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On bounded solutions to convolution equations

Authors: Enrico Priola and Jerzy Zabczyk
Journal: Proc. Amer. Math. Soc. 134 (2006), 3275-3286
MSC (2000): Primary 43A55, 68B15, 47D07, 31C05
Published electronically: May 8, 2006
MathSciNet review: 2231912
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Abstract: Periodicity of bounded solutions for convolution equations on a separable abelian metric group $ G$ is established, and related Liouville type theorems are obtained. A non-constant Borel and bounded harmonic function is constructed for an arbitrary convolution semigroup on any infinite-dimensional separable Hilbert space, generalizing a classical result by Goodman (1973).

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  • 1. R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Academic Press (1968) New York-London. MR 0264757 (41:9348)
  • 2. S. Cambanis and M. Marques, Admissible and singular translates of stable processes, Probability theory on vector spaces IV, Lecture Notes in Math. 1391 (1989) 239-257. MR 1020566 (91h:60044)
  • 3. G. Choquet and J. Deny, Sur l'equation de convolution $ \mu = \mu * \sigma $, C. R. Acad. Sci. Paris 250 (1960) 799-801. MR 0119041 (22:9808)
  • 4. C.-H. Chu and C.-W. Leung, The convolution equation of Choquet and Deny on [IN]-groups, Integral Equations Operator Theory 40 (2001) 391-402. MR 1839466 (2002e:43001)
  • 5. C.-H. Chu, T. Hilberdink, and J. Howroyd, A matrix-valued choquet deny theorem, Proc. Amer. Soc. 129 (2000) 229-235. MR 1784024 (2001j:43007)
  • 6. C.-H. Chu and A. T.-M. Lau, Harmonic functions on groups and Fourier algebras, Lecture Notes in Mathematics 1782, Springer-Verlag (2002) Berlin. MR 1914221 (2003i:43001)
  • 7. G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society Lecture Note Series 293, Cambridge University Press (2002). MR 1985790 (2004e:47058)
  • 8. N. Dunford, J. T. Schwartz, Linear operators. Part I. General theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc. (1958) New York. MR 1009162 (90g:47001a)
  • 9. E. B. Dynkin, Markov Processes, Vol. I-II, Springer-Verlag (1965) Berlin-Göttingen-Heidelberg. MR 0193671 (33:1887)
  • 10. W. Feller, An introduction to probability theory and its applications, Vol. II, second edition, John Wiley & Sons, Inc. (1971) New York-London-Sydney. MR 0270403 (42:5292)
  • 11. S. R. Foguel, The ergodic theory of Markov processes, Van Nostrand Mathematical Studies 21 (1969) New York-Toronto. MR 0261686 (41:6299)
  • 12. I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes, Vol. I, Springer-Verlag (1974) Berlin. MR 0346882 (49:11603)
  • 13. I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes, Vol. II, Springer-Verlag (1974) Berlin. MR 0375463 (51:11656)
  • 14. V. Goodman, A Liouville theorem for abstract Wiener spaces, Amer. J. Math. 95 (1973) 215-220. MR 0322971 (48:1329)
  • 15. B. E. Johnson, Harmonic functions on nilpotent groups, Integr. Equ. Oper. Theory 40 (2001) 454-464. MR 1839470 (2002c:43002)
  • 16. K. R. Parthasarathy, Probability measures on metric spaces, Academic Press (1967) New York and London. MR 0226684 (37:2271)
  • 17. R. G. Pinsky, Positive Harmonic Functions and Diffusion, Cambridge Studies in Advanced Mathematics 45 (1995). MR 1326606 (96m:60179)
  • 18. E. Priola and J. Zabczyk, Liouville Theorems for non-local operators, J. Funct. Anal. 216 (2004) 455-490. MR 2095690 (2005g:35315)
  • 19. E. Priola and J. Zabczyk, On bounded solutions to convolution equations, Preprint 11, Department of Mathematics, University of Turin (2005), see
  • 20. C. R. Rao and D. N. Shanbhag, Choquet-Deny type functional equations with applications to stochastic models, John Wiley & Sons (1994). MR 1329995 (97d:60020)
  • 21. A. Raugi, A general Choquet-Deny theorem for nilpotent groups, Ann. Inst. H. Poincaré Probab. Statist. 40 (2004) 677-683. MR 2096214 (2005h:60016)
  • 22. K-I. Sato, Lévy processes and infinite divisible distributions, Cambridge University Press (1999). MR 1739520 (2003b:60064)
  • 23. N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan, Probability distributions on Banach spaces, Mathematics and its Applications, D. Reidel Publishing Co. (1987) Dordrecht. MR 1435288 (97k:60007)
  • 24. J. Zinn, Admissible translates of stable measures, Studia Math. 54 (1976) 245-257. MR 0400376 (53:4210)

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Additional Information

Enrico Priola
Affiliation: Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123, Torino, Italy

Jerzy Zabczyk
Affiliation: Instytut Matematyczny, Polskiej Akademii Nauk, ul. Sniadeckich 8, 00-950, War- szawa, Poland

Keywords: Convolution equations on groups, bounded harmonic functions, L\'evy processes
Received by editor(s): May 25, 2005
Published electronically: May 8, 2006
Additional Notes: The first author was partially supported by Italian National Project MURST “Equazioni di Kolmogorov” and by Contract No ICA1-CT-2000-70024 between European Community and the Stefan Banach International Mathematical Center in Warsaw.
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2006 American Mathematical Society

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