On the excursion theory for linear diffusions

Paavo P. Salminen (1) , Pierre P. Vallois (2) , Marc Yor (3)

1 MD - Mathematical department [Abo Finland]
2 IECN - Institut Élie Cartan de Nancy
3 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We present a number of important identities related to the excursion theory of linear diffusions. In particular, excursions straddling an independent exponential time are studied in detail. Letting the parameter of the exponential time tend to zero it is seen that these results connect to the corresponding results for excursions of stationary diffusions (in stationary state). We characterize also the law of the diffusion prior and posterior to the last zero before the exponential time. It is proved using Krein's representations that, e.g., the law of the length of the excursion straddling an exponential time is infinitely divisible. As an illustration of the results we discuss Ornstein-Uhlenbeck processes.

Mots clés: Brownian motion; last exit decomposition; local time; infinite divisibility; spectral representation; Ornstein-Uhlenbeck process

Domaines: Probabilités [math.PR]

Paavo P. Salminen, Pierre P. Vallois, Marc Yor. On the excursion theory for linear diffusions. Japanese Journal of Mathematics, 2007, 2 (1), pp.97-127. ⟨10.1007/s11537-007-0662-y⟩. ⟨hal-00121803⟩