Ray-knight Theorems via Continuous Trees

In this paper, we are interested in obtaining the generalized Ray-Knight theorems (previously proved by Yor 10]) using branching processes. It was already observed in O'Connell's thesis 6] that the construction of the natural tree associated to Brownian excursion (Le Gall 4], Neveu-Pitman 5]) enables to obtain one of the "classical" Ray-Knight theorems. Our aim is to use the same ideas to get those theorems when the excursion is no longer Brownian but is associated to a Brownian motion perturbed by its local time. 1 The generalized Ray-Knight theorem. On a probability space ((; F; IP), we consider the following process: 8t 0; X t = jB t j + ?1 (2l t) where (B t) t0 is a Brownian motion starting at 0 l t is its local time at level 0 and time t is a strictly increasing C 1-function on 0; +1 such that (0) = 0 and lim t!+1 (t) = +1 We denote by L a t the local time of X at level a and time t. Yor ((10] Theorem 3.3) proved the following result: Theorem 1 the process (L a 1 ; a 0) is a "Bessel square process of dimension " starting at 0, i.e. an inhomogeneous diiusion process with generator ? t f(x) = 2x d 2 f dx 2 + 0 (t) df dx for every function f 2 C 2 b (IR). We propose here to prove this theorem via branching processes techniques.