[Paper Review] Quenched central limit theorem for the stochastic heat equation in weak disorder

Yannic Broeker, Chiranjib Mukherjee|arXiv (Cornell University)|Oct 2, 2017

Stochastic processes and statistical mechanics11 references4 citations

TL;DR

This paper establishes a quenched central limit theorem for the stochastic heat equation in $d \geq 3$ under weak disorder, proving that the diffusively rescaled polymer path measure converges almost surely to a standard Gaussian distribution. Using renormalized Feynman-Kac representation and moment convergence techniques, it shows that for small $\beta > 0$, the quenched law of the rescaled Brownian motion under the polymer measure converges to the standard normal distribution in the long-time limit.

ABSTRACT

We continue with the study of the mollified stochastic heat equation in $d\geq 3$ given by $d u_{ε,t}=\frac 12Δu_{ε,t}+ βε^{(d-2)/2} \,u_{ε,t} \,d B_{ε,t}$ with spatially smoothened cylindrical Wiener process $B$, whose (renormalized) Feynman-Kac solution describes the partition function of the continuous directed polymer. In an earlier work (\cite{MSZ16}), a phase transition was obtained, depending on the value of $β>0$ in the limiting object of the smoothened solution $u_ε$ as the smoothing parameter $ε o 0$ This partition function naturally defines a quenched polymer path measure and we prove that as long as $β>0$ stays small enough while $u_ε$ converges to a strictly positive non-degenerate random variable, the distribution of the diffusively rescaled Brownian path converges under the aforementioned polymer path measure to standard Gaussian distribution.

Motivation & Objective

• To establish a quenched central limit theorem for the continuous directed polymer in $d \geq 3$ under weak disorder.
• To analyze the long-time behavior of the polymer path measure defined by the solution of the mollified stochastic heat equation.
• To prove almost sure convergence of the rescaled polymer path distribution to the standard Gaussian under the quenched measure.
• To extend discrete polymer results to the continuum setting using rigorous stochastic PDE techniques.

Proposed method

• Uses the mollified stochastic heat equation with space-time white noise, regularized via a smooth mollifier $\phi_\varepsilon$.
• Applies the Feynman-Kac formula to express the solution as an expectation over Brownian motion with a random exponential functional.
• Employs time reversal to relate the solution to a partition function of a continuous directed polymer.
• Uses moment generating function techniques and multivariate Hermite polynomial expansions to analyze convergence of moments.
• Applies a renormalization procedure to handle the divergent noise, removing the $\varepsilon$-dependent $V_\varepsilon(0)$ term.
• Employs an induction argument on moments to show convergence to Gaussian moments under the quenched measure.

Experimental results

Research questions

• RQ1 Does the diffusively rescaled polymer path under the quenched measure converge to a Gaussian distribution in the long-time limit?
• RQ2 Can a quenched central limit theorem be established for the continuous stochastic heat equation in $d \geq 3$ under weak disorder?
• RQ3 How do the moments of the rescaled polymer path converge under the quenched measure?
• RQ4 What is the role of the disorder strength $\beta$ in determining the limiting distribution?
• RQ5 Can the convergence be proven almost surely for every realization of the noise?

Key findings

• For $d \geq 3$ and sufficiently small $\beta > 0$, the quenched distribution of the diffusively rescaled polymer path converges almost surely to the standard Gaussian distribution.
• The convergence is established via almost sure convergence of all moments to those of the standard normal distribution.
• The proof relies on showing that the moments of the rescaled path under the quenched measure converge to the corresponding moments of a standard normal random vector.
• The key technical step involves an induction argument on the moments, using Hermite polynomial expansions and coefficient matching.
• The result holds for every fixed realization of the space-time white noise, confirming a quenched limit theorem.
• The renormalized Feynman-Kac solution ensures $\mathbb{E}[u_\varepsilon(t,x)] = 1$, which is crucial for moment analysis.