Space-time fluctuation of the Kardar-Parisi-Zhang equation in d≥ 3 and the Gaussian free field

2019/05/08 by Francis Comets, Comets, Francis, Clément Cosco +3
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math-ph #math.AP #math.MP #math.PR

paper · pdf · doi:10.48550/arxiv.1905.03200

In the current version, Theorem 2.2 is new (in the earlier version it appeared as Remark 2.8) and gives the fluctuations of the stationary solution. Introduction also revised

openalex publication_date 2019/05/08 · arxiv created 2021/04/08 · arxiv updated 2021/04/09 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28

Abstract

We study the solution h_ε of the Kardar-Parisi-Zhang (KPZ) equation for d ≥ 3: (∂)/(∂ t) h_ε = \frac12 Δh_ε + [\frac12 |∇ h_ε |² - C_ε]+ βε^\fracd-22 ξ_ε with h_ε(0,x)=0. Here ξ_ε=ξ⋆ ϕ_ε is a spatially smoothened (at scale ε) Gaussian space-time white noise and C_ε is a divergent constant as ε→ 0. When the disorder β is sufficiently small and ε→ 0, h_ε(t,x)- \mathfrak h^st_ε(t,x)→ 0 in probability where \mathfrak h^st_ε(t,x) is the \emph stationary solution of the KPZ equation - more precisely, \mathfrak h^st_εsolves the above equation with a random initial condition (that is independent of the driving noise ξ) and its law is constant in (ε,t,x). In the present article we quantify the rate of the above convergence in this regime and show that the fluctuation \emph about the stationary solution (ε^{1-\frac d2} [h_ε(t,x) - \mathfrak h^st_ε(t,x)])_x,t converges pointwise (with finite dimensional distributions in space and time) to a Gaussian free field (GFF) evolved by the deterministic heat equation. We also identify the fluctuations \it of the stationary solution itself and show that the rescaled averages ∫_{\mathbb R^d} \mathrm d x φ(x) ε^{1-\frac d2} [\mathfrak h^st_ε(t,x)- \mathbb E(\mathfrak h^st_ε(t,x))] converge to that of the \emph stationary solution of the stochastic heat equation with additive noise, but with (random) \emph GFF marginals (instead of flat initial condition).

Citations

• Gaussian fluctuations for the directed polymer partition function for d≥ 3 and in the whole L²-region

Cited by

• Fluctuation and Rate of Convergence for the Stochastic Heat Equation in Weak Disorder

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