This article is cited in 5 scientific papers (total in 5 papers)
Interpretations of Probability and Their $p$-Adic Extensions
A. Yu. Khrennikov
This paper is devoted to foundations of probability theory. We discuss interpretations of probability, corresponding mathematical formalisms, and applications to quantum physics. One of the aims of this paper is to show that the probability model based on Kolmogorov's axiomatics cannot describe all stochastic phenomena, i.e., that quantum physics induces natural restrictions of the use of Kolmogorov's theory and that we need to develop non-Kolmogorov models for describing some quantum phenomena. The physical motivations are presented in a clear and brief manner. Thus the reader does not need to have preliminary knowledgeof quantum physics. Our main idea is that we cannot develop non-Kolmogorov models by the formal change of Kolmogorov's axiomatics. We begin with interpretations (classical, frequency, and proportional). Then we present a class of non-Kolmogorov models described by so-called $p$-adic numbers. Here, in particular, we obtain a quite natural realization of negative probabilities. These negative probability distributions might provide a solution of some quantum paradoxes.
$p$-adic, foundations of probability theory, probability model, Bell inequality.
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Theory of Probability and its Applications, 2002, 46:2, 256–273
A. Yu. Khrennikov, “Interpretations of Probability and Their $p$-Adic Extensions”, Teor. Veroyatnost. i Primenen., 46:2 (2001), 311–325; Theory Probab. Appl., 46:2 (2002), 256–273
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\paper Interpretations of Probability and Their $p$-Adic Extensions
\jour Teor. Veroyatnost. i Primenen.
\jour Theory Probab. Appl.
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Kotovich N.V., Khrennikov A.Y., “Representation and compression of images with the aid of $m$-adic coordinate systems”, Dokl. Math., 66:3 (2002), 330–334
Schmitt B.M., “The quantitation of buffering action I. A formal & general approach”, Theoretical Biology and Medical Modelling, 2 (2005), 8
Khrennikov A., “$p$-adic probability theory and its generalizations”, p-adic mathematical physics, AIP Conf. Proc., 826, Amer. Inst. Phys., Melville, NY, 2006, 105–120
Khrennikov A.Yu., “Generalized probabilities taking values in non-Archimedean fields and in topological groups”, Russ. J. Math. Phys., 14:2 (2007), 142–159
Milosevic M., “A Propositional P-Adic Probability Logic”, Publ. Inst. Math.-Beograd, 87:101 (2010), 75–83
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