Contributions to the Theory of Gaussian Measures and Processes with Applications

Zachary Selk, Purdue University

Abstract

This thesis studies infinite dimensional Gaussian measures on Banach spaces. Let µ0 be a centered Gaussian measure on Banach space B, and µ∗ is a measure equivalent to µ0. We are interested in approximating, in sense of relative entropy (or KL divergence) the quantity dµz/dµ∗ where µz is a mean shift measure of µ0 by an element z in the so-called “Cameron-Martin” space Hµ0. That is, we want to find the information projection 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.

Degree

Ph.D.

Advisors

Honnappa, Purdue University.

Subject Area

Mathematics|Quantum physics

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