Pointwise weak existence of distorted skew Brownian motion with respect to discontinuous Muckenhoupt weights

TL;DR

This paper characterizes the stochastic differential equation governing distorted Brownian motion with respect to certain discontinuous Muckenhoupt weights, especially focusing on the behavior at permeable membranes defined by level sets.

Contribution

It establishes the pointwise weak existence of solutions to the SDE for distorted Brownian motion with discontinuous weights, extending understanding to weights with complex discontinuity structures.

Findings

Identifies the SDE satisfied by distorted Brownian motion with specific weights.

Analyzes the behavior of the process at permeable membranes.

Provides conditions for the existence of solutions with discontinuous weights.

Abstract

For any starting point in $\mathbb{R}^d$, we identify the stochastic differential equation that is satisfied by distorted Brownian motion with respect to a certain discontinuous Muckenhoupt $A_2$-weight $\psi$. The discontinuities of $\psi$ typically take place on a sequence of level sets of the Euclidean norm $D_k:=\{x\in \mathbb{R}^d\, | \;\|x\|=d_k\}$, $k\in\mathbb{Z}$, where $(d_k)_{k\in\mathbb{Z}}\subset (0,\infty)$ may have accumulation points and each level set $D_k$ plays the role of a permeable membrane.