On the $l_p$ stability estimates for stochastic and deterministic difference equations and their application to SPDEs and PDEs

TL;DR

This paper develops an $l_p$-theory for stochastic difference equations, analogous to Krylov's $L_p$-theory for SPDEs, and establishes Calderon-Zygmund estimates for deterministic schemes with variable coefficients.

Contribution

It introduces a discrete $l_p$-theory for stochastic difference equations and proves Calderon-Zygmund estimates for finite difference schemes with relaxed assumptions.

Findings

Established $l_p$-stability estimates for stochastic difference equations.

Proved Calderon-Zygmund type estimates for deterministic finite difference schemes.

Extended the theory to variable coefficient scenarios.

Abstract

In this paper we develop the $l_p$-theory of space-time stochastic difference equations which can be considered as a discrete counterpart of N.V. Krylov's $L_p$-theory of stochastic partial differential equations. We also prove a Calderon-Zygmund type estimate for deterministic parabolic finite difference schemes with variable coefficients under relaxed assumptions on the coefficients, the initial data and the forcing term.