# American Institute of Mathematical Sciences

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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