Stochastic wave equation with Hölder noise coefficient: Well-posedness and small mass limit

We construct unique martingale solutions to the damped stochastic wave equation partial differential 2u partial differential u mu partial differential t2 (t, x) = Delta u(t, x) - partial differential t (t, x) + b(t, x, u(t, x)) +sigma(t, x, u(t, x)) dWt dt , where Delta is the Laplacian on [0, 1] with Dirichlet boundary condition, W is space-time white noise, sigma is 34 + -Holder continuous in u and uniformly non-degenerate, and b has linear growth. The same construction holds for the stochastic wave equation without damping term. More generally, the construction holds for SPDEs defined on separable Hilbert spaces with a densely defined operator A, and the assumed Holder regularity on the noise coefficient depends on the eigenvalues of A in a quantitative way. We further show the validity of the Smoluchowski-Kramers approximation: assume b is Holder continuous in u, then as mu tends to 0 the solution to the damped stochastic wave equation converges in distribution, on the space of continuous paths, to the solution of the corresponding stochastic heat equation. The latter result is new even in the case of additive noise. (c) 2023 The Author. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).

Automated summary

This article constructs unique martingale solutions to the damped stochastic wave equation and shows their applicability to a wider class of SPDEs. It also demonstrates the validity of the Smoluchowski-Kramers approximation.