On trace of Brownian motion on the boundary of a strip

TL;DR

This paper investigates the boundary traces of reflecting Brownian motions on strips, providing explicit Dirichlet form expressions and analyzing their limits as the strip width varies.

Contribution

It offers explicit formulas for the Dirichlet forms of boundary traces and studies their asymptotic behavior as the strip width approaches zero or infinity.

Findings

Explicit Dirichlet form expressions for boundary traces

Limits of traces as strip width tends to 0 or ∞

Asymptotic behavior of boundary traces

Abstract

The trace of a Markov process is the time changed process of the original process on the support of the Revuz measure used in the time change. In this paper, we will concentrate on the reflecting Brownian motions on certain closed strips. On one hand, we will formulate the concrete expression of the Dirichlet forms associated with the traces of such reflecting Brownian motions on the boundary. On the other hand, the limits of these traces as the distance between the upper and lower boundaries tends to $0$ or $\infty$ will be further obtained.