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Homogenization of non-symmetric jump processes

Part of: Partial differential equations Limit theorems Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  05 June 2023

Qiao Huang ,
Jinqiao Duan  and
Renming Song
Qiao Huang*
Affiliation:
Huazhong University of Science and Technology
Jinqiao Duan*
Affiliation:
Illinois Institute of Technology
Renming Song*
Affiliation:
University of Illinois at Urbana-Champaign
*
*Postal address: School of Mathematics and Statistics and Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P.R. China. Email address: hq932309@alumni.hust.edu.cn
**Postal address: Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA. Email address: duan@iit.edu
***Postal address:Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. Email address: rsong@illinois.edu
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Abstract

We study homogenization for a class of non-symmetric pure jump Feller processes. The jump intensity involves periodic and aperiodic constituents, as well as oscillating and non-oscillating constituents. This means that the noise can come both from the underlying periodic medium and from external environments, and is allowed to have different scales. It turns out that the Feller process converges in distribution, as the scaling parameter goes to zero, to a Lévy process. As special cases of our result, some homogenization problems studied in previous works can be recovered. We also generalize the approach to the homogenization of symmetric stable-like processes with variable order. Moreover, we present some numerical experiments to demonstrate the usage of our homogenization results in the numerical approximation of first exit times.

Keywords

Feller processesweak convergencenon-local operatorsstable-like processes

MSC classification

Primary: 35B27: Homogenization; equations in media with periodic structure
Secondary: 60F17: Functional limit theorems; invariance principles 35R09: Integro-partial differential equations
Type
Original Article
Information
Advances in Applied Probability , Volume 56 , Issue 1 , March 2024 , pp. 1 - 33
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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