On weak convergence of stochastic wave equation with colored noise on ℝ

2024/08/19 by Wenxuan Tao, Tao, Wenxuan
Economics, Econometrics and Finance · Computer Science · Engineering · #Stochastic processes and financial applications #Advanced Mathematical Modeling in Engineering #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2408.10326

Abstract

In this paper, we study the following stochastic wave equation on the real line ∂_t² u_α=∂_x² u_α+b(u_α)+σ(u_α)η_α. The noise η_α is white in time and colored in space with a covariance structure 𝔼[η_α(t,x)η_α(s,y)]=δ(t-s)f_α(x-y) where f_α is continuous with respect to α in Fourier mode, see Assumption 1.2. We prove the continuity of the probability measure induced by the solution u_α, in terms of α, with respect to the convergence in law in the topology of continuous functions with uniform metric on compact sets. We also give several examples of f_α such that our theorem applies to.

Related

• On weak convergence of stochastic heat equation with colored noise
• A Laplace principle for a stochastic wave equation in spatial dimension three
• Convergence in law for quasi-linear SPDEs
• A proof of a support theorem for stochastic wave equations in Holder norm with some general noises
• On Hölder continuity of the solution of stochastic wave equations