Hausdorff measures on the Wiener space

We construct a Hausdorff measure of finite co-dimension on the Wiener space. We then extend the Federer co-area Formula to this Wiener space for functions with the sole condition that they belong to the first Sobolev space. An explicit formula for the density of the images of the Wiener measure under such functions follows naturally from this. As a corollary, this yields a new and easy proof of the Krée-Watanabe theorem concerning the regularity of the images of the Wiener measure.     

Hausdorff dimension and doubling measures on metric spaces

Volberg and Konyagin have proved that a compact metric space carries a nontrivial doubling measure if and only if it has finite uniform metric dimension. Their construction of doubling measures requires infinitely many adjustments. We give a simpler and more direct construction, and also prove that for any , the doubling measure may be chosen to have full measure on a set of Hausdorff dimension at most .

     

Orlicz capacities and Hausdorff measures on metric spaces

In the setting of doubling metric measure spaces with a 1-Poincaré inequality, we show that sets of Orlicz Φ-capacity zero have generalized Hausdorff h-measure zero provided thatwhere Θ⁻¹ is the inverse of the function Θ(t)=Φ(t)/t, and s is the “upper dimension” of the metric measure space. This condition is a generalization of a well known condition in Rⁿ. For spaces satisfying the weaker q-Poincaré inequality, we obtain a similar but slightly more restrictive condition. Several examples are also provided.     

On the Hausdorff and packing measures of typical compact metric spaces

We study the Hausdorff and packing measures of typical compact metric spaces belonging to the Gromov–Hausdorff space (of all compact metric spaces) equipped with the Gromov–Hausdorff metric.     

Entropy of the space of twice smooth curves in the Hausdorff metric

Bound from above and below for the entropy of the space of twice smooth curves on a plane in the Hausdorff metric are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 113–118, January, 1990.     

Borel complexity of the space of probability measures

Using a technique developed by Louveau and Saint Raymond, we find the complexity of the space of probability measures in the Borel hierarchy: if is any non-Polish Borel subspace of a Polish space, then , the space of probability Borel measures on with the weak topology, is always true , where is the least ordinal such that is .

     

Computation of the ε entropy of the space of continuous functions with the Hausdorff metric

The number K_p,q, i.e., the number of (p, q) corridors of closed domains which are convex in the vertical direction, consist of elementary squares of the integral lattice, are situated within a rectangle of the size q × p, and completely cover the side of length p of this rectangle under projection is computed. The asymptotic (K_p,q/q²)^1/p → λ, as p, q → ∞, where λ = 0.3644255… is the maximum root of the equation₁F₁(-1/2 − 1/(16λ), 1/2, 1/(4λ)) = 0,₁F₁ being the confluence hypergeometric function, is established. These results allow us to compute the ε entropy of the space of continuous functions with the Hausdorff metric. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 39–50, January, 1977.     

A free probability analogue of the Wasserstein metric on the trace-state space

We define a free probability analogue of the Wasserstein metric, which extends the classical one. In dimension one, we prove that the square of the Wasserstein distance to the semi-circle distribution is majorized by a modified free entropy quantity. Submitted: August 2000.     

Hamiltonian ODEs in the Wasserstein space of probability measures

In this paper we consider a Hamiltonian H on ??₂(?^2d), the set of probability measures with finite quadratic moments on the phase space ?^2d = ?^d × ?^d, which is a metric space when endowed with the Wasserstein distance W₂. We study the initial value problem dμ_t/dt + ? · (??_d v _tμ_t) = 0, where ??_d is the canonical symplectic matrix, μ₀ is prescribed, and v _t is a tangent vector to ??₂(?^2d) at μ_t, belonging to ?H(μ_t), the subdifferential of H at μ_t. Two methods for constructing solutions of the evolutive system are provided. The first one concerns only the case where μ₀ is absolutely continuous. It ensures that μ_t remains absolutely continuous and v _t = ?H(μ_t) is the element of minimal norm in ?H(μ_t). The second method handles any initial measure μ₀. If we further assume that H is λ‐convex, proper, and lower‐semicontinuous on ??₂(?^2d), we prove that the Hamiltonian is preserved along any solution of our evolutive system, H(μ_t) = H(μ₀). © 2007 Wiley Periodicals, Inc.     

Entropy characteristics of spaces of probability measures on a metric compactum

In the paper one obtains the asymptotic behavior of the finite-dimensional diameters and of the -entropy for a space of probability measures on the compactum K=[0,1] and one gives upper bounds for these characteristics for an arbitrary metric compactum.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 65–68, 1983.     

On best-approximation polynomials in the Hausdorff metric

A definition of the Hausdorff alternance is given. In its terms we give a sufficient condition for an algebraic polynomial to have minimal deviation from a function f in the Hausdorff α-metric. A condition under which a polynomial P_n is the unique best-approximation polynomial for a function f and a necessary condition for P_n to have minimal deviation from f are given. Similar theorems for 2π-periodic functions are formulated. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 130–143.     

Singular measures and Hausdorff measures

An example is given of a family of singular probability measures on the unit interval which are supported on a set of fractional Hausdorff dimension but cannot be represented as Hausdorff measures.     

Topological dynamics of transformations induced on the space of probability measures

LetT be a continuous transformation of a compact metric spaceX. T induces in a natural way a transformationT _M on the spaceM (X) of probability measures onX, and a transformationT _K on the spaceK (X) of closed subsets ofX. This note investigates which of the topological properties ofT∶X→X (like distality, transitivity, mixing property etc. ...) are “inherited” byT _M∶M (X)→M (X) andT _K∶K (X)→K (X).     

Limit sets of dynamical systems on the space of probability measures

Let m be a dynamical system on the space of probability measures $\text{M}$₁(R^d), and let Λ + (?) be the positive limit set for ? ∈ $\text{M}$₁(R^d), where ? has compact support K ?R^d. The main result of this paper states that support of Λ⁺(?) ?

,support of Λ + (δ_x), where δ_x is the Dirac measure at point x.     

A note on the Poisson structures of the space of probability measures

The space of probability measures on a Riemannian manifold is endowed with the Fisher information metric. In [4] T. Friedrich showed that this space admits also Poisson structures {, }. In this note, we give directly another proof for the structure {, } being Poisson. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Invariant measures for equicontinuous semigroups of continuous transformations of a compact Hausdorff space

Equicontinuous semigroups of transformations of a compact Hausdorff space and their sets of all invariant (Borel, regular and probabilistic) measures are studied. Conditions equivalent to the existence of at least one invariant measure are given. The (algebraic and topological) structure of the set of invariant measures is researched.