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

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.