scholarly journals Sobolev spaces with respect to a weighted Gaussian measure in infinite dimensions

2019 ◽  
Vol 22 (04) ◽  
pp. 1950026 ◽  
Author(s):  
S. Ferrari
Keyword(s):  
Banach Space ◽  
Sobolev Spaces ◽  
Gaussian Measure ◽  
Separable Banach Space ◽  
Positive Function ◽  
Infinite Dimensions ◽  
Sobolev Function ◽  
Gradient Operator ◽  
Divergence Operator ◽  
Sobolev Functions

Let [Formula: see text] be a separable Banach space endowed with a nondegenerate centered Gaussian measure [Formula: see text] and let [Formula: see text] be a positive function on [Formula: see text] such that [Formula: see text] and [Formula: see text] for some [Formula: see text] and [Formula: see text]. In this paper, we introduce and study Sobolev spaces with respect to the weighted Gaussian measure [Formula: see text]. We obtain results regarding the divergence operator (i.e. the adjoint in [Formula: see text] of the gradient operator along the Cameron–Martin space) and the trace of Sobolev functions on hypersurfaces [Formula: see text], where [Formula: see text] is a suitable version of a Sobolev function.

scholarly journals The Topological Support of Gaussian Measure in Banach Space

1975 ◽  
Vol 57 ◽  
pp. 59-63 ◽  
Author(s):  
N. N. Vakhania
Keyword(s):  
Banach Space ◽  
Probability Measure ◽  
Gaussian Measure ◽  
Separable Banach Space ◽  
Covariance Operator ◽  
Probability Measures ◽  
Gaussian Measures ◽  
Linear Spaces

The main result of the present paper is the theorem 1, which describes the topological support of an arbitrary Gaussian measure in a separable Banach space. This theorem will be proved after some discussion of the notion of support itself. But we begin with the reminder of the notion of covariance operator of a probability measure. This notion has a great importance not only for the description of support of Gaussian measures but also for the study of other problems in the theory of probability measures in linear spaces (c.f. [1], [2]).

scholarly journals The Blum-Hanson Property

2019 ◽  
Vol 6 (1) ◽  
pp. 92-105
Author(s):  
Sophie Grivaux
Keyword(s):  
Banach Space ◽  
Hilbert Spaces ◽  
Separable Banach Space ◽  
Metric Spaces ◽  
Compact Metric Spaces ◽  
Theoretic Motivation

AbstractGiven a (real or complex, separable) Banach space, and a contraction T on X, we say that T has the Blum-Hanson property if whenever x, y ∈ X are such that Tnx tends weakly to y in X as n tends to infinity, the means{1 \over N}\sum\limits_{k = 1}^N {{T^{{n_k}}}x} tend to y in norm for every strictly increasing sequence (nk) k≥1 of integers. The space X itself has the Blum-Hanson property if every contraction on X has the Blum-Hanson property. We explain the ergodic-theoretic motivation for the Blum-Hanson property, prove that Hilbert spaces have the Blum-Hanson property, and then present a recent criterion of a geometric flavor, due to Lefèvre-Matheron-Primot, which allows to retrieve essentially all the known examples of spaces with the Blum-Hanson property. Lastly, following Lefèvre-Matheron, we characterize the compact metric spaces K such that the space C(K) has the Blum-Hanson property.

scholarly journals On Common Random Fixed Points of a New Iteration with Errors for Nonself Asymptotically Quasi-Nonexpansive Type Random Mappings

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
R. A. Rashwan ◽  
P. K. Jhade ◽  
Dhekra Mohammed Al-Baqeri
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Fixed Points ◽  
Separable Banach Space ◽  
Iterative Scheme ◽  
Random Fixed Points ◽  
Real Separable Banach Space ◽  
Random Mappings

We prove some strong convergence of a new random iterative scheme with errors to common random fixed points for three and then N nonself asymptotically quasi-nonexpansive-type random mappings in a real separable Banach space. Our results extend and improve the recent results in Kiziltunc, 2011, Thianwan, 2008, Deng et al., 2012, and Zhou and Wang, 2007 as well as many others.

scholarly journals Decompositions of Weakly Compact Valued Integrable Multifunctions

