Maximum principle for non-uniformly parabolic equations and applications

TL;DR

This paper establishes global boundedness for solutions to degenerate parabolic equations using De-Giorgi's iteration and applies these results to prove existence of weak solutions for stochastic differential equations with singular coefficients, also constructing associated Markov processes.

Contribution

It introduces a novel approach to analyze degenerate parabolic equations and extends the theory to stochastic differential equations with singular coefficients.

Findings

Proves global boundedness of solutions to degenerate parabolic equations.

Establishes existence of weak solutions for stochastic differential equations with singular diffusion and drift.

Constructs strong Markov families for these stochastic processes.

Abstract

In this paper we study the global boundedness for the solutions to a class of possibly degenerate parabolic equations by De-Giorgi's iteration. As applications, we show the existence of weak solutions for possibly degenerate stochastic differential equations with singular diffusion and drift coefficients. Moreover, by the Markov selection theorem of Krylov [8], we also establish the existence of the associated strong Markov family.