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

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.

Small changes according to the published version, J. Differential Equations (2016)

References in corpus (4)

• A relatively short proof of Itô's formula for SPDEs and its applications
• Stochastic De Giorgi Iteration and Regularity of Stochastic Partial Differential Equation
• Non-divergence Parabolic Equations of Second Order with Critical Drift in Lebesgue Spaces
• $L^2$-Theory of Linear Degenerate SPDEs and $L^p$ ($p>0$) Estimates for the Uniform Norm of Weak Solutions

Cited by in corpus (1)

• Supremum estimates for degenerate, quasilinear stochastic partial differential equations