A central limit theorem for the stochastic cable equation

2025/06/03 by Soma Nishino, Nishino, Soma
Economics, Econometrics and Finance · Engineering · Mathematics · #60H07 (Secondary) #60H15 (Primary) 60F05 #FOS: Mathematics #Nonlinear Differential Equations Analysis #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2506.02755

openalex publication_date 2025/06/03 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

Abstract

We study one-dimensional nonlinear stochastic cable equations driven by a multiplicative space-time white noise. Using the Malliavin-Stein method, we prove a central limit theorem for the spatial average of the solution. The convergence is established in the total variation distance with mild conditions. We also establish a functional central limit theorem with a technical assumption. Furthermore, we show that this assumption holds in a special case.

Citations

• Central limit theorems for stochastic wave equations in high dimensions
• Central limit theorems for nonlinear stochastic wave equations in dimension three
• Central limit theorems for heat equation with time-independent noise: the regular and rough cases
• Limit theorems for time-dependent averages of nonlinear stochastic heat equations
• Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Stein's method
• Quantitative normal approximations for the stochastic fractional heat\n equation
• Spatial stationarity, ergodicity and CLT for parabolic Anderson model with delta initial condition in dimension d≥ 1
• A Central Limit Theorem for the stochastic heat equation
• A central limit theorem for the stochastic heat equation

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• Central limit theorems for nonlinear stochastic wave equations in dimension three
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