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Reminiscences of some of Paul Lévy’s ideas in Brownian motion and in Markov chains. (English) Zbl 0665.60076

Les processus stochastiques, Coll. Paul Lévy, Palaiseau/Fr. 1987, Astérisque 157-158, 29-36 (1988).
[For the entire collection see Zbl 0649.00017.]
First of all the author discussed P. Lévy’s contributions to the structure of paths of Markov chains. The second point is Lévy’s ideas on local time and Brownian excursions. P. Lévy was the first who proposed the notions of Brownian sojourn time, local time, inverse local time and excursions and investigated their properties, which occurred to be very useful not only in exploration of Brownian motion, but also in the theory of processes with independent increments, diffusion processes and general Markov processes. As the author mentions, “despite later alternative approaches to these matters, Lévy’s original way should be preserved not as a museum piece, but as a monument conjuring up the past and beckoning to the future.”
Reviewer: N.M.Zinchenko

MSC:

60J65 Brownian motion
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J27 Continuous-time Markov processes on discrete state spaces
60J55 Local time and additive functionals

Citations:

Zbl 0649.00017