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

Abstract

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.