The Experts below are selected from a list of 153 Experts worldwide ranked by ideXlab platform
N. A. Evseev - One of the best experts on this subject based on the ideXlab platform.
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The Composition Operator on Mixed-Norm Lebesgue Spaces
Mathematical Notes, 2019Co-Authors: N. A. Evseev, A. V. MenovshchikovAbstract:It is known that the boundedness of the composition operator on Lebesgue spaces is equivalent to the integrability of the volume derivative of the Measurable Mapping inducing the given operator. In the present paper, we prove a similar result for mixed-norm Lebesgue spaces in the class of Mappings preserving the priority of the variables.
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Bounded composition operator on Lorentz spaces
Mathematical Notes, 2017Co-Authors: N. A. EvseevAbstract:We study composition operators on Lorentz spaces. In particular, we obtain necessary and sufficient conditions under which a Measurable Mapping induces a bounded composition operator.
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Bounded composition operator on Lorentz spaces
arXiv: Functional Analysis, 2017Co-Authors: N. A. EvseevAbstract:We study a composition operator on Lorentz spaces. In particular we provide necessary and sufficient conditions under which a Measurable Mapping induces a bounded composition operator.
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Isomorphisms of Sobolev spaces on Carnot groups and quasiconformal Mappings
Siberian Mathematical Journal, 2015Co-Authors: S. K. Vodop’yanov, N. A. EvseevAbstract:We prove that a Measurable Mapping of domains on a Carnot group induces by the corresponding change of variables an isomorphism of the Sobolev spaces whose integrability exponent is equal to the Hausdorff dimension of the group if and only if the Mapping coincides with a quasiconformal Mapping almost everywhere.
Volodymyr Mykhaylyuk - One of the best experts on this subject based on the ideXlab platform.
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On weakly Gibson $F_\sigma$-Measurable Mappings
arXiv: General Topology, 2014Co-Authors: Olena Karlova, Volodymyr MykhaylyukAbstract:A function $f:X\to Y$ between topological spaces is said to be a {\it weakly Gibson function} if $f(\overline{U})\subseteq \overline{f(U)}$ for any open connected set \mbox{$U\subseteq X$}. We prove that if $X$ is a locally connected hereditarily Baire space and $Y$ is a $T_1$-space then an $F_\sigma$-Measurable Mapping $f:X\to Y$ is weakly Gibson if and only if for any connected set $C\subseteq X$ with the dense connected interior the image $f(C)$ is connected. Moreover, we show that each weakly Gibson $F_\sigma$-Measurable Mapping $f:\mathbb R^n\to Y$, where $Y$ is a $T_1$-space, has a connected graph.
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On weakly Gibson $F_{\sigma} $-Measurable Mappings
Colloquium Mathematicum, 2013Co-Authors: Olena Karlova, Volodymyr MykhaylyukAbstract:A function f : X ! Y between topological spaces is said to be a weakly Gibson function if f(U) f(U) for any open connected set U X. We prove that if X is a locally connected hereditarily Baire space and Y is a T1-space then an F -Measurable Mapping f : X ! Y is weakly Gibson if and only if for any connected set C X with dense connected interior the image f(C) is connected. Moreover, we show that each weakly Gibson F -Measurable Mapping f : R n ! Y , where Y is a T1-space, has a connected graph.
Olena Karlova - One of the best experts on this subject based on the ideXlab platform.
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On weakly Gibson $F_\sigma$-Measurable Mappings
arXiv: General Topology, 2014Co-Authors: Olena Karlova, Volodymyr MykhaylyukAbstract:A function $f:X\to Y$ between topological spaces is said to be a {\it weakly Gibson function} if $f(\overline{U})\subseteq \overline{f(U)}$ for any open connected set \mbox{$U\subseteq X$}. We prove that if $X$ is a locally connected hereditarily Baire space and $Y$ is a $T_1$-space then an $F_\sigma$-Measurable Mapping $f:X\to Y$ is weakly Gibson if and only if for any connected set $C\subseteq X$ with the dense connected interior the image $f(C)$ is connected. Moreover, we show that each weakly Gibson $F_\sigma$-Measurable Mapping $f:\mathbb R^n\to Y$, where $Y$ is a $T_1$-space, has a connected graph.
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On weakly Gibson $F_{\sigma} $-Measurable Mappings
Colloquium Mathematicum, 2013Co-Authors: Olena Karlova, Volodymyr MykhaylyukAbstract:A function f : X ! Y between topological spaces is said to be a weakly Gibson function if f(U) f(U) for any open connected set U X. We prove that if X is a locally connected hereditarily Baire space and Y is a T1-space then an F -Measurable Mapping f : X ! Y is weakly Gibson if and only if for any connected set C X with dense connected interior the image f(C) is connected. Moreover, we show that each weakly Gibson F -Measurable Mapping f : R n ! Y , where Y is a T1-space, has a connected graph.
David Applebaum - One of the best experts on this subject based on the ideXlab platform.
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Second quantisation for skew convolution products of infinitely divisible measures
Infinite Dimensional Analysis Quantum Probability and Related Topics, 2020Co-Authors: David Applebaum, Jan Van NeervenAbstract:Suppose λ1 and λ2 are infinitely divisible Radon measures on real Banach spaces E1 and E2, respectively and let T : E1 → E2 be a Borel Measurable Mapping so that T(λ1) * ρ = λ2 for some Radon probability measure ρ on E2. Extending previous results for the Gaussian and the Poissonian case, we study the problem of representing the "transition operator" PT : Lp(E2, λ2) → Lp(E1, λ1) given by [Formula: see text] as the second quantisation of a contraction operator acting between suitably chosen "reproducing kernel Hilbert spaces" associated with λ1 and λ2.
