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An Application of a Functional Inequality to Quasi-Invariance in Infinite Dimensions

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An Application of a Functional Inequality to Quasi-Invariance in Infinite Dimensions An Application of a Functional Inequality to Quasi-Invariance in Infinite Dimensions Maria Gordina Abstract One way to interpret smoothness of a measure in infinite dimensions is quasi-invariance of the measure under a class of transformations. Usually such settings lack a reference measure such as the Lebesgue or Haar measure, and therefore we cannot use smoothness of a density with respect to such a measure. We describe how a functional inequality can be used to prove quasi-invariance results in several settings. In particular, this gives a different proof of the classical Cameron- Martin (Girsanov) theorem for an abstract Wiener space. In addition, we revisit several more geometric examples, even though the main abstract result concerns quasi-invariance of a measure under a group action on a measure space. Keywords and phrases Quasi-invariance • Group action • Functional inequalities 1991 Mathematics Subject Classification. Primary 58G32 58J35; Secondary 22E65 22E30 22E45 58J65 60B15 60H05. 1 Introduction Our goal in this paper is to describe how a functional inequality can be used to prove quasi-invariance of certain measures in infinite dimensions. Even though the original argument was used in a geometric setting, we take a slightly different approach in this paper. Namely, we formulate a method that can be used to prove quasi- invariance of a measure under a group action. Such methods are useful in infinite dimensions when usually there is no natural reference measure such as the Lebesgue measure. At the same time quasi-invariance of measures is a useful tool in proving regularity results when it is reformulated as an integration by parts formula. We do not discuss significance of such results, and M. Gordina () Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA e-mail: [email protected] © Springer Science+Business Media LLC 2017 251 E. Carlen et al. (eds.), Convexity and Concentration, The IMA Volumes in Mathematics and its Applications 161, DOI 10.1007/978-1-4939-7005-6_8 [email protected] 252 M. Gordina moreover we do not refer to the extensive literature on the subject, as it is beyond the scope of our paper. We start by describing an abstract setting of how finite-dimensional approxi- mations can be used to prove such a quasi-invariance. In [11] this method was applied to projective and inductive limits of finite-dimensional Lie groups acting on themselves by left or right multiplication. In that setting a functional inequality (integrated Harnack inequality) on the finite-dimensional approximations leads to a quasi-invariance theorem on the infinite-dimensional group space. As we pointed out in [11], this is an abstraction of results in [9] for loop groups, and while there were probably earlier results of similar flavor, the most relevant later publications include [1] and [13]. Similar methods were used in the elliptic setting on infinite- dimensional Heisenberg-like groups in [10], and on semi-infinite Lie groups in [17]. Note that the assumptions we make below in Section 3 have been verified in these settings, including the sub-elliptic case for infinite-dimensional Heisenberg group in [2]. Even though the integrated Harnack inequality we use in these situations has a distinctly geometric flavor, we show in this paper that it does not have to be. The paper is organized as follows. The general setting is described in Sections 2 and 3, where Theorem 3.2 is the main result. One of the ingredients for this result is quasi-invariance for finite-dimensional approximations which is described in Section 3. We review the connection between an integrated Harnack inequality and Wang’s Harnack inequality in Section 4. Finally, Section 5 gives several examples of how one can use Theorem 3.2. We describe in detail the case of an abstract Wiener space, where the group in question is identified with the Cameron-Martin subspace acting by translation on the Wiener space. In addition we discuss elliptic (Riemannian) and sub-elliptic (sub-Riemannian) infinite-dimensional groups which are examples of a subgroup acting on the group by multiplication. 