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Lévy processes and relativistic quantum dynamics. (English) Zbl 0835.60095
Garbaczewski, Piotr (ed.) et al., Chaos: the interplay between stochastic and deterministic behaviour. Proceedings of the XXXIst winter school of theoretical physics held in Karpacz, Poland, 13-24 February 1995. Karpacz: Springer-Verlag. Lect. Notes Phys. 457, 75-86 (1995).
Summary: The traditional Gaussian framework (Wiener process as the “free noise”, with the Laplacian as noise generator) is extended to encompass any infinitely divisible probability law covered by the Lévy-Khinchin formula. It implies a family of random environment models (of the fluctuating medium) governed by the generally non-Gaussian “free noises”. Since the so-called relativistic Hamiltonians $$|\nabla |$$ and $$\sqrt {- \Delta + m^2} - m$$ are known to generate such laws, we focus on them for the analysis of probabilistic phenomena, which are shown to be associated with the relativistic quantum propagation once an analytic continuation in time of the corresponding holomorphic semigroup is accomplished. The pertinent stochastic processes are identified to be spatial jump processes.
For the entire collection see [Zbl 0826.00026].
##### MSC:
 60K40 Other physical applications of random processes 81S25 Quantum stochastic calculus