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Second quantisation for skew convolution products of infinitely divisible measures
arXiv: Probability, 2014Co-Authors: David Applebaum, Jan Van NeervenAbstract:Suppose $\lambda_1$ and $\lambda_2$ are infinitely divisible Radon measures on real Banach spaces $E_1$ and $E_2$, respectively and let $T:E_{1} \rightarrow E_{2}$ be a Borel Measurable Mapping so that $T(\lambda_1) * \rho = \lambda_2 $ for some Radon probability measure $\rho$ on $E_{2}$. Extending previous results for the Gaussian and the Poissonian case, we study the problem of representing the `transition operator' $P_{T}:L^{p}(E_{2}, \lambda_{2}) \rightarrow L^{p}(E_{1}, \lambda_{1})$ given by $$ P_{T}f(x) = \int_{E_{2}}f(T(x) + y)d\rho(y) %% d\rho(y) instead of \rho(dy) in order to unify notations $$ as the second quantisation of a contraction operator acting between suitably chosen `reproducing kernel Hilbert spaces' associated with $\lambda_1$ and $\lambda_2$.
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Lévy Processes—From Probability to Finance and Quantum Groups
2004Co-Authors: David ApplebaumAbstract:1320 NOTICES OF THE AMS VOLUME 51, NUMBER 11 T he theory of stochastic processes was one of the most important mathematical developments of the twentieth century. Intuitively, it aims to model the interaction of “chance” with “time”. The tools with which this is made precise were provided by the great Russian mathematician A. N. Kolmogorov in the 1930s. He realized that probability can be rigorously founded on measure theory, and then a stochastic process is a family of random variables (X(t), t ≥ 0) defined on a probability space (Ω,F , P ) and taking values in a Measurable space (E,E) . Here Ω is a set (the sample space of possible outcomes), F is a σ-algebra of subsets of Ω (the events), and P is a positive measure of total mass 1 on (Ω,F ) (the probability). E is sometimes called the state space. Each X(t) is a (F ,E) Measurable Mapping from Ω to E and should be thought of as a random observation made on E made at time t . For many developments, both theoretical and applied, E is Euclidean space Rd (often with d = 1); however, there is also considerable interest in the case where E is an infinite dimensional Hilbert or Banach space, or a finite-dimensional Lie group or manifold. In all of these cases E can be taken to be the Borel σalgebra generated by the open sets. To model probabilities arising within quantum theory, the scheme described above is insufficiently general and must be embedded into a suitable noncommutative structure. Stochastic processes are not only mathematically rich objects. They also have an extensive range of applications in, e.g., physics, engineering, ecology, and economics—indeed, it is difficult to conceive of a quantitative discipline in which they do not feature. There is a limited amount that can be said about the general concept, and much of both theory and applications focusses on the properties of specific classes of process that possess additional structure. Many of these, such as random walks and Markov chains, will be well known to readers. Others, such as semimartingales and measure-valued diffusions, are more esoteric. In this article, I will give an introduction to a class of stochastic processes called Levy processes, in honor of the great French probabilist Paul Levy, who first studied them in the 1930s. Their basic structure was understood during the “heroic age” of probability in the 1930s and 1940s and much of this was due to Paul Levy himself, the Russian mathematician A. N. Khintchine, and to K. Ito in Japan. During the past ten years, there has been a great revival of interest in these processes, due to new theoretical developments and also a wealth of novel applications—particularly to option pricing in mathematical finance. As well as a vast number of research papers, a number of books on the subject have been published ([3], [11], [1], [2], [12]) and there have been annual international conferences devoted to these processes since 1998. Before we begin the main part of the article, it is worth David Applebaum is professor of probability and statistics at the University of Sheffield. His email address is D.Applebaum@sheffield.ac.uk. He is the author of Levy Processes and Stochastic Calculus, Cambridge University Press, 2004, on which part of this article is based.
Jan Van Neerven - One of the best experts on this subject based on the ideXlab platform.
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Second quantisation for skew convolution products of infinitely divisible measures
Infinite Dimensional Analysis Quantum Probability and Related Topics, 2020Co-Authors: David Applebaum, Jan Van NeervenAbstract:Suppose λ1 and λ2 are infinitely divisible Radon measures on real Banach spaces E1 and E2, respectively and let T : E1 → E2 be a Borel Measurable Mapping so that T(λ1) * ρ = λ2 for some Radon probability measure ρ on E2. Extending previous results for the Gaussian and the Poissonian case, we study the problem of representing the "transition operator" PT : Lp(E2, λ2) → Lp(E1, λ1) given by [Formula: see text] as the second quantisation of a contraction operator acting between suitably chosen "reproducing kernel Hilbert spaces" associated with λ1 and λ2.
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Second quantisation for skew convolution products of infinitely divisible measures
arXiv: Probability, 2014Co-Authors: David Applebaum, Jan Van NeervenAbstract:Suppose $\lambda_1$ and $\lambda_2$ are infinitely divisible Radon measures on real Banach spaces $E_1$ and $E_2$, respectively and let $T:E_{1} \rightarrow E_{2}$ be a Borel Measurable Mapping so that $T(\lambda_1) * \rho = \lambda_2 $ for some Radon probability measure $\rho$ on $E_{2}$. Extending previous results for the Gaussian and the Poissonian case, we study the problem of representing the `transition operator' $P_{T}:L^{p}(E_{2}, \lambda_{2}) \rightarrow L^{p}(E_{1}, \lambda_{1})$ given by $$ P_{T}f(x) = \int_{E_{2}}f(T(x) + y)d\rho(y) %% d\rho(y) instead of \rho(dy) in order to unify notations $$ as the second quantisation of a contraction operator acting between suitably chosen `reproducing kernel Hilbert spaces' associated with $\lambda_1$ and $\lambda_2$.