scholarly journals Intermediate spaces, Gaussian probabilities and exponential tightness

2021 ◽  
Author(s):  
Paolo Baldi
Keyword(s):  
Banach Space ◽  
Gaussian Measure ◽  
Wiener Measure ◽  
Intermediate Space ◽  
Exponential Tightness

AbstractWe prove the existence of an intermediate Banach space between the space where the Gaussian measure lives and its RKHS, thus extending what happens with Wiener measure, where the intermediate space can be chosen as a space of Hölder paths. From this result, it is very simple to deduce a result of exponential tightness for Gaussian probabilities.

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 On Abstract Wiener Measure

1972 ◽  
Vol 46 ◽  
pp. 155-160 ◽  
Author(s):  
Balram S. Rajput
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Gaussian Measure ◽  
Separable Hilbert Space ◽  
Reflexive Banach Space ◽  
Wiener Measure ◽  
Real Separable Hilbert Space ◽  
Cylinder Measure

In a recent paper, Sato [6] has shown that for every Gaussian measure n on a real separable or reflexive Banach space (X, ‖ • ‖) there exists a separable closed sub-space X〵 of X such that and is the σ-extension of the canonical Gaussian cylinder measure of a real separable Hilbert space such that the norm is contiunous on and is dense in The main purpose of this note is to prove that ‖ • ‖ x〵 is measurable (and not merely continuous) on .

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 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 Non-recurrence sets for weakly mixing linear dynamical systems

2012 ◽  
Vol 34 (1) ◽  
pp. 132-152 ◽  
Author(s):  
SOPHIE GRIVAUX
Keyword(s):  
Banach Space ◽  
Dynamical Systems ◽  
Linear Operator ◽  
Gaussian Measure ◽  
Bounded Linear Operator ◽  
Probability Space ◽  
Complex Banach Space ◽  
Bounded Linear ◽  
Linear Dynamical Systems ◽  
Weakly Mixing

AbstractWe study non-recurrence sets for weakly mixing dynamical systems by using linear dynamical systems. These are systems consisting of a bounded linear operator acting on a separable complex Banach space$X$, which becomes a probability space when endowed with a non-degenerate Gaussian measure. We generalize some recent results of Bergelson, del Junco, Lemańczyk and Rosenblatt, and show in particular that sets$\{n_{k}\}$such that$n_{k+1}/n_{k}\to +\infty $, or such that$n_{k}$divides$n_{k+1}$for each$k\ge 0$, are non-recurrence sets for weakly mixing linear dynamical systems. We also give examples, for each$r\ge 1$, of$r$-Bohr sets which are non-recurrence sets for some weakly mixing systems.

scholarly journals Domains of elliptic operators on sets in Wiener space

2020 ◽  
Vol 23 (01) ◽  
pp. 2050004
Author(s):  
Davide Addona ◽  
Gianluca Cappa ◽  
Simone Ferrari
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Quadratic Form ◽  
Gaussian Measure ◽  
Adjoint Operator ◽  
The Self ◽  
Affine Function ◽  
Surface Measure ◽  
Regular Convex

Let [Formula: see text] be a separable Banach space endowed with a non-degenerate centered Gaussian measure [Formula: see text]. The associated Cameron–Martin space is denoted by [Formula: see text]. Consider two sufficiently regular convex functions [Formula: see text] and [Formula: see text]. We let [Formula: see text] and [Formula: see text]. In this paper, we study the domain of the self-adjoint operator associated with the quadratic form [Formula: see text] and we give sharp embedding results for it. In particular, we obtain a characterization of the domain of the Ornstein–Uhlenbeck operator in Hilbert space with [Formula: see text] and on half-spaces, namely if [Formula: see text] and [Formula: see text] is an affine function, then the domain of the operator defined via (0.1) is the space [Formula: see text] where [Formula: see text] is the Feyel–de La Pradelle Hausdorff–Gauss surface measure.

scholarly journals Tightness and exponential tightness of Gaussian probabilities

2020 ◽  
Vol 24 ◽  
pp. 113-126
Author(s):  
Paolo Baldi
Keyword(s):  
Banach Space ◽  
Large Deviations ◽  
General Result ◽  
Separable Banach Space ◽  
Simple Criterion ◽  
Exponential Tightness

We prove a simple criterion of exponential tightness for sequences of Gaussian r.v.’s with values in a separable Banach space from which we deduce a general result of Large Deviations which allows easily to obtain LD estimates in various situations.

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