Kyoto Journal of Mathematics

Exponential convergence of Markovian semigroups and their spectra on Lp-spaces

Seiichiro Kusuoka and Ichiro Shigekawa

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Markovian semigroups on L2-space with suitable conditions can be regarded as Markovian semigroups on Lp-spaces for p[1,). When we additionally assume the ergodicity of the Markovian semigroups, the rate of convergence on Lp-space for each p is considerable. However, the rate of convergence depends on the norm of the space. The purpose of this paper is to investigate the relation between the rates on Lp-spaces for different p’s, to obtain some sufficient condition for the rates to be independent of p, and to give an example for which the rates depend on p. We also consider spectra of Markovian semigroups on Lp-spaces, because the rate of convergence is closely related to the spectra.

Article information

Kyoto J. Math., Volume 54, Number 2 (2014), 367-399.

First available in Project Euclid: 2 June 2014

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Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47A10: Spectrum, resolvent


Kusuoka, Seiichiro; Shigekawa, Ichiro. Exponential convergence of Markovian semigroups and their spectra on $L^{p}$ -spaces. Kyoto J. Math. 54 (2014), no. 2, 367--399. doi:10.1215/21562261-2642431.

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