Regularity of solutions to stochastic Volterra equations with infinite delay

Karczewska, Anna; Lizama, Carlos

doi:10.1090/S0002-9939-06-08487-5

Regularity of solutions to stochastic Volterra equations with infinite delay
HTML articles powered by AMS MathViewer

by Anna Karczewska and Carlos Lizama PDF

Proc. Amer. Math. Soc. 135 (2007), 531-540 Request permission

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

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Philippe Clément and Giuseppe Da Prato, Some results on stochastic convolutions arising in Volterra equations perturbed by noise, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 7 (1996), no. 3, 147–153 (English, with English and Italian summaries). MR 1454409
Ph. Clément and G. Da Prato, Existence and regularity results for an integral equation with infinite delay in a Banach space, Integral Equations Operator Theory 11 (1988), no. 4, 480–500. MR 950513, DOI 10.1007/BF01199303
Robert C. Dalang and N. E. Frangos, The stochastic wave equation in two spatial dimensions, Ann. Probab. 26 (1998), no. 1, 187–212. MR 1617046, DOI 10.1214/aop/1022855416
C. Gasquet and P. Witomski, Fourier analysis and applications, Texts in Applied Mathematics, vol. 30, Springer-Verlag, New York, 1999. Filtering, numerical computation, wavelets; Translated from the French and with a preface by R. Ryan. MR 1657104, DOI 10.1007/978-1-4612-1598-1
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.
G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, Encyclopedia of Mathematics and its Applications, vol. 34, Cambridge University Press, Cambridge, 1990. MR 1050319, DOI 10.1017/CBO9780511662805
Anna Karczewska and Jerzy Zabczyk, A note on stochastic wave equations, Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998) Lecture Notes in Pure and Appl. Math., vol. 215, Dekker, New York, 2001, pp. 501–511. MR 1818028
Anna Karczewska and Jerzy Zabczyk, Stochastic PDE’s with function-valued solutions, Infinite dimensional stochastic analysis (Amsterdam, 1999) Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet., vol. 52, R. Neth. Acad. Arts Sci., Amsterdam, 2000, pp. 197–216. MR 1832378
Anna Karczewska and Jerzy Zabczyk, Regularity of solutions to stochastic Volterra equations, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11 (2000), no. 3, 141–154 (2001) (English, with English and Italian summaries). MR 1841688
N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027, DOI 10.1007/978-3-642-65183-0
Annie Millet and Marta Sanz-Solé, A stochastic wave equation in two space dimension: smoothness of the law, Ann. Probab. 27 (1999), no. 2, 803–844. MR 1698971, DOI 10.1214/aop/1022677387
Carl Mueller, Long time existence for the wave equation with a noise term, Ann. Probab. 25 (1997), no. 1, 133–151. MR 1428503, DOI 10.1214/aop/1024404282
Jace W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29 (1971), 187–204. MR 295683, DOI 10.1090/S0033-569X-1971-0295683-6
Jan Prüss, Evolutionary integral equations and applications, Monographs in Mathematics, vol. 87, Birkhäuser Verlag, Basel, 1993. MR 1238939, DOI 10.1007/978-3-0348-8570-6
M. Reed, B. Simon, Methods of modern mathematical physics, Vol. II, Academic Press, New York, 1975.
Laurent Schwartz, Méthodes mathématiques pour les sciences physiques, Enseignement des Sciences, Hermann, Paris, 1961 (French). MR 0143360
Daniel W. Stroock, Probability theory, an analytic view, Cambridge University Press, Cambridge, 1993. MR 1267569