Homogenization of periodic diffusion with small jumps

Sandrić, Nikola

doi:10.1016/j.jmaa.2015.10.054

Cited by 14 publications

(16 citation statements)

References 35 publications

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“…limit equation. Since one can approximate diffusions through much simpler objects such as random walks or Markov chains, it is not surprising that there are also homogenization models for jump processes that generate a diffusion in the limit, see [San16] or [PZ17], An annealed convergence result for jump processes in random media is contained in [RV09, Theorem 5.3]. As in our quenched result, no corrector appears.…”

Section: Introductionmentioning

confidence: 82%

Homogenization of Lévy-type Operators with Oscillating Coefficients

Kaßmann¹,

Piatnitski²,

Жижина³

2019

SIAM J. Math. Anal.

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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

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Section: Introductionmentioning

confidence: 82%

Homogenization of Lévy-type Operators with Oscillating Coefficients

Kaßmann¹,

Piatnitski²,

Жижина³

2019

SIAM J. Math. Anal.

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“…The methods used in the these papers are analytic and called the corrector method. The probabilistic study of homogenization of periodic stable-like processes can be found in [20,25,37] via the characteristics of semimartingales, in [21,23,24] by SDE driven by Lévy processes or by Poisson random measures, and in [22,41] via the martingale problem method. 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.…”

Section: Introductionmentioning

confidence: 99%

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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“…For example, a non-local linear operator with a kernel of convolution type in periodic medium [25], concerning certain diffusion process with jumps have been considered. That is also known Feller process generated by an integro-differential operator [30]. Homogenization of a certain class of integrodifferential equations with Lévy operators [3], including scaling limits for symmetric Itô-Lévy processes in random medium [26] has been studied.…”

Section: Introductionmentioning

confidence: 99%

H-convergence and homogenization of non-local elliptic operators in both perforated and non-perforated domains

Bălilescu

Ghosh

2019

Z. Angew. Math. Phys.

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In this paper, we focus on the homogenization process of the non-local elliptic boundary value problem L s ε u ε = (−∇ · (A ε (x)∇)) s u ε = f in O, with 0 < s < 1, considering non-homogeneous Dirichlet type condition outside of the bounded domain O ⊆ R n . We find the homogenized problem by using the H-convergence method, as ε → 0, under standard uniform ellipticity, boundedness and symmetry assumptions on coefficients A ε (x), with the homogenized coefficients as the standard H-limit (cf. [22]) of the sequence {A ε } ε>0 . We also prove that the commonly referred to as the strange term in the literature (see [10, Chapter 4]) does not appear in the homogenized problem associated with the fractional Laplace operator (−∆) s in a perforated domain. Both of these results have been obtained in the class of general microstructures. Consequently, we could certify that the homogenization process, as ε → 0, is stable under s → 1 − in the non-perforated domains, but not necessarily in the case of perforated domains.

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Homogenization of periodic diffusion with small jumps

Cited by 14 publications

References 35 publications

Homogenization of Lévy-type Operators with Oscillating Coefficients

Homogenization of Lévy-type Operators with Oscillating Coefficients

Periodic homogenization of non-symmetric Lévy-type processes

H-convergence and homogenization of non-local elliptic operators in both perforated and non-perforated domains

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