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

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 can not 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.1991 Mathematics Subject Classification. Primary 58G32 58J35; Secondary 22E65 22E30 22E45 58J65 60B15 60H05.1 2 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 approximations can be used to prove such a quasi-invariance. In [9] 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. Similar methods were used in the elliptic setting on infinite-dimensional Heisenberg-like groups in [8], and on semi-infinite Lie groups in [14]. 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 [1]. Even though the integrated Harnack inequality we use in these situations have 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 Section 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.Acknowledgement. The author is grateful to Sasha Teplyaev and Tom Laetsch for useful discussions and helpful comments.

Quasi-invariance for infinite-dimensional Kolmogorov diffusions

“…We now state the Cameron-Martin type quasi-invariance result for ν t = Law(X t ). We prove this as an application of the main theorem in [11]. However, to see how a direct proof would work, the reader may see the proof of Theorem 3.8 in the next subsection.…”

“…Also, in the case of the generalized Kolmogorov diffusions we consider, there is no underlying group structure and so a shift in the initial state of the process does not correspond to any group action. Thus the results in the current paper can not be proven by directly applying the approach developed in [10,11].…”

“…Mariano study gradient bounds and other related functional inequalities for Kolmogorov-type diffusions in finite dimensions via both the generalized Γ-calculus techniques employed in the present paper as well as through coupling methods. The method we follow for proving quasi-invariance for measures in infinite dimensions via functional inequalities was developed in [5,10,11]. This approach relies on certain inequalities holding with dimension-independent coefficients on appropriate finite-dimensional approximations.…”

“…We first study quantitative functional inequalities for appropriate finite-dimensional approximations of these diffusions, and we prove these inequalities hold with dimension-independent coefficients. Applying an approach developed in [5, 10,11], these uniform bounds may then be used to prove that the heat kernel measure for certain of these infinite-dimensional diffusions is quasi-invariant under changes of the initial state. Contents 1.…”

Characteristic functionals of Dirichlet measures

“…Whereas Theorem 1.1 allows for Bochner-Minlos and Lévy Continuity related results to come into play, the non-multiplicativity of D σ (corresponding to the non-infinite-divisibility of the measure) immediately rules out the usual approach to quasi-invariance via Fourier transforms [2,20,42,43]. Other approaches to this problem rely on finite-dimensional approximation techniques, variously concerned with approximating the space [34,35], the σ-algebra [20] or the acting group [11,45]. The common denominator here is for the approximation to be a filtration (cf.…”