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 863 ◽  
Author(s):  
Luisa Di Piazza ◽  
Kazimierz Musiał
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Compact Convex ◽  
Weakly Compact ◽  
Seminal Paper ◽  
Decomposition Property ◽  
Short Overview ◽  
Decomposition Theorems ◽  
State Of Art

We give a short overview on the decomposition property for integrable multifunctions, i.e., when an “integrable in a certain sense” multifunction can be represented as a sum of one of its integrable selections and a multifunction integrable in a narrower sense. The decomposition theorems are important tools of the theory of multivalued integration since they allow us to see an integrable multifunction as a translation of a multifunction with better properties. Consequently, they provide better characterization of integrable multifunctions under consideration. There is a large literature on it starting from the seminal paper of the authors in 2006, where the property was proved for Henstock integrable multifunctions taking compact convex values in a separable Banach space X. In this paper, we summarize the earlier results, we prove further results and present tables which show the state of art in this topic.

Compactness in Lebesgue–Bochner spaces of random variables and the existence of mean-square random attractors

2019 ◽  
Vol 19 (04) ◽  
pp. 1950032
Author(s):  
Yejuan Wang ◽  
Xiangming Zhu ◽  
Peter Kloeden
Keyword(s):  
Banach Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Separable Banach Space ◽  
Additional Condition ◽  
Probability Space ◽  
Parabolic Partial Differential Equations ◽  
Mean Square ◽  
Probabilistic Argument ◽  
Random Attractors

Let [Formula: see text] be a probability space and let [Formula: see text] be a separable Banach space. It is shown a subset [Formula: see text] of [Formula: see text], where [Formula: see text], is relatively compact in [Formula: see text] if and only if it is uniformly [Formula: see text]-integrable and uniformly tight. The additional condition of scalarly relatively compact required in the literature is shown to hold by a probabilistic argument. The result is then used to establish the existence of a mean-square random attractor for dissipative stochastic differential equations and stochastic parabolic partial differential equations.

scholarly journals Gaussian Measure on a Banach Space and Abstract Winer Measure

1969 ◽  
Vol 36 ◽  
pp. 65-81 ◽  
Author(s):  
Hiroshi Sato
Keyword(s):  
Banach Space ◽  
Gaussian Measure ◽  
Reflexive Banach Space ◽  
Wiener Measure

In this paper, we shall show that any Gaussian measure on a separable or reflexive Banach space is an abstract Wiener measure in the sense of L. Gross [1] and, for the proof of that, establish the Radon extensibility of a Gaussian measure on such a Banach space. In addition, we shall give some remarks on the support of an abstract Wiener measure.

scholarly journals Properties of the trajectories of set-valued integrals in banach spaces

1990 ◽  
Vol 41 (2) ◽  
pp. 271-281
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
Hausdorff Metric ◽  
Density Theorem ◽  
Mosco Convergence ◽  
Convexity Theorem ◽  
Convergence Theorems ◽  
Convergence Of Sets ◽  
Differentiability Properties

Let F: T → 2x \ {} be a closed-valued multifunction into a separable Banach space X. We define the sets and We prove various convergence theorems for those two sets using the Hausdorff metric and the Kuratowski-Mosco convergence of sets. Then we prove a density theorem of CF and a corresponding convexity theorem for F(·). Finally we study the “differentiability” properties of K(·). Our work extends and improves earlier ones by Artstein, Bridgland, Hermes and Papageorgiou.

An Embedding Theorem for Separable Locally Convex Spaces

1971 ◽  
Vol 14 (1) ◽  
pp. 119-120 ◽  
Author(s):  
Robert H. Lohman
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Space ◽  
Separable Banach Space ◽  
Locally Convex Space ◽  
Embedding Theorem ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Topological Vector Spaces ◽  
Locally Convex

A well-known embedding theorem of Banach and Mazur [1, p. 185] states that every separable Banach space is isometrically isomorphic to a subspace of C[0, 1], establishing C[0, 1] as a universal separable Banach space. The embedding theorem one encounters in a course in topological vector spaces states that every Hausdorff locally convex space (l.c.s.) is topologically isomorphic to a subspace of a product of Banach spaces.

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