Researches in Stochastic Analysis

Fundamentally this book presents an analytic point of view on the probability theory of processes and emphasizes the strong connections between classical Potential Theory and Brownian motion. Since well-known superharmonic functions and supermartingales are so closely related that the filtering theory of processes can be seen as a particular Potential theory, they are analyzed here by the use of a common framework. This framework is the space L¹(c) where c is a capacity, that is, roughly speaking, a sublinear functional, as first considered by B. Fuglede. An associated notion of quasi-topology is defined that gives a very good account of the classical quasi-continuous functions in the finite or the infinite dimensional case of the Malliavin calculus on a Wiener space.

Methods and definitions differ from the usual ones. The white noise calculus of T. Hida is also considered. The existence of the Hausdorff measure of finite codimension on the Wiener space is established with the corresponding Federer coarea formula. Two chapters are devoted to the determinist calculus initiated by T. Lyons. It is based on a « sewing lemma » which leads to an easy proof of the existence and uniqueness of the solution of a differential equation driven by α-Holder signal greater than 1/3

This book concerns mathematicians who use stochastic methods in their research. It should also be of interest to students having some notions of functional analysis and of probability theory or at least of measure theory.

Denis Feyel