Analysis and Probability on Infinite-Dimensional Spaces

Abstract

I wrote these lecture notes for a graduate topics course I taught at Cornell University in Fall 2011 (Math 7770). The ostensible primary goal of the course was for the students to learn some of the fundamental results and techniques in the study of probability on infinite-dimensional spaces, particularly Gaussian measures on Banach spaces (also known as abstract Wiener spaces). As others who have taught such courses will understand, a nontrivial secondary goal of the course was for the instructor (i.e., me) to do the same. These notes only scratch the very surface of the subject, but I tried to use them to work through some of the basics and see how they fit together into a bigger picture. In addition to theorems and proofs, I've left in some more informal discussions that attempt to develop intuition.Most of the material here comes from the books [14, 16, 2], and the lecture notes prepared by Bruce Driver for the 2010 Cornell Probability Summer School [4,5]. If you are looking to learn more, these are great places to look. 1 Any text marked Question N is something that I found myself wondering while writing this, but didn't ever resolve. I'm not proposing them as open problems; the answers could be well-known, just not by me. If you know the answer to any of them, I'd be happy to hear about it! There are also still a few places where proofs are rough or have some gaps that I never got around to filling in.On the other hand, something marked Exercise N is really meant as an exercise. I would like to take this opportunity to thank the graduate students who attended the course. These notes were much improved by their questions and contributions. I'd also like to thank several colleagues who sat in on the course or otherwise contributed to these notes, particularly Clinton Conley, Bruce Driver, Leonard Gross, Ambar Sengupta, and Benjamin Steinhurst. Obviously, the many deficiencies in these notes are my responsibility and not theirs.Questions and comments on these notes are most welcome. I am now at the University of Northern Colorado, and you can email me at

Functional Variational Bayesian Neural Networks

“…Here KL[q p] is the KL divergence between two stochastic processes. As pointed out in Matthews et al (2016), it does not have a convenient form as log q(f ) p(f ) q(f )df due to there is no infinitedimensional Lebesgue measure (Eldredge, 2016). Since the KL divergence between stochastic processes is difficult to work with, we reduce it to a more familiar object: KL divergence between the marginal distributions of function values at finite sets of points, which we term measurement sets.…”

Quantum Field Theory and Functional Integrals

“…where F is a force acting on the particle with mass m. Hence, the trajectories of motion are given by solutions of (3). We note that (3) is a system of second order ordinary differential equations and is nonlinear in general 3 .…”

“…Thus, in Newtonian mechanics, we are interested in solving the equation (3). One way to try to solve (3) would be to try to find conserved quantities which may help simplifying the problem.…”

“…Proposition 2.1.1. Suppose a particle moving on R n satisfying Newton's law of the form (3). Then (6) d dt E(x(t), ẋ(t)) = 0, along any trajectory x(t) satisfying (3).…”

“…This procedure is mathematically well-defined. These notes are based on [1,2,3,5,6,7,8,9,10,11,12,13,14,15].…”