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Analysis and Probability on Infinite-Dimensional Spaces

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Analysis and Probability on Infinite-Dimensional Spaces Analysis and Probability on Infinite-Dimensional Spaces Nathaniel Eldredge September 8, 2016 Preface 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 arXiv:1607.03591v2 [math.PR] 7 Sep 2016 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 [email protected]. 1In reading these notes in conjunction with [16], you should identify Nualart’s abstract probability space (Ω, F, P ) with our Banach space (W, B, µ). His “Gaussian process” h 7→ W (h) should be viewed as corresponding to our map T defined in Section 4.3; his indexing Hilbert space H may be identified with the Cameron–Martin space, and his W (h) is the random variable, defined on the Banach space, that we have denoted by T h or hh, ·i. There’s a general principle in this area that all the “action” takes place on the Cameron–Martin space, so one doesn’t really lose much by dropping the Banach space structure on the space W and replacing it with an generic Ω (and moreover generality is gained). Nonetheless, I found it helpful in building intuition to work on a concrete space W ; this also gives one the opportunity to explore how the topologies of W and H interact. 1 1 Introduction 1.1 Why analysis and probability on Rn is nice Classically, real analysis is usually based on the study of real-valued functions on finite-dimensional Euclidean space Rn, and operations on those functions involving limits, differentiation, and inte- gration. Why is Rn such a nice space for this theory? Rn is a nice topological space, so limits behave well. Specifically, it is a complete separable • metric space, and it’s locally compact. Rn has a nice algebraic structure: it’s a vector space, so translation and scaling make sense. • This is where differentiation comes in: the derivative of a function just measures how it changes under infinitesimal translation. Rn has a natural measure space structure; namely, Lebesgue measure m on the Borel σ- • algebra. The most important property of Lebesgue measure is that it is invariant under translation. This leads to nice interactions between differentiation and integration, such as integration by parts, and it gives nice functional-analytic properties to differentiation opera- tors: for instance, the Laplacian ∆ is a self-adjoint operator on the Hilbert space L2(Rn,m). Of course, a lot of analysis only involves local properties, and so it can be done on spaces that are locally like Rn: e.g. manifolds. Let’s set this idea aside for now. 1.2 Why infinite-dimensional spaces might be less nice The fundamental idea in this course will be: how can we do analysis when we replace Rn by an infinite dimensional space? First we should ask: what sort of space should we use? Separable Banach spaces seem to be nice. They have a nice topology (complete separable metric spaces) and are vector spaces. But what’s missing is Lebesgue measure. Specifically: Theorem 1.1. “There is no infinite-dimensional Lebesgue measure.” Let W be an infinite- dimensional separable Banach space. There does not exist a translation-invariant Borel measure on W which assigns positive finite measure to open balls. In fact, any translation-invariant Borel measure m on W is either the zero measure or assigns infinite measure to every open set. Proof. Essentially, the problem is that inside any ball B(x, r), one can find infinitely many disjoint balls B(xi,s) of some fixed smaller radius s. By translation invariance, all the B(xi,s) have the same measure. If that measure is positive, then m(B(x, r)) = . If that measure is zero, then we ∞ observe that W can be covered by countably many balls of radius s (by separability) and so m is the zero measure. The first sentence is essentially Riesz’s lemma: given any proper closed subspace E, one can find a point x with x 1 and d(x, E) > 1/2. (Start by picking any y / E, so that d(y, E) > 0; || || ≤ ∈ then by definition there is an z E with d(y, z) < 2d(y, E). Now look at y z and rescale as ∈ − needed.) Now let’s look at B(0, 2) for concreteness. Construct x1,x2,... inductively by letting E = span x ,...,x (which is closed) and choosing x as in Riesz’s lemma with x 1 n { 1 n} n+1 || n+1|| ≤ and d(x , E ) > 1/2. In particular, d(x ,x ) > 1/2 for i n. Since our space is infinite n+1 n n+1 i ≤ dimensional, the finite-dimensional subspaces En are always proper and the induction can continue, producing a sequence x with d(x ,x ) > 1/2 for i = j, and thus the balls B(x , 1/4) are pairwise { i} i j 6 i disjoint. 2 Exercise 1.2. Prove the above theorem for W an infinite-dimensional Hausdorff topological vector space. (Do we need separability?) A more intuitive idea why infinite-dimensional Lebesgue measure can’t exist comes from con- sidering the effect of scaling. In Rn, the measure of B(0, 2) is 2n times larger than B(0, 1). When n = this suggests that one cannot get sensible numbers for the measures of balls. ∞ There are nontrivial translation-invariant Borel measures on infinite-dimensional spaces: for instance, counting measure. But these measures are useless for analysis since they cannot say anything helpful about open sets. So we are going to have to give up on translation invariance, at least for now. Later, as it turns out, we will study some measures that recover a little bit of this: they are quasi-invariant under some translations. This will be explained in due course. 1.3 Probability measures in infinite dimensions If we just wanted to think about Borel measures on infinite-dimensional topological vector spaces, we actually have lots of examples from probability, that we deal with every day. Example 1.3. Let (Ω, , P) be a probability space and X , X ,... a sequence of random variables. F 1 2 Consider the infinite product space R , thought of as the space of all sequences x(i) of real ∞ { }i∞=1 numbers. This is a topological vector space when equipped with its product topology. We can equip R∞ with its Borel σ-algebra, which is the same as the product Borel σ-algebra (verify). Then the map from Ω to R∞ which sends ω to the sequence x(i) = Xi(ω) is measurable. The pushforward of P under this map gives a Borel probability measure µ on R∞. The Kolmogorov extension theorem guarantees lots of choices for the joint distribution of the Xi, and hence lots of probability measures µ. Perhaps the simplest interesting case is when the Xi are iid with distribution ν, in which case µ is the infinite product measure µ = i∞=1 ν. Note that in general one can only take the infinite product of probability measures (essentially because the Q only number a with 0 < ∞ a< is a = 1). i=1 ∞ Example 1.4. Let (Ω, Q, P) be a probability space, and X : 0 t 1 be any stochastic process. F { t ≤ ≤ } We could play the same game as before, getting a probability measure on R[0,1] (with its product σ-algebra). This case is not as pleasant because nothing is countable. In particular, the Borel σ-algebra generated by the product topology is not the same as the product σ-algebra (exercise: verify this, perhaps by showing that the latter does not contain singleton sets.) Also, the product [0,1] topology on R is rather nasty; for example it is not first countable.
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