## On bounded solutions to convolution equations

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- by Enrico Priola and Jerzy Zabczyk PDF
- Proc. Amer. Math. Soc.
**134**(2006), 3275-3286 Request permission

## 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).## References

- R. M. Blumenthal and R. K. Getoor,
*Markov processes and potential theory*, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR**0264757** - Mauro Marques and Stamatis Cambanis,
*Admissible and singular translates of stable processes*, Probability theory on vector spaces, IV (Łańcut, 1987) Lecture Notes in Math., vol. 1391, Springer, Berlin, 1989, pp. 239–257. MR**1020566**, DOI 10.1007/BFb0083394 - Gustave Choquet and Jacques Deny,
*Sur l’équation de convolution $\mu =\mu \ast \sigma$*, C. R. Acad. Sci. Paris**250**(1960), 799–801 (French). MR**119041** - Cho-Ho Chu and Chi-Wai Leung,
*The convolution equation of Choquet and Deny on $\rm [IN]$-groups*, Integral Equations Operator Theory**40**(2001), no. 4, 391–402. MR**1839466**, DOI 10.1007/BF01198136 - Cho-Ho Chu, Titus Hilberdink, and John Howroyd,
*A matrix-valued Choquet-Deny theorem*, Proc. Amer. Math. Soc.**129**(2001), no. 1, 229–235. MR**1784024**, DOI 10.1090/S0002-9939-00-05694-X - Cho-Ho Chu and Anthony To-Ming Lau,
*Harmonic functions on groups and Fourier algebras*, Lecture Notes in Mathematics, vol. 1782, Springer-Verlag, Berlin, 2002. MR**1914221**, DOI 10.1007/b83280 - Giuseppe Da Prato and Jerzy Zabczyk,
*Second order partial differential equations in Hilbert spaces*, London Mathematical Society Lecture Note Series, vol. 293, Cambridge University Press, Cambridge, 2002. MR**1985790**, DOI 10.1017/CBO9780511543210 - Nelson Dunford and Jacob T. Schwartz,
*Linear operators. Part I*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR**1009162** - E. B. Dynkin,
*Markov processes. Vols. I, II*, Die Grundlehren der mathematischen Wissenschaften, Band 121, vol. 122, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. MR**0193671** - William Feller,
*An introduction to probability theory and its applications. Vol. II.*, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR**0270403** - Shaul R. Foguel,
*The ergodic theory of Markov processes*, Van Nostrand Mathematical Studies, No. 21, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR**0261686** - Ĭ. Ī. Gīhman and A. V. Skorohod,
*The theory of stochastic processes. I*, Die Grundlehren der mathematischen Wissenschaften, Band 210, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by S. Kotz. MR**0346882** - Ĭ. Ī. Gīhman and A. V. Skorohod,
*The theory of stochastic processes. II*, Die Grundlehren der mathematischen Wissenschaften, Band 218, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by Samuel Kotz. MR**0375463** - Victor Goodman,
*A Liouville theorem for abstract Wiener spaces*, Amer. J. Math.**95**(1973), 215–220. MR**322971**, DOI 10.2307/2373653 - B. E. Johnson,
*Harmonic functions on nilpotent groups*, Integral Equations Operator Theory**40**(2001), no. 4, 454–464. MR**1839470**, DOI 10.1007/BF01198140 - K. R. Parthasarathy,
*Probability measures on metric spaces*, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR**0226684** - Ross G. Pinsky,
*Positive harmonic functions and diffusion*, Cambridge Studies in Advanced Mathematics, vol. 45, Cambridge University Press, Cambridge, 1995. MR**1326606**, DOI 10.1017/CBO9780511526244 - Enrico Priola and Jerzy Zabczyk,
*Liouville theorems for non-local operators*, J. Funct. Anal.**216**(2004), no. 2, 455–490. MR**2095690**, DOI 10.1016/j.jfa.2004.04.001 - E. Priola and J. Zabczyk,
*On bounded solutions to convolution equations,*Preprint 11, Department of Mathematics, University of Turin (2005), see http://www2.dm.unito.it/paginepersonali/priola/index.htm. - C. Radhakrishna Rao and D. N. Shanbhag,
*Choquet-Deny type functional equations with applications to stochastic models*, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester, 1994. MR**1329995** - Albert Raugi,
*A general Choquet-Deny theorem for nilpotent groups*, Ann. Inst. H. Poincaré Probab. Statist.**40**(2004), no. 6, 677–683 (English, with English and French summaries). MR**2096214**, DOI 10.1016/j.anihpb.2003.06.004 - Ken-iti Sato,
*Lévy processes and infinitely divisible distributions*, Cambridge Studies in Advanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original; Revised by the author. MR**1739520** - N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan,
*Probability distributions on Banach spaces*, Mathematics and its Applications (Soviet Series), vol. 14, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian and with a preface by Wojbor A. Woyczynski. MR**1435288**, DOI 10.1007/978-94-009-3873-1 - Joel Zinn,
*Admissible translates of stable measures*, Studia Math.**54**(1975/76), no. 3, 245–257. MR**400376**, DOI 10.4064/sm-54-3-245-257

## Additional Information

**Enrico Priola**- Affiliation: Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123, Torino, Italy
- Email: priola@dm.unito.it
**Jerzy Zabczyk**- Affiliation: Instytut Matematyczny, Polskiej Akademii Nauk, ul. Sniadeckich 8, 00-950, War- szawa, Poland
- Email: zabczyk@impan.gov.pl
- 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
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**134**(2006), 3275-3286 - MSC (2000): Primary 43A55, 68B15, 47D07, 31C05
- DOI: https://doi.org/10.1090/S0002-9939-06-08608-4
- MathSciNet review: 2231912