Diagonalization of the Lévy Laplacian and related stable processes

Hui Hsiung Kuo; Nobuaki Obata; Kimiaki Saitô

doi:10.1142/S0219025702000882

Diagonalization of the Lévy Laplacian and related stable processes

Hui Hsiung Kuo, Nobuaki Obata, Kimiaki Saitô

Center for Data-driven Science and Artificial Intelligence (CDS)

Research output: Contribution to journal › Article › peer-review

18 Citations (Scopus)

Abstract

Eigenfunctions of the Lévy Laplacian with an arbitrary real number as an eigenvalue are constructed by means of a coordinate change of white noise distributions. The Lévy Laplacian is diagonalized on the direct integral Hilbert space of such eigenfunctions and the corresponding equi-continuous semigroup is obtained. Moreover, an infinite dimensional stochastic process related to the Lévy Laplacian is constructed from a one-dimensional stable process.

Original language	English
Pages (from-to)	317-331
Number of pages	15
Journal	Infinite Dimensional Analysis, Quantum Probability and Related Topics
Volume	5
Issue number	3
DOIs	https://doi.org/10.1142/S0219025702000882
Publication status	Published - 2002 Sept

Keywords

Diagonalization
Equi-continuous semigroup
Exponential coordinate change
Gaussian space
Gel'fand triple
Lévy Laplacian
S-transform
Self-adjoint extension
Stable processes
Wiener-Itô decomposition

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10.1142/S0219025702000882

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title = "Diagonalization of the L{\'e}vy Laplacian and related stable processes",

abstract = "Eigenfunctions of the L{\'e}vy Laplacian with an arbitrary real number as an eigenvalue are constructed by means of a coordinate change of white noise distributions. The L{\'e}vy Laplacian is diagonalized on the direct integral Hilbert space of such eigenfunctions and the corresponding equi-continuous semigroup is obtained. Moreover, an infinite dimensional stochastic process related to the L{\'e}vy Laplacian is constructed from a one-dimensional stable process.",

keywords = "Diagonalization, Equi-continuous semigroup, Exponential coordinate change, Gaussian space, Gel'fand triple, L{\'e}vy Laplacian, S-transform, Self-adjoint extension, Stable processes, Wiener-It{\^o} decomposition",

author = "Kuo, {Hui Hsiung} and Nobuaki Obata and Kimiaki Sait{\^o}",

note = "Funding Information: This work was supported in part by the Research Project \Quantum Information Theoretical Approach to Life Science{"} for the Academic Frontier in Science promoted by the Ministry of Education in Japan and was also supported in part by JSPS-PAN Joint Research Project \In nite Dimensional Harmonic Analysis{"}. The authors are grateful for the support.",

year = "2002",

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doi = "10.1142/S0219025702000882",

language = "English",

volume = "5",

pages = "317--331",

journal = "Infinite Dimensional Analysis, Quantum Probability and Related Topics",

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T1 - Diagonalization of the Lévy Laplacian and related stable processes

AU - Kuo, Hui Hsiung

AU - Obata, Nobuaki

AU - Saitô, Kimiaki

N1 - Funding Information: This work was supported in part by the Research Project \Quantum Information Theoretical Approach to Life Science" for the Academic Frontier in Science promoted by the Ministry of Education in Japan and was also supported in part by JSPS-PAN Joint Research Project \In nite Dimensional Harmonic Analysis". The authors are grateful for the support.

PY - 2002/9

Y1 - 2002/9

N2 - Eigenfunctions of the Lévy Laplacian with an arbitrary real number as an eigenvalue are constructed by means of a coordinate change of white noise distributions. The Lévy Laplacian is diagonalized on the direct integral Hilbert space of such eigenfunctions and the corresponding equi-continuous semigroup is obtained. Moreover, an infinite dimensional stochastic process related to the Lévy Laplacian is constructed from a one-dimensional stable process.

AB - Eigenfunctions of the Lévy Laplacian with an arbitrary real number as an eigenvalue are constructed by means of a coordinate change of white noise distributions. The Lévy Laplacian is diagonalized on the direct integral Hilbert space of such eigenfunctions and the corresponding equi-continuous semigroup is obtained. Moreover, an infinite dimensional stochastic process related to the Lévy Laplacian is constructed from a one-dimensional stable process.

KW - Diagonalization

KW - Equi-continuous semigroup

KW - Exponential coordinate change

KW - Gaussian space

KW - Gel'fand triple

KW - Lévy Laplacian

KW - S-transform

KW - Self-adjoint extension

KW - Stable processes

KW - Wiener-Itô decomposition

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JO - Infinite Dimensional Analysis, Quantum Probability and Related Topics

JF - Infinite Dimensional Analysis, Quantum Probability and Related Topics

IS - 3

ER -

Diagonalization of the Lévy Laplacian and related stable processes

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