Stochastic wave equation with H\"older noise coefficient: well-posedness and small mass limit

Abstract

We construct unique martingale solutions to the damped stochastic wave equation $$ \mu \frac{\partial^2u}{\partial t^2}(t,x)=\Delta u(t,x)-\frac{\partial u}{\partial t}(t,x)+b(t,x,u(t,x))+\sigma(t,x,u(t,x))\frac{dW_t}{dt},$$ where $\Delta$ is the Laplacian on $[0,1]$ with Dirichlet boundary condition, $W$ is space-time white noise, $\sigma$ is $\frac{3}{4}+\epsilon$ -H\"older 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 H\"older 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 H\"older 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.