On the stochastic regularity of distorted Brownian motions

TL;DR

This paper develops tools to analyze distorted Brownian motions, enabling the construction of weak solutions to complex stochastic differential equations with singular drifts and boundary conditions.

Contribution

It introduces general methods for applying Fukushima's absolute continuity condition to construct and analyze distorted Brownian motions with singularities and boundary reflections.

Findings

Constructed weak solutions for SDEs with unbounded, discontinuous drift.

Explicitly identified skew reflected and normally reflected Brownian motions.

Extended analysis to multi-dimensional domains with various boundary conditions.

Abstract

We systematically develop general tools to apply Fukushima's absolute continuity condition. These tools comprise methods to obtain a Hunt process on a locally compact separable metric state space whose transition function has a density w.r.t. the reference measure and methods to estimate drift potentials comfortably. We then apply our results to distorted Brownian motions and construct weak solutions to singular stochastic differential equations, i.e. equations with possibly unbounded and discontinuous drift and reflection terms which may be the sum of countably many local times. The solutions can start from any point of the explicitly specified state space. We consider different kind of weights, like Muckenhoupt $A_2$ weights and weights with moderate growth at singularities as well as different kind of (multiple) boundary conditions. Our approach leads in particular to the construction and explicit identification of countably skew reflected and normally reflected Brownian motions with singular drift in bounded and unbounded multi-dimensional domains