Some remarks on perturbed reflecting Brownian motion. (English) Zbl 0835.60072

Werner, Wendelin

Azéma, J. (ed.) et al., Séminaire de probabilités XXIX. Berlin: Springer-Verlag. Lect. Notes Math. 1613, 37-43 (1995).

The perturbed Brownian motion is the stochastic process \(X_t = |B_t |- \mu L_t\) where \(B\) is a real Brownian motion, \(\mu > 0\) a real constant and \(L\) the local time at zero of \(B\). This process has been studied by several authors, see e.g. J.-F. Le Gall and M. Yor [C. R. Acad. Sci., Paris, Sér. I 303, 73-76 (1986; Zbl 0589.60070)], or Ph. Carmona, F. Petit and M. Yor [Probab. Theory Relat. Fields 100, No. 1, 1-29 (1994; Zbl 0808.60066)]. The author gives new derivations of previously known results on this process. In particular, using a result of Lamperti on semi-stable Markov processes on the line, he gives a proof of the Ray-Knight type theorem, which expresses the distribution of the local times of the process \(X\) in terms of squares of Bessel processes.
For the entire collection see [Zbl 0826.00027].

Reviewer: Ph.Biane (Paris)

MSC:

60J65

Brownian motion

Keywords:

perturbed Brownian motion; local time; semi-stable Markov processes; Ray- Knight type theorem; Bessel processes

Citations:

Zbl 0589.60070; Zbl 0808.60066