KPZ equation tails for general initial data

TL;DR

This paper analyzes the tail probabilities of the KPZ equation's solution for general initial data, establishing bounds that reveal different decay behaviors in the upper and lower tails.

Contribution

It provides the first comprehensive tail bounds for the KPZ equation with broad initial conditions, highlighting a crossover in decay rates for the lower tail.

Findings

Lower tail exhibits a crossover from exponent 3 to 5/2.

Upper tail shows super-exponential decay with exponent 3/2.

Results apply to a very general class of initial data.

Abstract

We consider the upper and lower tail probabilities for the centered (by time$/24$) and scaled (according to KPZ time$^{1/3}$ scaling) one-point distribution of the Cole-Hopf solution of the KPZ equation when started with initial data drawn from a very general class. For the lower tail, we prove an upper bound which demonstrates a crossover from super-exponential decay with exponent $3$ in the shallow tail to an exponent $5/2$ in the deep tail. For the upper tail, we prove super-exponential decay bounds with exponent $3/2$ at all depth in the tail.