## Kyoto Journal of Mathematics

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

#### 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

#### 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

#### References

• [1] E. B. Davies, One-Parameter Semigroups, London Math. Soc. Monogr. Ser. 15, Academic Press, London, 1980.
• [2] E. B. Davies, Linear Operators and Their Spectra, Cambridge Stud. Adv. Math. 106, Cambridge Univ. Press, Cambridge, 2007.
• [3] J.-D. Deuschel and D. W. Stroock, Large Deviations, Pure Appl. Math. 137, Academic Press, Boston, 1989.
• [4] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math. 194, Springer, New York, 2000.
• [5] P.-A. Meyer, “Notes sur les processus d’Ornstein-Uhlenbeck” in Seminar on Probability, XVI, Lecture Notes in Math. 920, Springer, Berlin, 1982, 95–133.
• [6] M. Reed and B. Simon, Method of Modern Mathematical Physics, IV, Analysis of Operators, Academic Press, New York, 1978.
• [7] M. Röckner and F.-Y. Wang, Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds, Forum Math. 15 (2003), 893–921.
• [8] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973.
• [9] I. Shigekawa, “Non-symmetric diffusions on a Riemannian manifold” in Probabilistic Approach to Geometry, Adv. Stud. Pure Math. 57, Math. Soc. Japan, Tokyo, 2010, 437–461.
• [10] E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Ann. of Math. Stud. 63, Princeton Univ. Press, Princeton, 1970.
• [11] H. Tanabe, Equations of Evolution, Monogr. Stud. Math. 6, Pitman, Boston, 1979.