On the (strict) positivity of solutions of the stochastic heat equation

TL;DR

This paper provides a new proof that solutions to the stochastic heat equation with non-negative initial conditions remain strictly positive at positive times, using concentration of measure techniques from directed polymers and Gaussian environments.

Contribution

It introduces a novel proof method based on concentration of measure, improving understanding of positivity and tail bounds for solutions of the stochastic heat equation.

Findings

Solutions are strictly positive at positive times with non-negative initial conditions.

Improved bounds on the lower tail of solutions with delta initial condition.

New connections between stochastic PDEs and directed polymer concentration phenomena.

Abstract

We give a new proof of the fact that the solutions of the stochastic heat equation, started with non-negative initial conditions, are strictly positive at positive times. The proof uses concentration of measure arguments for discrete directed polymers in Gaussian environments, originated in M. Talagrand's work on spin glasses and brought to directed polymers by Ph. Carmona and Y. Hu. We also get slightly improved bounds on the lower tail of the solutions of the stochastic heat equation started with a delta initial condition.