Strict decomposition of diffusions associated to degenerate (sub)-elliptic forms

TL;DR

This paper proves the existence of weak solutions to certain degenerate stochastic differential equations associated with (sub)-elliptic Dirichlet forms, and discusses conditions for strong solutions and non-explosion criteria.

Contribution

It establishes the pointwise existence of weak solutions for degenerate diffusions with unbounded, discontinuous drifts using heat kernel estimates and Dirichlet form theory.

Findings

Existence of weak solutions from all points in 

Conditions for pathwise uniqueness and strong solutions

New non-explosion criterion for degenerate diffusions

Abstract

For given strongly local Dirichlet forms with possibly degenerate symmetric (sub)-elliptic matrix, we show the existence of weak solutions to the stochastic differential equations (associated with the Dirichlet forms) starting from all points in $\R^d$. More precisely, using heat kernel estimates, stochastic calculus, and Dirichlet form theory, we obtain the pointwise existence of weak solutions to the stochastic differential equations which have possibly unbounded and discontinuous drift. We also present some conditions that the weak solutions become pathwise unique strong solutions and provide a new non-explosion criterion.