Coming up from $-\infty$ for KPZ via stochastic control

TL;DR

This paper establishes initial-condition-independent bounds for the KPZ equation on the torus by linking it to a stochastic control problem, leading to a Harnack inequality for the rough heat equation.

Contribution

It introduces a novel stochastic control representation for the KPZ equation, enabling bounds independent of initial conditions and deriving a Harnack inequality.

Findings

Derived a lower bound for KPZ solutions independent of initial conditions.

Proved a bound on the oscillation of KPZ solutions.

Established a Harnack inequality for the rough heat equation.

Abstract

We derive a lower bound, independent of the initial condition, for the solution of the KPZ equation on the torus through its representation as the value function of a (conditional) stochastic control problem. With the same techniques, we also prove a bound for its oscillation, again independent of initial conditions, from which a Harnack type inequality for the rough heat equation (on the torus) can be obtained.