A probabilistic Harnack inequality and strict positivity of stochastic partial differential equations

TL;DR

This paper establishes a probabilistic Harnack inequality for solutions of certain stochastic PDEs and proves that solutions become strictly positive almost surely if starting from non-negative initial data.

Contribution

It introduces a probabilistic Harnack inequality for stochastic PDEs and demonstrates strict positivity of solutions under broad conditions.

Findings

Proves a probabilistic Harnack inequality for stochastic PDEs.

Shows solutions are almost surely strictly positive from non-negative initial data.

Provides a framework for analyzing positivity in stochastic PDEs.

Abstract

Under general conditions we show an a priori probabilistic Harnack inequality for the non-negative solution of a stochastic partial differential equation of the following form d_tu = div (A\nabla u) + f (t, x, u;w) + g_i(t, x, u;w)\dot{w}^i_t. We will also show that the solution of the above equation will be almost surely strictly positive if the initial condition is non-negative and not identically vanishing.