Measures In Banach Spaces Research Articles

A new fuzzy-valued integral and its convergence theorems

In this paper, we generalize Kluv?anek-Lewis type set-valued integral to fuzzy number measures in Banach spaces. Firstly, we introduce the new fuzzy-valued integral with some natural properties. And then, We establish Vitali type convergence theorem and dominated convergence theorem for this kind of integral.

Maximal inequalities for stochastic convolutions driven by compensated Poisson random measures in Banach spaces

Let $(E, \| \cdot\|)$ be a Banach space such that, for some $q\geq 2$, the function $x\mapsto \|x\|^q$ is of $C^2$ class and its first and second Fr\'{e}chet derivatives are bounded by some constant multiples of $(q-1)$-th power of the norm and $(q-2)$-th power of the norm and let $S$ be a $C_0$-semigroup of contraction type on $(E, \| \cdot\|)$. We consider the following stochastic convolution process \begin{align*} u(t)=\int_0^t\int_ZS(t-s)\xi(s,z)\,\tilde{N}(\mathrm{d} s,\mathrm{d} z), \;\;\; t\geq 0, \end{align*} where $\tilde{N}$ is a compensated Poisson random measure on a measurable space $(Z,\mathcal{Z})$ and $\xi:[0,\infty)\times\Omega\times Z\rightarrow E$ is an $\mathbb{F}\otimes \mathcal{Z}$-predictable function. We prove that there exists a c\`{a}dl\`{a}g modification a $\tilde{u}$ of the process $u$ which satisfies the following maximal inequality \begin{align*} \mathbb{E} \sup_{0\leq s\leq t} \|\tilde{u}(s)\|^{q^\prime}\leq C \mathbb{E} \left(\int_0^t\int_Z \|\xi(s,z) \|^{p}\,N(\mathrm{d} s,\mathrm{d} z)\right)^{\frac{q^\prime}{p}}, \end{align*} for all $ q^\prime \geq q$ and $1<p\leq 2$ with $C=C(q,p)$.

Existence and uniqueness of translation invariant measures in separable Banach spaces

It is shown that for the vector space $\mathbb{R^N}$ (equipped with the product topology and the Yamasaki-Kharazishvili measure) the group of linear measure preserving isomorphisms is quite rich. Using Kharazishvili's approach, we prove that every infinite-dimensional Polish linear space admits a $\sigma$-finite non-trivial Borel measure that is translation invariant with respect to a dense linear subspace. This extends a~recent result of Gill, Pantsulaia and Zachary on the existence of such measures in Banach spaces with Schauder bases. It is shown that each $\sigma$-finite Borel measure defined on an infinite-dimensional Polish linear space, which assigns the value 1 to a fixed compact set and is translation invariant with respect to a~linear subspace fails the uniqueness property. For Banach spaces with absolutely convergent Markushevich bases, a similar problem for the usual completion of the concrete $\sigma$-finite Borel measure is solved positively. The uniqueness problem for non-$\sigma$-finite semi-finite translation invariant Borel measures on a Banach space $X$ which assign the value 1 to the standard rectangle (i.e., the rectangle generated by an absolutely convergent Markushevich basis) is solved negatively. In addition, it is constructed an example of such a measure $\mu_B^0$ on $X$, which possesses a strict uniqueness property in the class of all translation invariant measures which are defined on the domain of $\mu_B^0$ and whose values on non-degenerate rectangles coincide with their volumes.

Second quantisation for skew convolution products of measures in Banach spaces

We study measures in Banach space which arise as the skew convolution product of two other measures where the convolution is deformed by a skew map. This is the structure that underlies both the theory of Mehler semigroups and operator self-decomposable measures. We show how that given such a set-up the skew map can be lifted to an operator that acts at the level of function spaces and demonstrate that this is an example of the well known functorial procedure of second quantisation. We give particular emphasis to the case where the product measure is infinitely divisible and study the second quantisation process in some detail using chaos expansions when this is either Gaussian or is generated by a Poisson random measure.

Continuous Disintegrations of Gaussian Processes

The goal of this paper is to understand the conditional law of a stochastic process once it has been observed over an interval. To make this precise, we introduce the notion of a continuous disintegration: a regular conditional probability measure which varies continuously in the conditioned parameter. The conditioning is infinite dimensional in character, which leads us to consider the general case of probability measures in Banach spaces. Our main result is that for a certain quantity $M$ based on the covariance structure, $M < \oo$ is a necessary and sufficient condition for a Gaussian measure to have a continuous disintegration. The condition $M < \oo$ is quite reasonable: for the familiar case of stationary processes, $M = 1$.

On uniform continuity of convex bodies with respect to measures in Banach spaces

Let μ be a probability measure on a separable Banach space X. A subset U⊂X is μ-continuous if μ(∂U)=0. In the paper the μ-continuity and uniform μ-continuity of convex bodies in X, especially of balls and half-spaces, is considered. The μ-continuity is interesting for study of the Glivenko–Cantelli theorem in Banach spaces. Answer to a question of F. Topsøe is given.

On graph measures in banach spaces and description of essential spectra of multidimensional transport equation

In this paper, we define new measures called respectively graph measure of noncompactness and graph measure of weak noncompactness. Moreover, we apply the obtained results to discuss the incidence of some perturbation results realized in [2] on the behavior of essential spectra of such closed densely defined linear operators on Banach spaces. These results are exploited to investigate the essential spectra of a multidimensional neutron transport operator on L1 spaces.

