Superpositions of Probability Distributions

Details Superpositions of Probability Distributions Superpositions of Probability Distributions

2008-02-05

arxiv.org/abs/0802.0695v2

Phys. Rev. E 78 (2008) 031122

Physics

Condensed Matter

Other Condensed Matter

23 pages, RevTeX, minor changes, accepted to Phys. Rev. E

Scientific paper

10.1103/PhysRevE.78.031122

Probability distributions which can be obtained from superpositions of Gaussian distributions of different variances v = \sigma ^2 play a favored role in quantum theory and financial markets. Such superpositions need not necessarily obey the Chapman-Kolmogorov semigroup relation for Markovian processes because they may introduce memory effects. We derive the general form of the smearing distributions in v which do not destroy the semigroup property. The smearing technique has two immediate applications. It permits simplifying the system of Kramers-Moyal equations for smeared and unsmeared conditional probabilities, and can be conveniently implemented in the path integral calculus. In many cases, the superposition of path integrals can be evaluated much easier than the initial path integral. Three simple examples are presented, and it is shown how the technique is extended to quantum mechanics.

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