Stochastic manifolds

TL;DR

This paper introduces stochastic manifolds, extending Malliavin Calculus to a geometric framework analogous to classical differential calculus on manifolds, with paths in Riemannian manifolds as a key example.

Contribution

It defines stochastic manifolds where Malliavin Calculus acts as differential calculus, generalizing the geometric structure of path spaces in Riemannian manifolds.

Findings

Defines stochastic manifolds with Malliavin Calculus as differential calculus

Shows paths in Riemannian manifolds form a special case

Establishes a geometric framework for stochastic analysis

Abstract

Malliavin Calculus can be seen as a differential calculus on Wiener spaces. We present the notion of stochastic manifold for which the Malliavin Calculus plays the same role as the classical differential calculus for the differential manifolds. The set of the paths in a Riemmanian compact manifold is then seen as a particular case of the above structure.