Spectral expansions of non-self-adjoint generalized Laguerre semigroups

We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a subset of negative definite functions...

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Bibliographische Detailangaben
Hauptverfasser: Patie, Pierre, Savov, Mladen
Format: E-Book Buch
Sprache:Englisch
Veröffentlicht: Providence, Rhode Island American Mathematical Society 2021
Ausgabe:1
Schriftenreihe:Memoirs of the American Mathematical Society
Schlagworte:
Laguerre polynomials
Nonselfadjoint operators
Spectral theory (Mathematics)
intertwining
functional equations
Spectral theory
Laguerre polynomials
asymptotic analysis
infinitely divisible distribution
Markov semigroups
convergence to equilibrium
non-self-adjoint integro-differential operators
Bernstein functions
Hilbert sequences
special functions
ISBN:9781470449360, 1470449366
ISSN:0065-9266, 1947-6221
Online-Zugang:Volltext
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Beschreibung
Zusammenfassung:We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between this class and a self-adjoint Markov semigroup, whose spectral expansion is expressed in terms of the classical Laguerre polynomials. As a by-product, we derive smoothness properties for the solution to the associated Cauchy problem as well as for the heat kernel. Our methodology also reveals a variety of possible decays, including the hypocoercivity type phenomena, for the speed of convergence to equilibrium for this class and enables us to provide an interpretation of these in terms of the rate of growth of the weighted Hilbert space norms of the spectral projections. Depending on the analytic properties of the aforementioned negative definite functions, we are led to implement several strategies, which require new developments in a variety of contexts, to derive precise upper bounds for these norms.
Bibliographie:Includes bibliographical references (p. 177-182)
July 2021, volume 272, number 1336 (sixth of 7 numbers)
ISBN:9781470449360
1470449366
ISSN:0065-9266
1947-6221
DOI:10.1090/memo/1336