2019/05/08 by Comets, Francis, Cosco, Clément, Mukherjee, Chiranjib
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.1905.03200
We study the solution hε of the Kardar-Parisi-Zhang (KPZ) equation for d ≥ 3: (∂)/(∂ t) hε = \frac12 Δhε + [\frac12 |∇ hε |2 - Cε]+ βε^\fracd-22 ξε with hε(0,x)=0. Here ξε=ξ⋆ ϕε is a spatially smoothened (at scale ε) Gaussian space-time white noise and Cε is a divergent constant as ε→ 0. When the disorder β is sufficiently small and ε→ 0, hε(t,x)- \mathfrak hstε(t,x)→ 0 in probability where \mathfrak hstε(t,x) is the \emph stationary solution of the KPZ equation - more precisely, \mathfrak hstεsolves 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 (ε1-\frac d2 [hε(t,x) - \mathfrak hstε(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 Rd \mathrm d x φ(x) ε1-\frac d2 [\mathfrak hstε(t,x)- \mathbb E(\mathfrak hstε(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).