2 Notation Suppose G is a topological group with the identity e, X is a topological space, .X; B;/ is a measure space, where B is the Borel -algebra, and is a probability measure. We assume that G is endowed with the structure of a Hilbert Lie group (e.g., [8]), and further that its Lie algebra g WD Lie .G/ D TeG is equipped with a Hilbertian inner product, h; i. The corresponding distance on G is denoted by d .; /. In addition, we assume that G is separable, and therefore we can use what is known about Borel actions of Polish groups [3, 4]. Once we have an inner product on the Lie algebra g, we can define the length of a path in G as follows. Suppose k 2 C1 .Œ0; 1; G/, k .0/ D e, then Z 1 P lG .k .// WD jLk.t/1k .t/ jdt; (2.1) 0 where Lg is the left translation by g 2 G. [email protected] An Application of a Functional Inequality to Quasi-Invariance in Infinite Dimensions 253 We assume that G acts measurably on X, that is, there is a (Borel) measurable map ˆ W G X ! X such that ˆ .e; x/ D x; for all x 2 X; ˆ .g1;ˆ.g2; x// D ˆ .g1g2; x/ ; for all x 2 X; g1; g2 2 G: We often will use ˆg WD ˆ .g; / for g 2 G. Definition 2.1 Suppose ˆ is a measurable group action of G on X. (1) In this case we denote by ˆg the pushforward measure defined by 1 B ˆg .A/ WD ˆ g ; A ; for all A 2 ; g 2 GI (2) the measure is invariant under the action ˆ if ˆg D for all g 2 GI (3) the measure is quasi-invariant with respect to the action ˆ if ˆg and are mutually absolutely continuous for all g 2 G. Notation 2.2 For a topological group G acting measurably on the measure space .X; B;/ in such a way that is quasi-invariant under the action by G, the Radon- Nikodym derivative of ˆg with respect to is denoted by ˆg .dx/ J .x/ WD for all g 2 G; x 2 X: g .dx/ For a thorough discussion of the Radon-Nikodym derivative in this setting we refer to [6, Appendix D] 3 Finite-Dimensional Approximations and Quasi-Invariance We start by describing approximations to both the group G and the measure space .X; B;/. At the end we also need to impose certain conditions to have consistency of the group action defined on these approximations. As X is a topological space, we denote by Cb .X/ the space of continuous bounded real-valued functions. Assumption 1 (Lie group assumptions). Suppose fGngn2N is a collection of finite- dimensional unimodular Lie subgroups of G such that Gn Gm for all n < m. We assume that there exists a smooth section fsn W G ! Gngn2N, thatS is, sn ı in D idGn , where in W Gn ! G is the smooth injection. We suppose that n2N Gn is a dense subgroup of G. In addition, we assume that the length of a path in G can be 1 approximated by the lengths in Gn, namely, if k 2 C .Œ0; 1; G/,k.0/ D e, then [email protected] 254 M. Gordina lG .k .// D lim lG .sn .k ./// : (3.1) n!1 n Note that sn does not have to be a group homomorphism. Assumption 2 (Measure space assumptions). We assume that X is a separable topological space with a sequence of topological spaces Xn X which come with corresponding continuous maps n W X ! Xn satisfying the following properties. For any f 2 Cb .X/ Z Z fd D lim f ı jndn; (3.2) n!1 X Xn where jn W Xn ! X is the continuous injection map, and n is the pushforward measure .n/ . Our last assumption concerns the group action for these approximations. Assumption 3 (Group action assumptions). The approximations to group G and the measure space .X;/ are consistent with the group action in the following way ˆ .Gn Xn/ Xn for each n 2 N; ˆg W X ! X is a continuous map for each g 2 G: n n We denote by ˆ the restriction of ˆ to Gn Xn. Observe that ˆ D ˆ ı .in; jn/ which together with Assumption 3, it is clear that ˆn is a measurable group action of Gn on .Xn; Bn;n/. n n Suppose now that n is quasi-invariant under the group action ˆ , and let Jg be the Radon-Nikodym derivative ˆn with respect to . We assume that there g n n is a positive constant C D C .p/ such that for any p 2 Œ1; 1/ and g 2 Gn n 2 J p 6 C .p/ d .e; g/ : k g kL .Xn;n/ exp Gn (3.3) Note that the constant C .p/ does not depend on n. Remark 3.1 The fact that this estimate is Gaussian (with the square of the distance) does not seem to be essential. But as we do not have examples with a different exponent, we leave (3.3) as is. Moreover, we could consider a more general function on the right-hand side than an exponential of the distance squared. Theorem 3.2 (Quasi-invariance of ) Suppose we have a group G and a measure space .X; B;/ satisfying Assumptions 1, 2, and 3.
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