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

Yi Han

Abstract

We construct unique martingale solutions to the damped stochastic wave equation $$ μ\frac{\partial^2u}{\partial t^2}(t,x)=Δu(t,x)-\frac{\partial u}{\partial t}(t,x)+b(t,x,u(t,x))+σ(t,x,u(t,x))\frac{dW_t}{dt},$$ where $Δ$ is the Laplacian on $[0,1]$ with Dirichlet boundary condition, $W$ is space-time white noise, $σ$ is $\frac{3}{4}+ε$ -Hölder 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 Hölder 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 Hölder continuous in $u$, then as $μ$ 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.

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

Abstract

We construct unique martingale solutions to the damped stochastic wave equation where is the Laplacian on with Dirichlet boundary condition, is space-time white noise, is -Hölder continuous in and uniformly non-degenerate, and 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 , and the assumed Hölder regularity on the noise coefficient depends on the eigenvalues of in a quantitative way. We further show the validity of the Smoluchowski-Kramers approximation: assume is Hölder continuous in , then as tends to 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.
Paper Structure (20 sections, 24 theorems, 204 equations)

This paper contains 20 sections, 24 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Related literature
  3. Works on regularization by noise
  4. Works on stochastic wave equation
  5. Works on Smoluchowski-Kramers approximation
  6. Plan of this paper
  7. Stochastic estimates of the semigroup
  8. Functional analytic preliminaries
  9. Estimates in shifted norm
  10. Maximal inequalities of stochastic integrals
  11. Existence and uniqueness of solutions
  12. Proof of weak existence: step 1
  13. Lipschitz approximation of continuous maps
  14. Proof of weak uniqueness
  15. Proof of weak existence: step 2
  16. ...and 5 more sections

Key Result

Theorem 1.2

Given a separable Hilbert space $H$. Suppose the operator $A$, the nonlinear mapping $B:[0,\infty)\times H\to H$ and $G:[0,\infty)\times H\to \mathcal{L}(H_0,H)$ (with $H_0$ specified below)In this paper if the subscript of $|\cdot|$ is not specified, then it refers to the norm of $H$. satisfy the f Moreover, there exists a Banach space $H_0\subset H$ with a continuous embedding, such that each ei

Theorems & Definitions (46)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 36 more