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19 results on '"David Applebaum"'

2. The positive maximum principle on symmetric spaces

Author

David Applebaum and Trang Le Ngan

Subjects
47G20, 47D07, 43A85, 47G30, 60B15, Pure mathematics, General Mathematics, 010102 general mathematics, Scalar (mathematics), Linear operators, Operator theory, 01 natural sciences, Lévy process, Potential theory, Functional Analysis (math.FA), Theoretical Computer Science, Mathematics - Functional Analysis, 010104 statistics & probability, symbols.namesake, Maximum principle, Fourier analysis, Symmetric space, FOS: Mathematics, symbols, 0101 mathematics, Analysis, Mathematics
Abstract

We investigate the Courrège theorem in the context of linear operators that satisfy the positive maximum principle on a space of continuous functions over a symmetric space. Applications are given to Feller–Markov processes. We also introduce Gangolli operators, which satisfy the positive maximum principle, and generalise the form associated with the generator of a Lévy process on a symmetric space. When the space is compact, we show that Gangolli operators are pseudo-differential operators having scalar symbols.

Published
2020
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3. $$L^2$$ Properties of Lévy Generators on Compact Riemannian Manifolds

Author

Rosemary Shewell Brockway and David Applebaum

Subjects
Statistics and Probability, Pure mathematics, Semigroup, General Mathematics, Isotropy, Statistics, Probability and Uncertainty, Riemannian manifold, Lévy process, Contraction (operator theory), Eigenvalues and eigenvectors, Brownian motion, Mathematics, Discrete spectrum
Abstract

We consider isotropic Lévy processes on a compact Riemannian manifold, obtained from an $${\mathbb {R}}^d$$ R d -valued Lévy process through rolling without slipping. We prove that the Feller semigroups associated with these processes extend to strongly continuous contraction semigroups on $$L^p$$ L p , for $$1\le p 1 ≤ p < ∞ . When $$p=2$$ p = 2 , we show that these semigroups are self-adjoint. If, in addition, the motion has a non-trivial Brownian part, we prove that the generator has a discrete spectrum of eigenvalues and that the semigroup is trace-class.

Published
2020
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4. Preface

Author

Shigeki Aida, David Applebaum, Yasushi Ishikawa, Arturo Kohatsu-Higa, and Nicolas Privault

Published
2021
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5. The positive maximum principle on Lie groups

Author

Trang Le Ngan and David Applebaum

Subjects
Pure mathematics, Semigroup, Group (mathematics), General Mathematics, 010102 general mathematics, Lie group, Type (model theory), 01 natural sciences, Functional Analysis (math.FA), Convolution, Mathematics - Functional Analysis, 010104 statistics & probability, Operator (computer programming), Maximum principle, FOS: Mathematics, 0101 mathematics, Constant (mathematics), 22E30, Mathematics
Abstract

We extend a classical theorem of Courr\`{e}ge to Lie groups in a global setting, thus characterising all linear operators on the space of smooth functions of compact support that satisfy the positive maximum principle. We show that these are L\'{e}vy type operators (with variable characteristics), and pseudo--differential operators when the group is compact. If the characteristics are constant, then the operator is the generator of the contraction semigroup associated to a convolution semigroup of sub--probability measures.

Published
2019
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7. Markov and Feller Semigroups

Author

David Applebaum

Subjects
Pure mathematics, Maximum principle, Mathematics::Probability, Markov chain, Stochastic process, Linear operators, Statistics::Other Statistics, Martingale (probability theory), Representation (mathematics), Mathematics, Variable (mathematics)
Abstract

Markov and Feller semigroups are introduced, together with the corresponding stochastic processes. As all generators of Feller semigroups satisfy the positive maximum principle, we focus on that property and discuss the associated Hille–Yosida–Ray theorem. The main result of the chapter is proof of the Courrege theorem, which gives a Levy–Khinchine representation (but with variable coefficients) for all linear operators satisfying the positive maximum principle. We conclude with a brief discussion of the martingale problem and sub-Feller semigroups.

Published
2019
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9. Semigroups of Linear Operators

Author

David Applebaum

Subjects
Algebra, Sobolev space, Linear map, Partial differential equation, Dynamical systems theory, Semigroup, Quantum, Resolvent, Generator (mathematics)
Abstract

The theory of semigroups of operators is one of the most important themes in modern analysis. Not only does it have great intellectual beauty, but also wide-ranging applications. In this book the author first presents the essential elements of the theory, introducing the notions of semigroup, generator and resolvent, and establishes the key theorems of Hille–Yosida and Lumer–Phillips that give conditions for a linear operator to generate a semigroup. He then presents a mixture of applications and further developments of the theory. This includes a description of how semigroups are used to solve parabolic partial differential equations, applications to Levy and Feller–Markov processes, Koopmanism in relation to dynamical systems, quantum dynamical semigroups, and applications to generalisations of the Riemann–Liouville fractional integral. Along the way the reader encounters several important ideas in modern analysis including Sobolev spaces, pseudo-differential operators and the Nash inequality.

