Pointwise weak existence for diffusions associated with degenerate elliptic forms and 2-admissible weights

TL;DR

This paper develops a method to construct weak solutions for a broad class of possibly degenerate stochastic differential equations with irregular drifts, using weighted Sobolev space analysis and stochastic calculus, leading to new criteria for solution existence and non-explosion.

Contribution

It introduces a novel approach combining weighted Sobolev spaces and stochastic calculus to handle degenerate elliptic forms with irregular drifts, enabling weak solution construction from a specified subset of Euclidean space.

Findings

Constructed weak solutions for degenerate SDEs with unbounded, discontinuous drifts.

Provided new non-explosion criteria for strong solutions.

Extended solution existence to points with infinite drift values.

Abstract

Using analysis for 2-admissible functions in weighted Sobolev spaces and stochastic calculus for possibly degenerate symmetric elliptic forms, we construct weak solutions to a wide class of stochastic differential equations starting from an explicitly specified subset in Euclidean space. The solutions have typically unbounded and discontinuous drift but may still in some cases start from all points of $\mathbb{R}^d$ and thus in particular from those where the drift terms are infinite. As a consequence of our approach we are able to provide new non-explosion criteria for the unique strong solutions of \cite{Zh}.