Spacetime limit shapes of the KPZ equation in the upper tails

2023/04/27 by Yier Lin, Lin, Yier, Li‐Cheng Tsai +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2304.14380

openalex publication_date 2023/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the n-point, fixed-time large deviations of the KPZ equation with the narrow wedge initial condition. The scope consists of concave-configured, upper-tail deviations and a wide range of scaling regimes that allows time to be short, unit-order, and long. We prove the n-point large deviation principle and characterize, with proof, the corresponding spacetime limit shape. Our proof is based on the results -- from the companion paper Tsai (2023) -- on moments of the stochastic heat equation and utilizes ideas coming from a tree decomposition. Behind our proof lies the phenomenon where the major contribution of the noise concentrates around certain corridors in spacetime, and we explicitly describe the corridors.

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