On singular stochastic differential equations and Dirichlet forms

TL;DR

This paper surveys recent advances in understanding when diffusions related to certain Dirichlet forms can be explicitly represented as solutions to stochastic differential equations, focusing on stochastic regularity, existence, and uniqueness.

Contribution

It provides a structured summary of new results on stochastic regularity and the conditions for strong existence and pathwise uniqueness of singular SDEs linked to non-symmetric Dirichlet forms.

Findings

Diffusions associated with regular strongly local Dirichlet forms can be explicitly identified as solutions to SDEs.

Conditions for strong existence of solutions to singular SDEs are established.

Criteria for pathwise uniqueness of solutions are discussed.

Abstract

This survey paper is a structured concise summary of four of our recent papers on the stochastic regularity of diffusions that are associated to regular strongly local (but not necessarily symmetric) Dirichlet forms. Here by stochastic regularity we mean the question whether a diffusion associated to a Dirichlet form as mentioned above can be started and identified as a solution to an explicit stochastic differential equation for explicitly given starting points. Beyond the stochastic regularity, we consider its applications to strong existence and pathwise uniqueness of singular stochastic differential equations.