Published
2019
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10. On the Spectrum of Self-Adjoint Lévy Generators

Author

David Applebaum

Subjects
Statistics and Probability, Physics, Spectrum (functional analysis), Self-adjoint operator, Mathematical physics
Abstract

We investigate the spectrum of the generator of a self-adjoint transition semigroup of a (symmetric) Lévy process taking values in d–dimensional space.

Published
2019
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12. Transition Densities and Traces for Invariant Feller Processes on Compact Symmetric Spaces

Author

Trang Le Ngan and David Applebaum

Subjects
Pure mathematics, Semigroup, 010102 general mathematics, Mathematical analysis, Probability (math.PR), 01 natural sciences, Lévy process, Potential theory, Gelfand pair, 010104 statistics & probability, Symmetric space, FOS: Mathematics, 0101 mathematics, Invariant (mathematics), Fourier series, Mathematics - Probability, Analysis, Brownian motion, 60B15, Mathematics
Abstract

We find necessary and sufficient conditions for a finite $K$-bi-invariant measure on a compact Gelfand pair $(G, K)$ to have a square-integrable density. For convolution semigroups, this is equivalent to having a continuous density in positive time. When $(G,K)$ is a compact Riemannian symmetric pair, we study the induced transition density for $G$-invariant Feller processes on the symmetric space $X = G/K$. These are obtained as projections of $K$-bi-invariant L\'{e}vy processes on $G$, whose laws form a convolution semigroup. We obtain a Fourier series expansion for the density, in terms of spherical functions, where the spectrum is described by Gangolli's L\'evy-Khintchine formula. The density of returns to any given point on $X$ is given by the trace of the transition semigroup, and for subordinated Brownian motion, we can calculate the short time asymptotics of this quantity using recent work of Ba\~nuelos and Baudoin. In the case of the sphere, there is an interesting connection with the Funk-Hecke theorem.

Published
2017
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13. Convolution Semigroups of Probability Measures on Gelfand Pairs, Revisited

Author

David Applebaum

Subjects
Statistics and Probability, Discrete mathematics, Semigroup, Probability (math.PR), Cartan decomposition, Lie group, Convolution power, Circular convolution, Convolution, FOS: Mathematics, Special classes of semigroups, Convolution theorem, Mathematics - Probability, 60B15, Mathematics
Abstract

Our goal is to find classes of convolution semigroups on Lie groups G that give rise to interesting processes in symmetric spaces G/K. The K–bi–invariant convolution semigroups are a well–studied example. An appealing direction for the next step is to generalise to right K–invariant convolution semigroups, but recent work of Liao has shown that these are in one–to–one correspondence with K–bi–invariant convolution semigroups. We investigate a weaker notion of right K–invariance, but show that this is, in fact, the same as the usual notion. Another possible approach is to use generalised notions of negative definite functions, but this also leads to nothing new. We finally find an interesting class of convolution semigroups that are obtained by making use of the Cartan decomposition of a semisimple Lie group, and the solution of certain stochastic differential equations. Examples suggest that these are well–suited for generating random motion along geodesics in symmetric spaces.

Published
2016
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14. Séminaire de Probabilités XLVIII

