LEADER 06443nam 22006975 450 001 9910585774303321 005 20251113182621.0 010 $a3-031-05296-X 024 7 $a10.1007/978-3-031-05296-5 035 $a(MiAaPQ)EBC7052881 035 $a(Au-PeEL)EBL7052881 035 $a(CKB)24286023400041 035 $a(PPN)263896986 035 $a(DE-He213)978-3-031-05296-5 035 $a(BIP)85218439 035 $a(BIP)83732284 035 $a(EXLCZ)9924286023400041 100 $a20220727d2022 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aConvolution-like Structures, Differential Operators and Diffusion Processes /$fby Rúben Sousa, Manuel Guerra, Semyon Yakubovich 205 $a1st ed. 2022. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2022. 215 $a1 online resource (269 pages) 225 1 $aLecture Notes in Mathematics,$x1617-9692 ;$v2315 311 08$a3-031-05295-1 320 $aIncludes bibliographical references and index. 327 $aIntro -- Preface -- Contents -- List of Symbols -- 1 Introduction -- 1.1 Motivation and Scope -- 1.2 Organization of the Book -- 2 Preliminaries -- 2.1 Continuous-Time Markov Processes -- 2.2 Sturm-Liouville Theory -- 2.2.1 Solutions of the Sturm-Liouville Equation -- 2.2.2 Eigenfunction Expansions -- 2.2.3 Diffusion Semigroups Generated by Sturm-Liouville Operators -- 2.2.4 Remarkable Particular Cases -- 2.3 Generalized Convolutions and Hypergroups -- 2.4 Harmonic Analysis with Respect to the Kingman Convolution -- 3 The Whittaker Convolution -- 3.1 A Special Case: The Kontorovich-Lebedev Convolution -- 3.2 The Product Formula for the Whittaker Function -- 3.3 Whittaker Translation -- 3.4 Index Whittaker Transforms -- 3.5 Whittaker Convolution of Measures -- 3.5.1 Infinitely Divisible Distributions -- 3.5.2 Lévy-Khintchine Type Representation -- 3.6 Lévy Processes with Respect to the Whittaker Convolution -- 3.6.1 Convolution Semigroups -- 3.6.2 Lévy and Gaussian Processes -- 3.6.3 Some Auxiliary Results on the Whittaker Translation -- 3.6.4 Moment Functions -- 3.6.5 Lévy-Type Characterization of the Shiryaev Process -- 3.7 Whittaker Convolution of Functions -- 3.7.1 Mapping Properties in the Spaces Lp(r?) -- 3.7.2 The Convolution Banach Algebra L?,? -- 3.8 Convolution-Type Integral Equations -- 4 Generalized Convolutions for Sturm-Liouville Operators -- 4.1 Known Results and Motivation -- 4.2 Laplace-Type Representation -- 4.3 The Existence Theorem for Sturm-Liouville Product Formulas -- 4.3.1 The Associated Hyperbolic Cauchy Problem -- 4.3.2 The Time-Shifted Product Formula -- 4.3.3 The Product Formula for w? as the Limit Case -- 4.4 Sturm-Liouville Transform of Measures -- 4.5 Sturm-Liouville Convolution of Measures -- 4.5.1 Infinite Divisibility and Lévy-Khintchine Type Representation -- 4.5.2 Convolution Semigroups. 327 $a4.5.3 Additive and Lévy Processes -- 4.6 Sturm-Liouville Hypergroups -- 4.6.1 The Nondegenerate Case -- 4.6.2 The Degenerate Case: Degenerate Hypergroups of Full Support -- 4.7 Harmonic Analysis on Lp Spaces -- 4.7.1 A Family of L1 Spaces -- 4.7.2 Application to Convolution-Type Integral Equations -- 5 Convolution-Like Structures on Multidimensional Spaces -- 5.1 Convolutions Associated with Conservative Strong Feller Semigroups -- 5.2 Nonexistence of Convolutions: Diffusion Processes on Bounded Domains -- 5.2.1 Special Cases and Numerical Examples -- 5.2.2 Some Auxiliary Results -- 5.2.3 Eigenfunction Expansions, Critical Points and Nonexistence Theorems -- 5.3 Nonexistence of Convolutions: One-Dimensional Diffusions -- 5.4 Families of Convolutions on Riemannian Structures with Cone-Like Metrics -- 5.4.1 The Eigenfunction Expansion of the Laplace-Beltrami Operator -- 5.4.2 Product Formulas and Convolutions -- 5.4.3 Infinitely Divisible Measures and Convolution Semigroups -- 5.4.4 Special Cases -- 5.4.5 Product Formulas and Convolutions Associated with Elliptic Operators on Subsets of R2 -- A Some Open Problems -- References -- Index. 330 $aThis book provides an introduction to recent developments in the theory of generalized harmonic analysis and its applications. It is well known that convolutions, differential operators and diffusion processes are interconnected: the ordinary convolution commutes with the Laplacian, and the law of Brownian motion has a convolution semigroup property with respect to the ordinary convolution. Seeking to generalize this useful connection, and also motivated by its probabilistic applications, the book focuses on the following question: given a diffusion process Xt on a metric space E, can we construct a convolution-like operator * on the space of probability measures on E with respect to which the law of Xt has the *-convolution semigroup property? A detailed analysis highlights the connection between the construction of convolution-like structures and disciplines such as stochastic processes, ordinary and partial differential equations, spectral theory,special functions and integral transforms. The book will be valuable for graduate students and researchers interested in the intersections between harmonic analysis, probability theory and differential equations. 410 0$aLecture Notes in Mathematics,$x1617-9692 ;$v2315 606 $aProbabilities 606 $aOperator theory 606 $aFunctions, Special 606 $aMathematical analysis 606 $aProbability Theory 606 $aOperator Theory 606 $aSpecial Functions 606 $aIntegral Transforms and Operational Calculus 615 0$aProbabilities. 615 0$aOperator theory. 615 0$aFunctions, Special. 615 0$aMathematical analysis. 615 14$aProbability Theory. 615 24$aOperator Theory. 615 24$aSpecial Functions. 615 24$aIntegral Transforms and Operational Calculus. 676 $a512.86 700 $aSousa$b Ru?ben$c(Mathematician),$01252162 702 $aGuerra$b Manuel$g(Manuel Cidraes Castro), 702 $aYakubovich$b S. B$g(Semen B.),$f1961- 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910585774303321 996 $aConvolution-like structures, differential operators and diffusion processes$92999680 997 $aUNINA