On a measure in Wiener space and applications

In this article, we consider a measure in Wiener space, induced by the sum of measures associated with an uncountable set of positive real numbers, and investigate the basic properties of this measure. We apply this measure to the various theories related to Wiener space. In particular, we can obtain a partial answer to Johnson and Skoug's open problems, raised in their 1979 paper. Moreover, we can improve and clarify some theories related to Wiener space.

     

Ergodicity of canonical Gibbs measures with respect to the diffeomorphism group

For general potentials we prove that every canonical Gibbs measure on configurations over a manifold X is quasi‐invariant w.r.t. the group of diffeomorphisms on X. We show that this quasi‐invariance property also characterizes the class of canonical Gibbs measures. From this we conclude that the extremal canonical Gibbs measures are just the ergodic ones w.r.t. the diffeomorphism group. Thus we provide a whole class of different irreducible representations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Construction of Gibbs measures for 1-dimensional continuum fields

We study 1-dimensional continuum fields of Ginzburg-Landau type under the presence of an external and a long-range pair interaction potentials. The corresponding Gibbs states are formulated as Gibbs measures relative to Brownian motion [17]. In this context we prove the existence of Gibbs measures for a wide class of potentials including a singular external potential as hard-wall ones, as well as a non-convex interaction. Our basic methods are: (i) to derive moment estimates via integration by parts; and (ii) in its finite-volume construction, to represent the hard-wall Gibbs measure on C(ℝ;ℝ⁺) in terms of a certain rotationally invariant Gibbs measure on C(ℝ;ℝ³).     

Ray Hölder-continuity for fractional Sobolev spaces in infinite dimensions and applications

We prove H?lder-continuity on rays in the direction of vectors in the (generalized) Cameron-Martin space for functions in Sobolev spaces in L ^p of fractional order α∈ (, 1) over infinite dimensional linear spaces. The underlying measures are required to satisfy some easy standard structural assumptions only. Apart from Wiener measure they include Gibbs measures on a lattice and Euclidean interacting quantum fields in infinite volume. A number of applications, e.g., to the two-dimensional polymer measure, are presented. In particular, irreducibility of the Dirichlet form associated with the latter measure is proved without restrictions on the coupling constant. Received: 9 November 1998 / Published online: 30 March 2000     

Gibbs Measures on Mutually Interacting Brownian Paths under Singularities

We are interested in the analysis of Gibbs measures defined on two independent Brownian paths in ?^d interacting through a mutual self‐attraction. This is expressed by the Hamiltonian with two probability measures μ and ν representing the occupation measures of two independent Brownian motions. We will be interested in a class of potentials V that are singular , e.g., Dirac‐ or Coulomb‐type interactions in ?³, or the correlation function of the parabolic Anderson problem with white noise potential. The mutual interaction of the Brownian paths inspires a compactification of the quotient space of orbits of product measures, which is structurally different from the self‐interacting case introduced in [27], owing to the lack of shift‐invariant structure in the mutual interaction. We prove a strong large‐deviation principle for the product measures of two Brownian occupation measures in such a compactification and derive asymptotic path behavior under Gibbs measures on Wiener paths arising from mutually attracting singular interactions. For the spatially smoothened parabolic Anderson model with white noise potential, our analysis allows a direct computation of the annealed Lyapunov exponents, and a strict ordering of them implies the intermittency effect present in the smoothened model. © 2017 Wiley Periodicals, Inc.     

Gibbs <Emphasis Type="Italic">u</Emphasis>-states for the foliated geodesic flow and transverse invariant measures

This paper is devoted to the study of Gibbs u-states for the geodesic flow tangent to a foliation F of a manifold M having negatively curved leaves. By definition, they are the probability measures on the unit tangent bundle to the foliation that are invariant under the foliated geodesic flow and have Lebesgue disintegration in the unstable manifolds of this flow. p]On the one hand we give sufficient conditions for the existence of transverse invariant measures. In particular we prove that when the foliated geodesic flow has a Gibbs su-state, i.e. an invariant measure with Lebesgue disintegration both in the stable and unstable manifolds, then this measure has to be obtained by combining a transverse invariant measure and the Liouville measure on the leaves. p]On the other hand we exhibit a bijective correspondence between the set of Gibbs u-states and a set of probability measure on M that we call φ ^u -harmonic. Such measures have Lebesgue disintegration in the leaves and their local densities have a very specific form: they possess an integral representation analogue to the Poisson representation of harmonic functions.     

On <Emphasis Type="Italic">p</Emphasis>-adic quasi Gibbs measures for <Emphasis Type="Italic">q</Emphasis> + 1-state Potts model on the Cayley tree

In the present paper we introduce a new kind of p-adic measures, associated with q + 1-state Potts model, called p-adic quasi Gibbs measure, which is totally different from the p-adic Gibbs measure. We establish the existence of p-adic quasi Gibbs measures for the model on a Cayley tree. If q is divisible by p, then we prove the occurrence of a strong phase transition. If q and p are relatively prime, then there is a quasi phase transition. These results are totally different from the results of [F. M. Mukhamedov and U. A. Rozikov, Indag. Math. N. S. 15, 85–100 (2005)], since when q is divisible by p, which means that q + 1 is not divided by p, so according to a main result of the mentioned paper, there is a unique and bounded p-adic Gibbs measure (different from p-adic quasi Gibbs measure)     

