June  2019, 11(2): 123-137. doi: 10.3934/jgm.2019006

Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena

1. 

Institut für Angewandte Mathematik and Hausdorff Center for Mathematics, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany

2. 

Dipartimento di Matematica, Università degli Studi di Trento and INFN-TIFPA, via Sommarive, 14 – 38123 Povo (Trento), Italy

* Corresponding author: Sergio Albeverio

Received  September 2018 Revised  April 2019 Published  May 2019

We present a survey of the relations between infinite dimensional integrals, both of the probabilistic type (e.g. Wiener path integrals) and of oscillatory type (e.g. Feynman path integrals).

Besides their mutual relations (analogies and differences) we also discuss their relations with certain types of partial differential equations (parabolic resp. hyperbolic), describing time evolution with or without stochastic terms.

The connection of these worlds of deterministic and stochastic evolutions with the world of quantum phenomena is also briefly illustrated. The survey spans a bridge from basic concepts and methods in these areas to recent developments concerning their relations.

Citation: Sergio Albeverio, Sonia Mazzucchi. Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena. Journal of Geometric Mechanics, 2019, 11 (2) : 123-137. doi: 10.3934/jgm.2019006
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show all references

References:
[1]

S. AlbeverioP. Blanchard and R. Høegh-Krohn, Feynman path integrals and the trace formula for the Schrödinger operators, Communications in Mathematical Physics, 83 (1982), 49-76. doi: 10.1007/BF01947071.

[2]

S. AlbeverioP. Blanchard and R. Høegh-Krohn, A stochastic model for the orbits of planets and satellites: an interpretation of Titius-Bode law, Expositiones Mathematicae. International Journal for Pure and Applied Mathematics, 1 (1983), 365-373.

[3]

S. Albeverio, N. Cangiotti and S. Mazzucchi, On the mathematical definition of Feynman path integrals for the Schrodinger equation with magnetic field, In preparation.

[4]

S. AlbeverioN. Cangiotti and S. Mazzucchi, Generalized Feynman path integrals and applications to higher-order heat-type equations, Expositiones Mathematicae, 36 (2018), 406-429. doi: 10.1016/j.exmath.2018.09.001.

[5]

S. Albeverio, B. K. Driver, M. Gordina and A. M. Vershik, Equivalence of the Brownian and energy representations, Rossiĭskaya Akademiya Nauk. Sankt-Peterburgskoe Otdelenie. Matematicheskiĭ Institut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI), 441 (2015), 17–44. doi: 10.1007/s10958-016-3134-1.

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S. Albeverio and S. Kusuoka, The invariant measure and the flow associated to the $\phi^4_3$-quantum field, Annali della Scuola Normale Superiore di Pisa – Classe die Scienze, 2018, p61. doi: 10.2422/2036-2145.201809_008.

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S. Albeverio and S. Mazzucchi, Feynman path integrals for polynomially growing potentials, Journal of Functional Analysis, 221 (2005), 83-121. doi: 10.1016/j.jfa.2004.07.014.

[10]

S. Albeverio and S. Mazzucchi, Generalized Fresnel integrals, Bulletin des Sciences Mathématiques, 129 (2005), 1-23. doi: 10.1016/j.bulsci.2004.05.005.

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[12]

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[13]

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