Weighted Aleksandrov estimates: PDE and stochastic versions

TL;DR

This paper establishes weighted pointwise estimates for solutions of linear elliptic and parabolic PDEs with measurable coefficients, accommodating boundary blow-up, and extends these results to occupation times of diffusion processes.

Contribution

It introduces new weighted estimates for PDE solutions and diffusion occupation times, handling boundary blow-up scenarios.

Findings

Weighted estimates for PDE solutions with boundary blow-up

Extension of estimates to diffusion occupation times

Applicable to equations with measurable coefficients

Abstract

We prove several pointwise estimates for solutions of linear elliptic (parabolic) equations with measurable coefficients in smooth domains (cylinders) through the weighted $L_{d}$ ($L_{d+1}$)-norm of the free term. The weights allow the free term to blow up near the (latteral) boundary. We also present weighted estimates for occupation times of diffusion processes.