Renormalization flow for the 2D nonlinear stochastic heat equation: pointwise statistics and universality

2023/08/23 by Alexander Dunlap, Dunlap, Alexander, Cole Graham +1 · 3 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Advanced Thermodynamics and Statistical Mechanics

paper · pdf · doi:10.48550/arxiv.2308.11850

Abstract

We consider a two-dimensional stochastic heat equation with noise correlated at scale ρ≪ 1 and of strength |logρ|^-1/2σ(v) depending nonlinearly on the solution v. Under certain conditions, the first author and Gu have shown that the one-point statistics of v converge in law as ρ→ 0 to the terminal value of an associated forward-backward SDE. Here, we show that the 2D stochastic heat equation is stable under renormalization with a new effective nonlinearity tied to the decoupling function of the forward-backward SDE. This allows us to extend the pointwise results to a much broader class of nonlinearities. We also show that these limiting pointwise statistics are insensitive to the fine details of the noise, and thus universal.

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• A critical stochastic heat equation with long-range noise
• Edwards-Wilkinson limit for a stochastic advection-diffusion PDE
• McKean-Vlasov limits of scaling-critical reaction-diffusion equations with random initial data

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