Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Regularity of solutions to stochastic Volterra equations with infinite delay

Authors: Anna Karczewska and Carlos Lizama
Journal: Proc. Amer. Math. Soc. 135 (2007), 531-540
MSC (2000): Primary 60H20; Secondary 60H05, 45D05.
Published electronically: August 2, 2006
MathSciNet review: 2255300
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this article we give necessary and sufficient conditions providing regularity of solutions to stochastic Volterra equations with infinite delay on a $ d$-dimensional torus. The harmonic analysis techniques and stochastic integration in function spaces are used. The work applies to both the stochastic heat and wave equations.

References [Enhancements On Off] (What's this?)

  • 1. R. Adams, Sobolev Spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
  • 2. Ph. Clément, G. Da Prato, Some results on stochastic convolutions arising in Volterra equations perturbed by noise, Rend. Math. Acc. Lincei. s. 9, 7, (1996) 147-153. MR 1454409 (98e:60106)
  • 3. Ph. Clément, G. Da Prato, Existence and regularity results for an integral equation with infinite delay in a Banach space, Integral Equations and Operator Theory, 11, (1988) 480-500. MR 0950513 (89f:45019)
  • 4. R. Dalang, N. Frangos, The stochastic wave equation in two spatial dimensions, The Annals of Probability 26, (1998) 187-212. MR 1617046 (99c:60127)
  • 5. C. Gasquet, P. Witomski, Fourier Analysis and Applications, Springer-Verlag, New York, Berlin, 1999. MR 1657104 (99h:42003)
  • 6. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products, Corrected and Enlarged Revision, Academic Press, San Diego, New York, London, Tokyo, 1980.
  • 7. G. Gripenberg, S.-O. Londen, O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications 34, Cambridge University Press, Cambridge-New York, 1990. MR 1050319 (91c:45003)
  • 8. A. Karczewska, J. Zabczyk, A note on stochastic wave equations, G. Lumer and L. Weis (eds.), Evolution equations and their applications in physical and life sciences, Marcel Dekker, Lect. Notes Pure Appl. Math. 215, 501-511 (2001). MR 1818028 (2002b:60112)
  • 9. A. Karczewska, J. Zabczyk, Stochastic PDE's with function-valued solutions. Ph. Clément (ed.) et al., Infinite dimensional stochastic analysis. Royal Netherlands Academy of Arts and Sciences. Verh. Afd. Natuurkd., 1. Reeks, K. Ned. Akad. Wet. 52, 197-216 (2000). MR 1832378 (2002h:60132)
  • 10. A. Karczewska, J. Zabczyk, Regularity of solutions to stochastic Volterra equations, Rend. Math. Acc. Lincei. s. 9, 11, (2001) 141-154. MR 1841688 (2002f:60124)
  • 11. N.S. Landkof, Foundations of modern potential theory, Springer-Verlag, Berlin, 1975. MR 0350027 (50:2520)
  • 12. A. Millet, M. Sanz-Solé, A stochastic wave equation in two space dimension: smoothness of the law, The Annals of Probability 27 (2), (1999) 803-844. MR 1698971 (2001e:60130)
  • 13. C. Mueller, Long time existence for the wave equation with a noise term, The Annals of Probability 25 (1), (1997) 133-151. MR 1428503 (98b:60113)
  • 14. J.W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29, (1971) 187-204. MR 0295683 (45:4749)
  • 15. J. Prüss, Evolutionary Integral Equations and Applications. Monographs Math., vol. 87, Birkhäuser Verlag, Basel, 1993. MR 1238939 (94h:45010)
  • 16. M. Reed, B. Simon, Methods of modern mathematical physics, Vol. II, Academic Press, New York, 1975.
  • 17. L. Schwartz, Méthodes mathématiques pour les sciences physiques, Hermann, Paris, 1965. MR 0143360 (26:919)
  • 18. D. Stroock, Probability Theory, an analytic view, Cambridge University Press, Cambridge, 1993. MR 1267569 (95f:60003)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 60H20, 60H05, 45D05.

Retrieve articles in all journals with MSC (2000): 60H20, 60H05, 45D05.

Additional Information

Anna Karczewska
Affiliation: Department of Mathematics, University of Zielona Góra, ul. Szafrana 4a, 65-246 Zielona Góra, Poland

Carlos Lizama
Affiliation: Departamento de Matemática, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307-Correo 2, Santiago, Chile

Keywords: Stochastic Volterra equation, function-valued solutions, equations on a torus, spatially homogeneous Wiener process
Received by editor(s): April 15, 2005
Received by editor(s) in revised form: August 25, 2005
Published electronically: August 2, 2006
Additional Notes: The second author was supported in part by FONDECYT Grant #1050084
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society