On Weak Convergence of Stochastic Wave Equation with Colored Noise on $$\mathbb {R}$$ R

Wenxuan Tao ()
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Wenxuan Tao: University of Birmingham

Journal of Theoretical Probability, 2025, vol. 38, issue 3, 1-23

Abstract: Abstract In this paper, we study the following stochastic wave equation on the real line: $$\partial _t^2 u_{\alpha }=\partial _x^2 u_{\alpha }+b\left( u_\alpha \right) +\sigma \left( u_\alpha \right) \eta _{\alpha }$$ ∂ t 2 u α = ∂ x 2 u α + b u α + σ u α η α . The noise $$\eta _\alpha $$ η α is white in time and colored in space with a covariance structure $$\mathbb {E}[\eta _\alpha (t,x)\eta _\alpha (s,y)]=\delta (t-s)f_\alpha (x-y)$$ E [ η α ( t , x ) η α ( s , y ) ] = δ ( t - s ) f α ( x - y ) where $$f_\alpha $$ f α is continuous with respect to $$\alpha $$ α in Fourier mode, see Assumption 1.2. We prove the continuity of the probability measure induced by the solution $$u_\alpha $$ u α , in terms of $$\alpha $$ 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 $$f_{\alpha }$$ f α to which our theorem applies.

Keywords: Stochastic wave equation; Finite-dimensional convergence; Tightness; Convergence in law; 60B10; 60H15 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-025-01427-8

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