A Smoluchowski-Kramers approximation for an infinite dimensional system with state-dependent damping. (English) Zbl 1487.35033

Cerrai, Sandra; Xi, Guangyu

Ann. Probab. 50, No. 3, 874-904 (2022).

Summary: We study the validity of a Smoluchowski-Kramers approximation for a class of wave equations in a bounded domain of \({\mathbb{R}^n}\) 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.

MSC:

35B25	Singular perturbations in context of PDEs
35K59	Quasilinear parabolic equations
35L20	Initial-boundary value problems for second-order hyperbolic equations
35L71	Second-order semilinear hyperbolic equations
35R60	PDEs with randomness, stochastic partial differential equations
60H15	Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

singular perturbation of SPDEs; Smoluchowski-Kramers approximation; stochastic damped wave equations; stochastic quasilinear equations; Smoluchowski-Kramers approximation

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