On the chaotic character of the stochastic heat equation, II

2011/11/21 by Daniel Conus, Conus, Daniel, Mathew Joseph +5 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1111.4728

openalex publication_date 2011/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the stochastic heat equation ∂_t u = ((\varkappa)/(2))Δu+σ(u)F, where the solution u:=u_t(x) is indexed by (t,x)∈ (0, ∞)×\R^d, and F is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large-|x| fixed-t behavior of the solution u in different regimes, thereby study the effect of noise on the solution in various cases. Among other things, we show that if the spatial correlation function f of the noise is of Riesz type, that is f(x)∝ ‖x‖^-α, then the "fluctuation exponents" of the solution are ψ for the spatial variable and 2ψ-1 for the time variable, where ψ:=2/(4-α). Moreover, these exponent relations hold as long as α∈(0, d\wedge 2); that is precisely when Dalang's theory implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions.

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