  The classical Ray-Knight theorems for 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 Brownian motion. We extend these results by describing the local time
process jointly for all a and all b, by means of 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
[10]), whereas a Feller CSBP with immigration satisfies a stochastic
differential equation driven by a white noise (Dawson and Li [7]); our result
gives an explicit relation between these two descriptions and shows that the
stochastic differential equation in question is a reformulation of Tanaka's
formula.
