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SOME FELLER SEMIGROUPS ON $C_{\infty }(\mathbb{R}^{n}\times \mathbb{Z}^{m})$ GENERATED BY PSEUDO-DIFFERENTIAL OPERATORS

Part of: Groups and semigroups of linear operators, their generalizations and applications Miscellaneous topics - Partial differential equations Markov processes

Published online by Cambridge University Press:  15 April 2015

Kristian P. Evans ,
Niels Jacob  and
Chenglin Shen
Kristian P. Evans
Affiliation:
Mathematics Department, Swansea University, Singleton Park, Swansea SA2 8PP, U.K. email [email protected]
Niels Jacob
Affiliation:
Mathematics Department, Swansea University, Singleton Park, Swansea SA2 8PP, U.K. email [email protected]
Chenglin Shen
Affiliation:
Mathematics Department, Swansea University, Singleton Park, Swansea SA2 8PP, U.K. email [email protected]
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Abstract

In this paper we construct some Feller semigroups, hence Feller processes, with state space $\mathbb{R}^{n}\times \mathbb{Z}^{m}$ starting with pseudo-differential operators having symbols defined on $\mathbb{R}^{n}\times \mathbb{R}^{n}\times \mathbb{Z}^{m}\times \mathbb{T}^{m}$.

MSC classification

Primary: 35S99: None of the above, but in this section 47D07: Markov semigroups and applications to diffusion processes 60J35: Transition functions, generators and resolvents
Type
Research Article
Information
Mathematika , Volume 61 , Issue 2 , May 2015 , pp. 402 - 413
Copyright
Copyright © University College London 2015 

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References

Berg, Chr. and Forst, G., Potential Theory on Locally Compact Abelian Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete (Ser. II) 87), Springer (Berlin, 1975).CrossRefGoogle Scholar
Böttcher, B., Schilling, R. and Wang, J., Lévy Type Processes: Construction, Approximation and Sample Path Properties (Lecture Notes in Mathematics 2099), Springer (Cham, 2013).Google Scholar
Courrège, Ph., Sur la forme intégro-différentielle des opérateurs de $C_{K}^{\infty }$ dans $C$ satisfaisant au principe du maximum. In: Sém. Théorie du Potentiel 1965/66. Exposé 2, 38pp.Google Scholar
Ethier, S. N. and Kurtz, Th. G., Markov Processes – Characterization and Convergence (Wiley Series in Probability and Mathematical Statistics), Wiley (New York, 1986).CrossRefGoogle Scholar
Evans, K. and Jacob, N., Q-matrices as pseudo-differential operators with negative definite symbols. Math. Nachr. 286 2013, 631–640.CrossRefGoogle Scholar
Evans, K., Jacob, N. and Morris, O., On a class of pseudo-differential operators on ℝn ×ℤm generating Feller semigroups. Rev. Roumaine Math. Pures Appl. 59 2014, 55–75.Google Scholar
Jacob, N., A class of Feller semigroups generated by pseudo-differential operators. Math. Z. 215 1994, 151–166.CrossRefGoogle Scholar
Jacob, N., Pseudo-differential Operators and Markov Processes I. Fourier Analysis and Semigroups, Imperial College Press (London, 2001).Google Scholar
Jacob, N., Pseudo-differential Operators and Markov Processes II. Generators and their Potential Theory, Imperial College Press (London, 2002).CrossRefGoogle Scholar
Jacob, N., Knopova, V., Landwehr, S. and Schilling, R. L., A geometric interpretation of the transition density of a symmetric Lévy process. Sci. China Math. 55 2012, 1099–1126.CrossRefGoogle Scholar
Mao, X. and Yuan, C., Stochastic Differential Equations with Markov Switching, Imperial College Press (London, 2006).CrossRefGoogle Scholar
Maz’ya, V., Sobolev Spaces with Applications to Elliptic Partial Differential Equations, 2nd edn, revised and augmented (Grundlchren der mathematischen Wissenschaften 342), Springer (Berlin, 2011).CrossRefGoogle Scholar
Morris, O. C., On a class of one-parameter operator semigroups with state space $\mathbb{R}^{n}\times \mathbb{Z}^{m}$ generated by pseudo-differential operators. PhD Thesis, Swansea University, 2013.Google Scholar
Shen, C., PhD Thesis, Swansea University (in preparation).Google Scholar