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May 2022 A Smoluchowski–Kramers approximation for an infinite dimensional system with state-dependent damping
Sandra Cerrai, Guangyu Xi
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Ann. Probab. 50(3): 874-904 (May 2022). DOI: 10.1214/21-AOP1549

Abstract

We study the validity of a Smoluchowski–Kramers approximation for a class of wave equations in a bounded domain of Rn subject to a state-dependent damping and perturbed by a multiplicative noise. We prove that in the small mass limit the solution converges to the solution of a stochastic quasilinear parabolic equation where a noise-induced extra drift is created.

Funding Statement

The authors were partially supported by NSF Grants DMS-1712934—Analysis of Stochastic Partial Differential Equations with Multiple Scales and DMS-1954299—Multiscale Analysis of Infinite-Dimensional Stochastic Systems.

Acknowledgments

The first named author would like to thank Viorel Barbu, Zdzisław Brzeźniak, Martina Hofmanovà and Irena Lasiecka for several interesting and useful conversations about this problem.

Citation

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Sandra Cerrai. Guangyu Xi. "A Smoluchowski–Kramers approximation for an infinite dimensional system with state-dependent damping." Ann. Probab. 50 (3) 874 - 904, May 2022. https://doi.org/10.1214/21-AOP1549

Information

Received: 1 November 2020; Revised: 1 September 2021; Published: May 2022
First available in Project Euclid: 27 April 2022

Digital Object Identifier: 10.1214/21-AOP1549

Subjects:
Primary: 35K57 , 35R60 , 60F99 , 60H15

Keywords: singular perturbation of SPDEs , Smoluchowski–Kramers approximation , stochastic damped wave equations , stochastic quasilinear equations

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 3 • May 2022
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