Local $L_\infty$-estimates, weak Harnack inequality, and stochastic continuity of solutions of SPDEs

TL;DR

This paper develops local supremum estimates, a weak Harnack inequality, and proves almost sure continuity for solutions of SPDEs with minimal assumptions, including scaling-critical diffusion terms.

Contribution

It introduces stochastic adaptations of classical PDE techniques to handle minimal assumptions and critical scaling in SPDEs, advancing regularity theory.

Findings

Established local supremum estimates for SPDE solutions.

Proved a weak Harnack inequality in the stochastic setting.

Demonstrated almost sure pointwise continuity of solutions.

Abstract

We consider stochastic partial differential equations under minimal assumptions: the coefficients are merely bounded and measurable and satisfy the stochastic parabolicity condition. In particular, the diffusion term is allowed to be scaling-critical. We derive local supremum estimates with a stochastic adaptation of De Giorgi's iteration and establish a weak Harnack inequality for the solutions. The latter is then used to obtain pointwise almost sure continuity.