faggionato¹

2020

Preprint

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We consider random walks on the support of a random pure point measure on R d with random jump probability rates. The jump range can be unbounded. The pure point measure is reversible for the random walk and stationary for the action of the group G = R d or G = Z d . By combining two-scale convergence and Palm theory for G-stationary random measures and by developing a cut-off procedure for long jumps, under suitable second moment conditions we prove for almost all environments the homogenization for the massive Poisson equation of the associated Markov generators. In addition, we obtain the quenched convergence of the L 2 -Markov semigroup of the diffusively rescaled random walk to the one of Brownian motion with covariance matrix 2D. For symmetric jump rates, the above convergence plays a crucial role in the derivation of hydrodynamic limits when considering multiple random walks with hard core or zero range interaction. We do not require any ellipticity assumption, neither non-degeneracy of the homogenized matrix D. Our results cover a large family of models, including e.g. random conductance models on Z d and on general lattices (possibly with long conductances), Mott variable range hopping, simple random walks on Delaunay triangulations, simple random walks on the supercritical percolation cluster.

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Homogenization of symmetric stable-like processes in stationary ergodic medium

Cited by 2 publications

References 33 publications

Periodic homogenization of non-symmetric Lévy-type processes

Periodic homogenization of non-symmetric Lévy-type processes

Stochastic homogenization of random walks on point processes

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