Variational formula for dirichlet forms and estimates of principal eigenvalues for symmetric α-stable processes

Yuichi Shiozawa; Masayoshi Takeda

doi:10.1007/s11118-004-5392-7

Variational formula for dirichlet forms and estimates of principal eigenvalues for symmetric α-stable processes

Yuichi Shiozawa, Masayoshi Takeda

Graduate School of Natural Science and Technology

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

5 Citations (Scopus)

Abstract

In this paper, we prove a variational formula for Dirichlet forms generated by general symmetric Markov processes. As its applications, we obtain lower bound estimates of the bottom of spectrum for symmetric Markov processes. Moreover, for a positive measure μ charging no set of zero capacity, we give a new proof of an L ²(μ)-estimate of functions in Dirichlet spaces. Finally, we calculate the principal eigenvalues for absorbing and time changed α-stable processes and obtain conditions for some measures being gaugeable.

Original language	English
Pages (from-to)	135-151
Number of pages	17
Journal	Potential Analysis
Volume	23
Issue number	2
DOIs	https://doi.org/10.1007/s11118-004-5392-7
Publication status	Published - Sep 1 2005

Keywords

Dirichlet form
Principal eigenvalue
Symmetric α-stable process
Time change
Variational formula

ASJC Scopus subject areas

Analysis

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AB - In this paper, we prove a variational formula for Dirichlet forms generated by general symmetric Markov processes. As its applications, we obtain lower bound estimates of the bottom of spectrum for symmetric Markov processes. Moreover, for a positive measure μ charging no set of zero capacity, we give a new proof of an L 2(μ)-estimate of functions in Dirichlet spaces. Finally, we calculate the principal eigenvalues for absorbing and time changed α-stable processes and obtain conditions for some measures being gaugeable.

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KW - Principal eigenvalue

KW - Symmetric α-stable process

KW - Time change

KW - Variational formula

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Variational formula for dirichlet forms and estimates of principal eigenvalues for symmetric α-stable processes

Abstract

Keywords

ASJC Scopus subject areas

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