COVER OF THE BROWNIAN BRIDGE AND STOCHASTIC SYMPLECTIC ACTION

Abstract

We define the universal cover of the Brownian bridge of a symplectic manifold. This allows us to define a non-trivial functional over it called the stochastic symplectic action, and to define local Sobolev spaces over the universal cover, such that the symplectic action belong to them. This is done in the purpose of an analytical Morse theory in the sense of Witten over the loop space associated to this symplectic action. In this purpose, a stochastic Witten complex is constructed over the universal cover of the loop space, and modulo some weights, it is shown that its cohomology is equal to the stochastic cohomology of the universal cover.