Integral functionals for the exponential of the wiener process and the Brownian bridge: Exact asymptotics and legendre functions

We prove results concerning the exact asymptotics of the probabilities $$P\left\{ {\int_0^1 {e^{\varepsilon \xi (t)} dt < b} } \right\}, P\left\{ {\int_0^1 {e^{\varepsilon \left| {\xi (t)} \right|} dt < b} } \right\}$$ as ɛ → 0 and 0 < b < 1 for two Gaussian processes ξ(t), the Wiener process and the Brownian bridge. We also obtain asymptotic formulas for integrals of Laplace type. Our study is based on the Laplace method for Gaussian measures in Banach spaces. The calculations of the constants are reduced to the solution of an extremal problem for the action functional and to the study of the spectrum of a second-order differential operator of Sturm-Liouville type using the Legendre functions.

On ordinary and standard products of infinite family of σ-finite measures and some of their applications

We introduce notions of ordinary and standard products of σ-finite measures and prove their existence. This approach allows us to construct invariant extensions of ordinary and standard products of Haar measures. In particular, we construct translation-invariant extensions of ordinary and standard Lebesgue measures on ℝ∞ and Rogers-Fremlin measures on l∞, respectively, such that topological weights of quasi-metric spaces associated with these measures are maximal (i.e., 2c). We also solve some Fremlin problems concerned with an existence of uniform measures in Banach spaces.

A Stochastic Poisson Structure

We define a Poisson structure on the Nualart-Pardoux test algebra associated to the path space of a finite dimensional Lie algebra.

Exact asymptotics of distributions of integral functionals of the geometric Brownian motion and other related formulas

We prove results on exact asymptotics of the probabilities $$P\left\{ {\int\limits_0^1 {e^{\varepsilon \xi (t)} dt > b} } \right\},P\left\{ {\int\limits_0^1 {e^{\varepsilon |\xi (t)|} dt > b} } \right\},\varepsilon \to 0,$$ where b > 1, for two Gaussian processes ?(t), namely, a Wiener process and a Brownian bridge. We use the Laplace method for Gaussian measures in Banach spaces. Evaluation of constants is reduced to solving an extreme value problem for the rate function and studying the spectrum of a second-order differential operator of the Sturm-Liouville type with the use of Legendre functions.

A numerical approximation of parabolic stochastic partial differential equations driven by a Poisson random measure

The paper deals with SPDEs driven by Poisson random measures in Banach spaces and its numerical approximation. We investigate the accuracy of space and time approximation. As the space approximation we consider spectral methods and as time approximation the implicit Euler scheme and the explicit Euler scheme.

The effect of projections on fractal sets and measures in Banach spaces

We study the extent to which the Hausdorff dimension of a compact subset of an infinite-dimensional Banach space is affected by a typical mapping into a finite-dimensional space. It is possible that the dimension drops under all such mappings, but the amount by which it typically drops is controlled by the ‘thickness exponent’ of the set, which was defined by Hunt and Kaloshin (Nonlinearity12 (1999), 1263–1275). More precisely, let .

Exact asymptotics of large deviations of stationary Ornstein-Uhlenbeck processes for L p-functionals, p &gt; 0

We prove a general result on the exact asymptotics of the probability $$P\left\{ {\int\limits_0^1 {\left| {\eta _\gamma (t)} \right|^p dt > u^p } } \right\}$$ as u ? ?, where p > 0, for a stationary Ornstein-Uhlenbeck process ? ?(t), i.e., a Gaussian Markov process with zero mean and with the covariance function E??(t)??(s), t, s ? ?, ? > 0. We use the Laplace method for Gaussian measures in Banach spaces. Evaluation of constants is reduced to solving an extreme value problem for the rate function and studying the spectrum of a second-order differential operator of the Sturm-Liouville type. For p = 1 and p = 2, explicit formulas for the asymptotics are given.

Radon–Nikodym theorem and Vitali–Hahn–Saks theorem on fuzzy number measures in Banach spacers

Radon–Nikodym theorem is established on fuzzy number measures in Banach spaces, Vitali–Hahn–Saks theorem is generalized to set valued measures and fuzzy number measures.

Parallelepipeds and Decomposition of Ranges of Vector Measures in Banach Spaces

Parallelepipeds and Decomposition of Ranges of Vector Measures in Banach Spaces

Fusions of Stable Measures in Banach Spaces

We show that in a separable Banach space a symmetric p-stable (0 < p < 1), probability measure can be fused to obtain any Borel probability measure.Keywordsfusion of a probabilitybarycenterp-stablemeasures in Hilbert and Banach spaces

Fusions of stable measures in Banach spaces

Fusions of stable measures in Banach spaces

On the extension of the fuzzy number measures in Banach spaces: Part I. Representation of the fuzzy number measures

In this paper, the definition of the fuzzy number measures which extends the concepts of the set-valued measures in Banach spaces is introduced. The relations between the fuzzy number measures and the set-valued measures are studied. The definition of the indefinite integral of the random fuzzy numbers is also introduced and its properties are discussed.

Measures in banach spaces are determined by their values on balls

The statement from the title is proved.