Stochastic wave equation with additive fractional noise: solvability and global Hölder continuity

2023/05/03 by Liu, Shuhui, Hu, Yaozhong, Wang, Xiong · 1 citation
#60G15 #60G22 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 60H15 #Probability (math.PR) #secondary 60H05

paper · doi:10.48550/arxiv.2305.02425

Abstract

We determine the range of Hurst parameters that provide the necessary and sufficient conditions for the solvability, in L²(Ω), of the stochastic wave equation: (∂² )/(∂ t²)u(t,x) =Δu(t,x)+W(t,x), where \ W(t,x), t≥ 0, x∈ ℝ^d\ is a fractional Brownian field with temporal Hurst parameter H₀∈[\tfrac12,1] and spatial Hurst parameters H_i∈(0,1) for i=1,⋯,d. In particular, the solvability condition exhibits a phase transition at H₀ = 1. We also obtain the sharp growth rate and the sharp Hölder continuity of the solution on the real line in the case H₀=1/2.

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