[Paper Review] A Central Limit Theorem for the stochastic heat equation

Jingyu Huang, David Nualart+1 more|arXiv (Cornell University)|Oct 22, 2018

Stochastic processes and financial applications8 references10 citations

TL;DR

This paper establishes a central limit theorem for the spatial integral of the solution to the one-dimensional stochastic heat equation driven by space-time white noise. Using Malliavin calculus and Stein's method, it proves that the normalized integral converges in total variation distance to a standard normal distribution as the spatial domain expands, with a convergence rate of $ O(1/ar{R}) $, and further establishes a functional central limit theorem in the space of continuous functions.

ABSTRACT

We consider the one-dimensional stochastic heat equation driven by a multiplicative space-time white noise. We show that the spatial integral of the solution from $-R$ to $R$ converges in total variance distance to a standard normal distribution as $R$ tends to infinity, after renormalization. We also show a functional version of this central limit theorem.

Motivation & Objective

• To establish a quantitative central limit theorem for the spatial integral of the solution to the stochastic heat equation with multiplicative space-time white noise.
• To analyze the asymptotic behavior of the normalized spatial integral as the spatial domain $[-R, R]$ expands to infinity.
• To extend the result to a functional central limit theorem, showing weak convergence of the process to a Brownian motion.
• To provide a rigorous convergence rate in total variation distance using Stein's method and Malliavin calculus.

Proposed method

• Uses the mild solution representation of the stochastic heat equation involving the Itô-Walsh integral with the heat kernel.
• Applies Malliavin calculus to compute the Malliavin derivative and divergence of the normalized integral functional.
• Employs Stein's method for normal approximation, specifically bounding the total variation distance via the Stein kernel and Malliavin calculus tools.
• Establishes a key technical lemma on the $ L^p $-norm of the Malliavin derivative of the solution, showing it is bounded by a scaled heat kernel.
• Uses the representation of the normalized integral as a Skorohod integral to leverage the duality between divergence and derivative operators.
• Applies a functional limit theorem framework by proving convergence in law on $ C([0,T]) $, using tightness and finite-dimensional convergence.

Experimental results

Research questions

• RQ1 Does the spatial integral of the solution to the stochastic heat equation satisfy a central limit theorem as the domain expands?
• RQ2 What is the rate of convergence in total variation distance for the normalized spatial integral to a standard normal distribution?
• RQ3 Can the central limit theorem be extended to a functional limit theorem for the process indexed by time?
• RQ4 How does the variance of the spatial integral scale with $ R $, and what is its limiting behavior?

Key findings

• The total variation distance between the normalized spatial integral $ F_R(t) $ and a standard normal random variable is bounded by $ C / \sqrt{R} $, where $ C $ depends only on $ t $.
• The variance $ \sigma_R^2 $ of the spatial integral satisfies $ \lim_{R \to \infty} \sigma_R^2 / R = 2 \int_0^t \mathbb{E}[\sigma(u(s,y))^2] \, ds $, showing linear growth in $ R $.
• The functional central limit theorem holds: the process $ \frac{1}{\sqrt{R}} \left( \int_{-R}^R u(t,x) \, dx - 2R \right) $ converges in law to $ \int_0^t \sqrt{2\xi(s)} \, dB_s $ in $ C([0,T]) $, where $ \xi(s) = \mathbb{E}[\sigma(u(s,y))^2] $.
• The convergence rate in total variation is sharp, with the bound $ d_{TV}(F_R(t), Z) \leq C / \sqrt{R} $, and this rate is derived using Stein's method and Malliavin calculus.
• The Malliavin derivative of the solution satisfies a key $ L^p $-bound: $ \|D_{s,y}u(r,z)\|_p \leq C_{t,p} p_{r-s}(z-y) $, which is essential for controlling the Stein method error.