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Stochastic Aspects of Classical and Quantum Systems: Proceedings of the 2nd French-German Encounter in Mathematics and Physics, held in Marseille, France, ... Notes in Mathematics) (French Edition): S. Albeverio, P. Combe, M. Sirugue-Collin

In recent years it has been realized by research workers in many different research areas that the traditional rather strict separation between deterministic and stochastic phenomena had to undergo a profound revision. Systems that used to be described and interpreted purely in classical deterministic terms have turned out to have essential stochastic features, and there have been on the other hand serious efforts to give a classical interpretation to some basic stochastic phenomena. As an example of the former we might mention the detailed study of simple dynamical systems exhibiting "strange attractors", as an example of the latter we think of the efforts to construct mechanical models of diffusion processes and hence, via stochastic mechanics, of quantum phenomena. The present workshop, the second in a series of Encounters between mathematicians and mathematical physicists operating in France and in the Federal Republic of Germany, was organized around a central subject belonging to the above area of interactions between classical and probabilistic descriptions of certain "chaotic" natural phenomena.

The relation between quantum theory and classical mechanics has many different aspects. Some of the contributions of this volume are concerned with this study. In recent years methods of functional integration and probabilistic techniques have been used to derive asymptotic expansions in Planck's constant for1various quantum mechanical quantities, the coefficient in the expansion being describahle in terms of classical mechanics (like e.g. in the work of Jona-Lasinio, Martinelli and Scoppola, based on methods of Ventzell and Freidlin; many references in this area are given in the Albeverio-Arede contribution to the Proceedings of the Como '83 Conference "Quantum Chaos", Ed. G. Casati, J. Ford, Plenum). The present volume contains new contribution to the subject by Azencott and Doss, which in particular unifies and extends by probabilistic technique results previously obtained separately for the Schrodinger and heat equations. A study of quantum stochastic differential equations, in connection with a problem of quantum statistics, starting from the corresponding classical limit is presented in the contribution by von Waldenfels. This paper gives a bridge between traditional probabilistic techniques in quantum theory and the new field of "quantum probability", in which there has been much activity in recent years.

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