Gideon R. Chiusole

The goal of this thesis is to set up a measure theoretic and functional analytic framework for a differential and integral calculus on infinite-dimensional topological vector spaces (TVSs). In the introductory part, we make some naive approaches and immediately see why they are doomed to fail. We give further motivation from pure mathematics, physics, and financial economics.

In the second part, we will introduce the basic notions of measures on locally convex TVSs and consider the problem of choosing "the right" sigma-algebra. We then define Gaussian measures and discuss the celebrated Theorem of Fernique and its consequences. After that we study the associated Cameron--Martin space and consider the example of the finite-dimensional real space R^n and the classical Wiener space. In the last part of the chapter we state and prove the Theorems of Cameron and Martin and summarize the theory in its most natural setting, separable Frechet spaces.

Finally, we consider the dual viewpoint, stemming from Quantum Field Theory, in which we start from a formal density w.r.t. a (hypothetical) Lebesgue measure and subsequently develop the corresponding functional analytic framework. In the final chapter we employ parts of the theory to obtain a generalized version of the classical Theorem of Schilder from the Theory of Large Deviations.

Fractional Brownian motion (fBM) is a one parameter generalization of Brownian motion which can be seen as the convolution of white noise with a power kernel t^(H - 1/2), splitting fBM into three quite distinct classes: 0 < H < 1/2, H = 1/2, and 1/2 < H < 1. Originally, fBM was introduced by B. Mandelbrot and J. Van Ness as a continuous time model for a long-range dependent stochastic process, specifically for the study of economics, hydraulics, and fluctuation in solids. From a probabilistic point of view, fBM is particularly interesting since it is neither a Markov process nor a semi-martingale. We will show both of these results alongside some other probabilistic and analytic properties.