Distorted Brownian motions on space with varying dimension

TL;DR

This paper introduces distorted Brownian motions on a space with varying dimension, analyzing their properties and heat kernel estimates using Dirichlet forms, thus advancing understanding of stochastic processes on complex geometries.

Contribution

It defines and studies distorted Brownian motions on a space with varying dimension, providing foundational properties and heat kernel estimates.

Findings

Established basic properties of dBMVDs using Dirichlet forms.

Proved joint continuity of transition density functions.

Derived short-time heat kernel estimates for dBMVDs.

Abstract

Roughly speaking, a space with varying dimension consists of at least two components with different dimensions. In this paper we will concentrate on the one, which can be treated as $\mathbb{R}^3$ tying a half line not contained by $\mathbb{R}^3$ at the origin. The aim is twofold. On one hand, we will introduce so-called distorted Brownian motions on this space with varying dimension (dBMVDs in abbreviation) and study their basic properties by means of Dirichlet forms. On the other hand, we will prove the joint continuity of the transition density functions of these dBMVDs and derive the short-time heat kernel estimates for them.