## Kyoto Journal of Mathematics

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

### Exponential convergence of Markovian semigroups and their spectra on ${L}^{p}$-spaces

Seiichiro Kusuoka and Ichiro Shigekawa

#### Abstract

Markovian semigroups on ${L}^{2}$-space with suitable conditions can be regarded as Markovian semigroups on ${L}^{p}$-spaces for $p\in [1,\infty )$. When we additionally assume the ergodicity of the Markovian semigroups, the rate of convergence on ${L}^{p}$-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 ${L}^{p}$-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 ${L}^{p}$-spaces, because the rate of convergence is closely related to the spectra.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 2 June 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1401741283

**Digital Object Identifier**

doi:10.1215/21562261-2642431

**Mathematical Reviews number (MathSciNet)**

MR3215572

**Zentralblatt MATH identifier**

1295.65061

**Subjects**

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

#### Citation

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. https://projecteuclid.org/euclid.kjm/1401741283