Malliavin Calculus and Euclidean Quantum Mechanics II. Variational Principle for Infinite Dimensional Processes

A class of time reversible non-stationary diffusion processes with values on the classical Wiener space is constructed. These processes should be relevant to ("2-dimensional") Euclidean quantum field theory since they generalize those constructed before for non-relativistic quantum mechanics, along the lines of a strategy suggested by Schrِdinger. Those processes are shown to be characterized by a stochastic variational principle and provide a probabilistic representation of the solutions of some infinite-dimensional heat equation. The Feynman-Kac formula on the Wiener space needed for this construction is also proved.