Author

Mathias Beiglböck, Martin Huesmann, Makoto Maejima, Nicolas Privault, Franck Maunoury, Anna Aksamit, Ismael Bailleul, Nicolas Juillet, David Applebaum, Christophe Profeta, Dai Taguchi, Peter Kern, Kilian Raschel, Alexis Devulder, Libo Li, Camille Tardif, Anita Behme, Thomas Simon, Gilles Pagès, Florian Stebegg, Oleskiy Khorunzhiy, Jürgen Angst, Stéphane Laurent, Cédric Lecouvey, Songzi Li, Wendelin Werner, Alexander Lindner, Matyas Barczy, Vienna University of Technology (TU Wien), Rheinische Friedrich-Wilhelms-Universität Bonn, Columbia University [New York], Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), TO Simulate and CAlibrate stochastic models (TOSCA), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Institute of Mathematics [Debrecen], University of Debrecen Egyetem [Debrecen], Technische Universität Dresden = Dresden University of Technology (TU Dresden), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Nanyang Technological University [Singapour], Universität Ulm - Ulm University [Ulm, Allemagne], Keio University, Laboratoire de Mathématiques et Physique Théorique (LMPT), Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et Modélisation d'Evry (LaMME), Institut National de la Recherche Agronomique (INRA)-Université d'Évry-Val-d'Essonne (UEVE)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), North Dakota State University (NDSU), Laboratoire Traitement et Communication de l'Information (LTCI), Télécom ParisTech-Institut Mines-Télécom [Paris] (IMT)-Centre National de la Recherche Scientifique (CNRS), Institut Fourier (IF ), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Mathematical Institute [Oxford] (MI), University of Oxford, Université d'Évry-Val-d'Essonne (UEVE), School of Mathematics and Statistics [Sheffield] (SoMaS), University of Sheffield [Sheffield], Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology [Zürich] (ETH Zürich), Donati-Martin, Catherine, Lejay, Antoine, Rouault, Alain, ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), Technische Universität Wien (TU Wien), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Université de Versailles Saint-Quentin-en-Yvelines (UVSQ), University of Debrecen, Technische Universität Dresden (TUD), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), Université de Tours-Centre National de la Recherche Scientifique (CNRS), Institut National de la Recherche Agronomique (INRA)-Université d'Évry-Val-d'Essonne (UEVE)-ENSIIE-Centre National de la Recherche Scientifique (CNRS), Laboratoire Paul Painlevé - UMR 8524 (LPP), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), University of Oxford [Oxford], Eidgenössische Technische Hochschule - Swiss Federal Institute of Technology in Zürich [Zürich] (ETH Zürich), Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Laboratoire de Mathématiques et Modélisation d'Evry, Centre National de la Recherche Scientifique (CNRS)-Université de Lille, Technische Universität Wien ( TU Wien ), Bonn Universität [Bonn], Institut de Recherche Mathématique Avancée ( IRMA ), Université de Strasbourg ( UNISTRA ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Probabilités et Modèles Aléatoires ( LPMA ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), TO Simulate and CAlibrate stochastic models ( TOSCA ), Inria Sophia Antipolis - Méditerranée ( CRISAM ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut Élie Cartan de Lorraine ( IECL ), Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Lorraine ( UL ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques de Versailles ( LMV ), Université Paris-Saclay-Centre National de la Recherche Scientifique ( CNRS ) -Université de Versailles Saint-Quentin-en-Yvelines ( UVSQ ), Technische Universität Dresden ( TUD ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Universität Ulm, Laboratoire de Mathématiques et Physique Théorique ( LMPT ), Université de Tours-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques et Modélisation d'Evry ( LaMME ), Institut National de la Recherche Agronomique ( INRA ) -Université d'Évry-Val-d'Essonne ( UEVE ) -ENSIIE-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire Paul Painlevé - UMR 8524 ( LPP ), Université de Lille-Centre National de la Recherche Scientifique ( CNRS ), North Dakota State University ( NDSU ), Laboratoire Traitement et Communication de l'Information ( LTCI ), Télécom ParisTech, Institut Fourier ( IF ), Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ), Mathematical Institute [Oxford] ( MI ), Université d'Évry-Val-d'Essonne ( UEVE ), School of Mathematics and Statistics [Sheffield] ( SoMaS ), and Eidgenössische Technische Hochschule [Zürich] ( ETH Zürich )

Subjects
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2016
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15. Probabilistic trace and Poisson summation formulae on locally compact abelian groups

Author

David Applebaum

Subjects
Pure mathematics, Semigroup, Discrete group, Applied Mathematics, General Mathematics, Probability (math.PR), 010102 general mathematics, Poisson summation formula, Torus, 01 natural sciences, 010104 statistics & probability, symbols.namesake, FOS: Mathematics, symbols, Locally compact space, 0101 mathematics, Abelian group, Invariant (mathematics), Quotient, Mathematics - Probability, Mathematics
Abstract

We investigate convolution semigroups of probability measures with continuous densities on locally compact abelian groups, which have a discrete subgroup such that the factor group is compact. Two interesting examples of the quotient structure are the d-dimensional torus, and the adèlic circle. Our main result is to show that the Poisson summation formula for the density can be interpreted as a probabilistic trace formula, linking values of the density on the factor group to the trace of the associated semigroup on L 2 ${L^{2}}$ -space. The Gaussian is a very important example. For rotationally invariant α-stable densities, the trace formula is valid, but we cannot verify the Poisson summation formula. To prepare to study semistable laws on the adèles, we first investigate these on the p-adics, where we show they have continuous densities which may be represented as series expansions. We use these laws to construct a convolution semigroup on the adèles whose densities fail to satisfy the probabilistic trace formula.

