We study the validity of a Smoluchowski–Kramers approximation for a class of wave equations in a bounded domain of 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.
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.
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.
"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