Contributions to the theory of Gaussian Measures and Processes with Applications

This thesis studies infinite dimensional Gaussian measures on Banach spaces. Let $\mu_0$ be a centered Gaussian measure on Banach space $\mathcal B$, and $\mu^\ast$ is a measure equivalent to $\mu_0$. We are interested in approximating, in sense of relative entropy (or KL divergence) the quantity $\frac{d\mu^z}{d\mu^\ast}$ where $\mu^z$ is a mean shift measure of $\mu_0$ by an element $z$ in the so-called ``Cameron-Martin" space $\mathcal H_{\mu_0}$. That is, we want to find the information projection

$$\inf_{z\in \mathcal H_{\mu_0}} D_{KL}(\mu^z||\mu_0)=\inf_{z\in \mathcal H_{\mu_0}} E_{\mu^z} \left(\log \left(\frac{d\mu^z}{d\mu^\ast}\right)\right).$$

We relate this information projection to a mode computation, to an ``open loop" control problem, and to a variational formulation leading to an Euler-Lagrange equation. Furthermore, we use this relationship to establish a kind of Feynman-Kac theorem for systems of ordinary differential equations. We demonstrate that the solution to a system of second order linear ordinary differential equations is the mode of a diffusion, analogous to the result of Feynman-Kac for parabolic partial differential equations.