Published
2017
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16. A generalised Gangolli–Lévy–Khintchine formula for infinitely divisible measures and Lévy processes on semi-simple Lie groups and symmetric spaces

Author

David Applebaum and Anthony H. Dooley

Subjects
Statistics and Probability, Extended Gangolli Lévy–Khintchine formula, Lie algebra, Generalised Eisenstein integral, Symmetric space, Measure (mathematics), Representation theory, Hyperbolic space, symbols.namesake, 60E07, 43A05, Probability measure, Mathematics, Discrete mathematics, Eisenstein transform, Lévy process, Lie group, 53C35, Fourier transform, 43A30, symbols, Statistics, Probability and Uncertainty, 60B15, 60G51, 22E30
Abstract

In 1964 R. Gangolli published a Lévy–Khintchine type formula which characterised $K$-bi-invariant infinitely divisible probability measures on a symmetric space $G/K$. His main tool was Harish-Chandra’s spherical functions which he used to construct a generalisation of the Fourier transform of a measure. In this paper we use generalised spherical functions (or Eisenstein integrals) and extensions of these which we construct using representation theory to obtain such a characterisation for arbitrary infinitely divisible probability measures on a non-compact symmetric space. We consider the example of hyperbolic space in some detail.

Published
2015
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17. Infinite dimensional Ornstein-Uhlenbeck processes driven by Lévy processes

Author

David Applebaum

Subjects
Statistics and Probability, 60H15, 60G51, Lévy process, Noise (electronics), Mehler semigroup, symbols.namesake, Mathematics::Probability, 60E07, FOS: Mathematics, Statistical physics, Mathematics, 60J80, invariant measure, Mathematical analysis, Probability (math.PR), Probabilistic logic, Hilbert space, Ornstein–Uhlenbeck process, Stochastic partial differential equation, branching property, operator self–decomposability, skew–convolution semigroup, Urbanik semigroup, symbols, 60H15, Invariant measure, 60H10, Ornstein-Uhlenbeck process, Mathematics - Probability, 60G51, cylindrical process
Abstract

We review the probabilistic properties of Ornstein-Uhlenbeck processes in Hilbert spaces driven by Levy processes. The emphasis is on the different contexts in which these processes arise, such as stochastic partial differential equations, continuous-state branching processes, generalised Mehler semigroups and operator self-decomposable distributions. We also examine generalisations to the case where the driving noise is cylindrical.

Published
2015

18. Second quantisation for skew convolution products of infinitely divisible measures

Author

Jan van Neerven and David Applebaum

Subjects
Statistics and Probability, Discrete mathematics, Kernel (set theory), Applied Mathematics, Gaussian, Probability (math.PR), 010102 general mathematics, 60B11, 60E07, 60G51, 60G57, 60H05, Banach space, Hilbert space, Statistical and Nonlinear Physics, Poisson random measure, 01 natural sciences, Convolution, 010104 statistics & probability, symbols.namesake, FOS: Mathematics, symbols, 0101 mathematics, Borel measure, Mathematics - Probability, Mathematical Physics, Probability measure, Mathematics
Abstract

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 $\lambda_2$., Comment: Some typos have been corrected. To appear in IDAQP

Published
2015
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19. The positive maximum principle on Lie groups and symmetric spaces

Author

Le Ngan, Trang and David, Applebaum

Subjects
510
Abstract

In this thesis we will use harmonic analysis to get new results in probability on Lie groups and symmetric spaces. We will establish necessary and sufficient conditions for the existence of a square integrable K-bi-invariant density of a K-bi-invariant measure. We will show that there is a topological isomorphism between K-bi-invariant smooth functions and a subspace of the Sugiura space of rapidly decreasing functions. Furthermore, we will extend Courrège's classical results to Lie groups and symmetric spaces, this consists of characterizing all linear operators on the space of smooth functions with compact support, that satisfy the positive maximum principle, as Lévy- type operators. We will specify some conditions under which such operators map to the Banach space of continuous functions vanishing at infinity, this allows us to study Feller semigroups and their generator in this context. We will show that on compact Lie groups all linear operators satisfying the positive maximum principle can be represented as pseudo-differential operators and on compact symmetric spaces they have analogous representations called spherical pseudo-differential operators.

Published
2019

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