An excursion approach to Ray-Knight theorems for perturbed Brownian motion

Abstract

Perturbed Brownian motion in this paper is defined as X t = | B t | - μl t where B is standard Brownian motion, ( l t : t ⩾ 0) is its local time at 0 and μ is a positive constant. Carmona et al. (1994) have extended the classical second Ray-Knight theorem about the local time processes in the space variable taken at an inverse local time to perturbed Brownian motion with the resulting Bessel square processes having dimensions depending on μ. In this paper a proof based on splitting the path of perturbed Brownian motion at its minimum is presented. The derivation relies mostly on excursion theory arguments.