Distorted Brownian motions on space with varying dimension

Abstract

In this paper we introduce and study distorted Brownian motion on state spaces with varying dimension (dBMV in abbreviation). Roughly speaking, the state space of dBMV is embedded in ${\mathbb{R}}^4$ and consists of two components: a 3-dimensional component and a 1-dimensional component. These two parts are joined together at the origin. 3-dimensional dBMV models homopolymer with attractive potential at the origin and has been studied in [9], [8], [7]. dBMV restricted on the 1-dimensional component can be viewed as a Brownian motion with drift of Kato-class type. Such a process with varying dimensional can be concisely characterized in terms of Dirichlet forms. Using the method of radial process developed in [5] combined with some calculation specifically for dBMV, we get its short-time heat kernel estimates.