Brownian Motion with Drift on Spaces with Varying Dimension

TL;DR

This paper investigates Brownian motion with drift on spaces with varying dimensions, establishing its relation to driftless cases via Girsanov transform and deriving Gaussian bounds and Green function estimates.

Contribution

It introduces a framework for BMVD with drift using Dirichlet forms and proves sharp Gaussian bounds and Green function estimates for this process.

Findings

BMVD with drift can be related to BMVD without drift via Girsanov transform.

Transition density of BMVD with drift has Gaussian bounds similar to the driftless case.

Green function estimates are derived for BMVD with drift.

Abstract

Many properties of Brownian motion on spaces with varying dimension (BMVD in abbreviation) have been explored in [5]. In this paper, we study Brownian motion with drift on spaces with varying dimension (BMVD with drift in abbreviation). Such a process can be conveniently defined by a regular Dirichlet form that is not necessarily symmetric. The drift term is in some type of $L^{p}$ space with $p$ depending on the region of the state space. We show BMVD with drift can be related to a BMVD without drift via Girsanov transform. Through the method of Duhamel's principle, it is established in this paper that the transition density of BMVD with drift has the same type of sharp two-sided Gaussian bounds as that for BMVD (without drift). As a corollary, we derive Green function estimate for BMVD with drift.