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.
