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Remarks on the paper “Transient Markov convolution semi-groups and the associated negative definite functions”

Published online by Cambridge University Press:  22 January 2016

Masayuki Itô
Masayuki Itô*
Affiliation:
Department of Mathematics, Faculty of Sciences, Nagoya University, Furô-chô, Chikusa-ku, Nagoya, 464, Japan
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Let X be a locally compact and σ-compact abelian group and let denote the dual group of X. We denote by ξ a fixed Haar measure on X and by the Haar measure associated with ξ. In [2], we show the following

Theorem. Let (αt)t≧0 be a sub-Markov convolution semi-group on X and let ψ be the negative definite function associated with (αt)t≧0. Then (αt)t≧0 is transient if and only if Re (1/ψ) is locally -summable.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 102 , June 1986 , pp. 181 - 184
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1986

References

[ 1 ] Beurling, A., J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U. S. A., 45 (1959), 208–215.CrossRefGoogle Scholar
[ 2 ] Ito, M., Transient Markov convolution semi-groups and the associated negative definite functions, Nagoya Math. J., 92 (1983), 153–161.CrossRefGoogle Scholar