Estimating Mixture of Dirichlet Process Models

Abstract

Current Gibbs sampling schemes in mixture of Dirichlet process (MDP) models are restricted to using “conjugate” base measures that allow analytic evaluation of the transition probabilities when resampling configurations, or alternatively need to rely on approximate numeric evaluations of some transition probabilities. Implementation of Gibbs sampling in more general MDP models is an open and important problem because most applications call for the use of nonconjugate base measures. In this article we propose a conceptual framework for computational strategies. This framework provides a perspective on current methods, facilitates comparisons between them, and leads to several new methods that expand the scope of MDP models to nonconjugate situations. We discuss one in detail. The basic strategy is based on expanding the parameter vector, and is applicable for MDP models with arbitrary base measure and likelihood. Strategies are also presented for the important class of normal-normal MDP models and for problems with fixed or few hyperparameters. The proposed algorithms are easily implemented and illustrated with an application.     

p-adic Gibbs measures and Markov random fields on countable graphs

The notions of the Gibbs measure and of the Markov random field are known to coincide in the real case. But in the p-adic case, the class of p-adic Markov random fields is broader than that of p-adic Gibbs measures. We construct p-adic Markov random fields (on finite graphs) that are not p-adic Gibbs measures. We define a p-adic Markov random field on countable graphs and show that the set of such fields is a nonempty closed subspace in the set of all p-adic probability measures     

The Monge problem in Wiener space

We address the Monge problem in the abstract Wiener space and we give an existence result provided both marginal measures are absolutely continuous with respect to the infinite dimensional Gaussian measure γ.     

Functional Integrals for the Schrodinger Equation on Compact Riemannian Manifolds

In this paper, we represent the solution of the Cauchy problem for the Schrodinger equation on compact Riemannian manifolds in terms of functional integrals with respect to the Wiener measure corresponding to the Brownian motion in a manifold and with respect to the Smolyanov surface measures constructed from the Wiener measure on trajectories in the underlying space. The representation of the solution is obtained for the case of analytic (on some sets) potential and analytic initial condition under certain assumptions on the geometric characteristics of the manifold. In the proof, we use a method due to Doss and the representations via functional integrals of the solution to the Cauchy problem for the heat equation in a compact Riemannian manifold.     

Geometric Gibbs theory

We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials. We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials. We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures. We prove that with this complex Banach manifold structure, the space is complete and, moreover, is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures. There is a maximum metric on the space, which is incomplete. We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same. We prove that a geometric Gibbs measure is an equilibrium state, and the infimum of the metric entropy function on the space is zero.

     

Geometric Gibbs theory In Memory of Professor Shantao Liao

We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials. We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials. We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures. We prove that with this complex Banach manifold structure, the space is complete and, moreover, is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures. There is a maximum metric on the space,which is incomplete. We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same. We prove that a geometric Gibbs measure is an equilibrium state, and the infimum of the metric entropy function on the space is zero.     

Nonanticipative transformations of the two-parameter Wiener process and a Girsanov theorem

Nonanticipative linear transformations of the two-parameter Wiener process W are studied. It is shown that they induce measures equivalent to two-parameter Wiener measure and the corresponding Radon-Nikodym derivatives are calculated. A two-parameter extension of Girsanov's theorem is established for a class of nonanticipative, possibly nonlinear transformations of W.     

Sobolev estimates for optimal transport maps on Gaussian spaces

In this work, we will take the standard Gaussian measure as the reference measure and study the variation of optimal transport maps in Sobolev spaces with respect to it; as a by-product, an inequality which gives a precise link between the variation of entropy, Fisher information between source and target measures, with the Sobolev norm of the optimal transport map will be given. As applications, we will construct strong solutions to Monge–Ampère equations in finite dimension, as well as on the Wiener space, when the target measure satisfies the strong log-concavity condition. A result on the regularity on the optimal transport map on the Wiener space will be obtained.     

Weakly Periodic Gibbs Measures for HC-Models on Cayley Trees

We study hard-core (HC) models on Cayley trees. Given a 2-state HC-model, we prove that exactly two weakly periodic (aperiodic) Gibbs measures exist under certain conditions on the parameters. Moreover, we consider fertile 4-state HC-models with the activity parameter λ > 0. The three types of these models are known to exist. For one of the models we show that the translationinvariant Gibbs measure is not unique.     

A class of infinite dimensional stochastic processes with unbounded diffusion

The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron–Martin space. In particular, it is shown that under a class of changes of the reference measure, quasi-regularity of the form is preserved. We also show that under these changes of the reference measure, derivative and divergence are closable with certain closable inverses. We first treat the case of the classical Wiener space and then we transfer the results to the Wiener space over a Riemannian manifold.     

On the Wiener semigroup and harmonic analysis on the infinite dimensional torus

We present a model for which certain difficulties often associated with analysis on infinite-dimensional spaces do not occur. In this situation, the convolution semigroup of Wiener measures constructed by Gross becomes a self-adjoint contraction semigroup. We generalize a facet of Sobolev theory to our infinite-dimensional context, and consider the differentiability of Wiener measure in this new weak sense.     

Non reversible stationary measures for infinite interacting particle systems

Summary Two results concerning the local conditional distributions of a stationary measure for a spin flip process with strictly positive and continuous rates are obtained: 1) The local conditional distributions and the rates of the reversed process determine each other. 2) Either all shift invariant stationary measures are Gibbs with the same potential or no shift invariant stationary Gibbs measure exist.Research supported by the Japan Society for the Promotion of Science     

Dimension-Independent Estimates on the Densities of Wiener Functionals via the Log-Sobolev Inequality

Using the log-Sobolev inequality, we shall present in this note some estimates on the density of finite dimensional non-degenerate Wiener functionals which are independent on the dimension. We shall take the Gaussian measure as the reference measure, contrary to the customary choice of Lebesgue measure in the literature. As an application, we show that the limit in probability of a uniformly bounded sequence of non-degenerate Wiener functionals has a density with respect to the Gaussian measure.