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The heat equation with time-correlated random potential in d=2: Edwards-Wilkinson fluctuations

Sotirios Kotitsas

TL;DR

The paper analyzes the two-dimensional stochastic heat equation with a time-space mollified Gaussian potential, using a delicate renormalization that scales the coupling as $eta_ ho= frac{eta}{ oot 2 rom ext{log}(1/ ho)}$. It proves that in the subcritical regime $eta<eta_c(R)$, the centered fluctuations of the mollified solution converge to Edwards–Wilkinson fluctuations with an explicit effective variance $v_{eff}^2(eta)$ and unit diffusivity, i.e. to a solution $ ext{U}$ of $ ext{d}_t ext{U}= frac12 abla ext{U} + v_{eff}(eta)oldsymbol{\xi}$, while the mean is renormalized by an exponential factor $oldsymbol{ u}^{( ho)}$. The core methodology combines a Feynman–Kac representation with a Markov chain on path space to capture the time correlations, yielding a regenerative structure that leads to a Kallianpur–Robbins-type law and a CLT for the fluctuations. Key steps include constructing a Doeblin-coupled Markov chain on $oldsymbol{ ext{Ω}}_1$, proving mixing and renewal properties, and establishing precise controls and limits for the additive functionals via Proposition ef{thm:prop2.4} and Corollary ef{cor6.1}. The results extend Edwards–Wilkinson universality to the critical dimension with time correlations in the mollified noise, and provide explicit formulas for the effective variance and the renormalization constants that govern the limiting fluctuations.

Abstract

We consider the stochastic PDE: $\partial_tu(t,x)=\frac{1}{2}Δu(t,x)+β{}u(t,x)V(t,x),$ in dimension $d=2$, where the potential V is the space and time mollification of the two-dimensional space-time white noise. We show that after renormalizing, the fluctuations of the solution converge to the Edwards-Wilkinson limit with an explicit effective variance and constant effective diffusivity. Our main tool is a Markov chain on the space of paths which we use to establish an extension of the Kallianpur-Robbins law to a specific regenerative process.

The heat equation with time-correlated random potential in d=2: Edwards-Wilkinson fluctuations

TL;DR

The paper analyzes the two-dimensional stochastic heat equation with a time-space mollified Gaussian potential, using a delicate renormalization that scales the coupling as . It proves that in the subcritical regime , the centered fluctuations of the mollified solution converge to Edwards–Wilkinson fluctuations with an explicit effective variance and unit diffusivity, i.e. to a solution of , while the mean is renormalized by an exponential factor . The core methodology combines a Feynman–Kac representation with a Markov chain on path space to capture the time correlations, yielding a regenerative structure that leads to a Kallianpur–Robbins-type law and a CLT for the fluctuations. Key steps include constructing a Doeblin-coupled Markov chain on , proving mixing and renewal properties, and establishing precise controls and limits for the additive functionals via Proposition ef{thm:prop2.4} and Corollary ef{cor6.1}. The results extend Edwards–Wilkinson universality to the critical dimension with time correlations in the mollified noise, and provide explicit formulas for the effective variance and the renormalization constants that govern the limiting fluctuations.

Abstract

We consider the stochastic PDE: in dimension , where the potential V is the space and time mollification of the two-dimensional space-time white noise. We show that after renormalizing, the fluctuations of the solution converge to the Edwards-Wilkinson limit with an explicit effective variance and constant effective diffusivity. Our main tool is a Markov chain on the space of paths which we use to establish an extension of the Kallianpur-Robbins law to a specific regenerative process.
Paper Structure (16 sections, 25 theorems, 422 equations)

This paper contains 16 sections, 25 theorems, 422 equations.

Table of Contents

  1. Introduction
  2. Comments and related results
  3. Future Directions
  4. Idea of the proof and outline
  5. Notation
  6. Proof of Theorem \ref{['thm:Main']}
  7. The Markov Chain on $\Omega_1$
  8. The construction
  9. The coupling in $d=2$
  10. The Mixing property
  11. Intersections of independent paths
  12. Proof of Proposition \ref{['thm:prop2.4']}
  13. Estimates on the Total Path Increments
  14. Kallianpur-Robbins Estimates
  15. A Khas’minskii-type Lemma
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $g\in C_c({\mathbb{R}^2})$ and: where $||R||_1=\int_{\mathbb{R}^{1+2}}R(s,y)dyds$. There exists $\zeta^{(\epsilon)}:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ such that $\zeta^{(\epsilon)}_{t/\epsilon^2}\rightarrow+\infty$ as $\epsilon\rightarrow0$, and such that for all $\hat{\beta}<\hat{\beta}_c(R)$ in probability as $\epsilon\rightarrow0$. Moreover, in distribution, where $\mathcal{U}$ so

Theorems & Definitions (53)

  • Theorem 1.1
  • Proposition 1.1
  • proof
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • ...and 43 more