Estimates on Green functions of second order differential operators with singular coefficients

TL;DR

This paper studies Green functions of second order differential operators with singular coefficients, showing they are more regular on singular hyperplanes when expressed via Brownian motion.

Contribution

It provides a probabilistic representation of Green functions for operators with singular coefficients and demonstrates increased regularity on certain hyperplanes.

Findings

Green functions expressed through Brownian motion.

Enhanced regularity of Green functions on singular hyperplanes.

Applicable to differential operators with coefficients depending on one coordinate.

Abstract

We investigate the Green functions G(x,x^{\prime}) of some second order differential operators on R^{d+1} with singular coefficients depending only on one coordinate x_{0}. We express the Green functions by means of the Brownian motion. Applying probabilistic methods we prove that when x=(0,{\bf x}) and x^{\prime}=(0,{\bf x}^{\prime}) (here x_{0}=0) lie on the singular hyperplanes then G(0,{\bf x};0,{\bf x}^{\prime}) is more regular than the Green function of operators with regular coefficients.