An infinite-dimensional representation of the Ray-Knight theorems

The classical Ray-Knight theorems for the Brownian motion determine the law of its local time process either at the first hitting time of a given value a by the local time at the origin, or at the first hitting time of a given position b by the Brownian motion. We extend these results by describing the local time process jointly for all a and b, by means of the stochastic integral with respect to an appropriate white noise. Our result applies to mu-processes, and has an immediate application: a mu-process is the height process of a Feller continuous-state branching process (CSBP) with immigration (Lambert (2002)), whereas a Feller CSBP with immigration satisfies a stochastic differential equation (SDE) driven by a white noise (Dawson and Li (2012)); our result gives an explicit relation between these two descriptions and shows that the SDE in question is a reformulation of Tanaka's formula.