A central limit theorem for the stochastic cable equation

TL;DR

This paper proves a central limit theorem for the spatial average of solutions to a nonlinear stochastic cable equation driven by space-time white noise, using the Malliavin-Stein method, with results on convergence and conditions.

Contribution

It introduces a novel application of the Malliavin-Stein method to establish CLTs for nonlinear stochastic PDEs with mild conditions and verifies assumptions in specific cases.

Findings

Proves a CLT for the spatial average of the solution.

Establishes convergence in total variation distance.

Validates the technical assumption in a special case.

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.