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 Electronic Journal of Probability > Vol. 10 (2005) > Paper 46 open journal systems 


Existence, Uniqueness and Regularity of Parabolic SPDEs Driven by Poisson Random Measure

ERIKA HAUSENBLAS, University Salzburg, Austria


Abstract
In this paper we investigate SPDEs in certain Banach spaces driven by a Poisson random measure. We show existence and uniqueness of the solution, investigate certain integrability properties and verify the cadlag property.


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Pages: 1496-1546

Published on: December 22, 2005


Bibliography
  1. S. Albeverio, J. Wu, and T. Zhang. Parabolic SPDEs driven by Poisson white noise. Stochastic Processes Appl., 74:21--36, 1998. Math. Review 89a:94023
  2. D.Applebaum and Jiang-Lun Wu. Stochastic partial differential equations driven by L'evy space-time white noise. Random Oper. Stochastic Equations, 8 245--259, 2000. Math. Review 2002e:60099
  3. P. Assouad. Burgess Davis inequalities in Banach spaces. In Proceedings of the Seminar on Random Series, Convex Sets and Geometry of Banach Spaces (Mat. Inst., Aarhus Univ., Aarhus, 1974; dedicated to the memory of E. Asplund), pages 1--20. Various Publ. Ser., No. 24, Aarhus Univ., Aarhus, 1975. Mat. Inst.Random Oper. Stochastic Equations, 8(3):245--259, 2000. Math. Review 0402910
  4. R.F. Bass and M.Cranston. The Malliavin calculus for pure jump processes and applications to local time. Ann. Probab., 14(2):490--532, 1986. Math. Review 88b:60113
  5. J. Bergh and J. Löfström. Interpolation spaces: An introduction, volume 223 of Die Grundlehren der mathematischen Wissenschaften. Springer Verlag, 1976. Math. Review 0482275
  6. K. Bichteler. Stochastic integration and Lp-theory of semimartingales. Ann. Probab., 9(1):49--89, 1981. Math. Review MR82g:60071
  7. K. Bichteler. Stochastic integration with jumps , volume~89 of {em Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2002. Math. Review 2003d:60002
  8. E. Bie. Etude d'une EDPS conduite par un bruit poissonnien. Probab. Theory Relat. Fields, 111:287--321, 1998. Math. Review 99j:60096
  9. J.K. Brooks and N.Dinculeanu. Stochastic integration in Banach spaces. In Seminar on Stochastic Processes , 1990 (Vancouver, BC, 1990), volume 24 of Progr. Probab., pages 27--115. Birkhauser Boston, Boston, MA, 2000. Math. Review 2004a:28022
  10. Z.Brzezniak. Stochastic partial differential equations in M-type 2 Banach spaces. Potential Anal.,4(1):1--45, 1995. Math. Review 95m:35213
  11. Z. Brzezniak. On stochastic convolution in Banach spaces and applications. Stochastics Stochastics Rep., 61(3-4):245--295, 1997. Math. Review 1488138
  12. Z. Brzezniak and K. D. Elworthy. Stochastic differential equations on Banach manifolds. Methods Funct. Anal. Topology, 6(1):43--84, 2000. Math. Review MR1784435
  13. D. Burkholder. Martingales and Fourier analysis in Banach spaces. in Probability and analysis, Lect. Sess. C.I.M.E., Varenna/Italy 1985, Lect. Notes Math. 1206, 61-108. Springer, 1986. Math. Review 88c:42017
  14. D. L. Burkholder. Distribution function inequalities for martingales. Ann. Probability, 1:19--42, 1973. Math. Review MR0365692
  15. G. Da Prato and J. Zabczyk. Stochastic equations in infinite dimensions., volume 44 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, 1992. Math. Review 95g:60073
  16. E. Dettweiler. A characterization of the Banach spaces of type p by Levy measures. Math. Z., 157:121--130, 1977. Math. Review 0482921
  17. E. Dettweiler. Poisson measures on Banach lattices. In Probability measures on groups, Proc. 6th Conf., Oberwolfach 1981, Lect. Notes Math. 928, pages 16--24. Springer, 1982. Math. Review 84m:60013
  18. E. Dettweiler. Banach space valued processes with independent increments and stochastic integration. In Probability in Banach spaces IV, Proc. Semin., Oberwolfach 1982, Lect. Notes Math., pages 54--83. Springer, 1983. Math. Review 84j:60013
  19. E. Dettweiler. Representation of Banach space valued martingales as stochastic integrals. In Probability in Banach spaces 7, Proc. 7th Int. Conf., Oberwolfach/FRG 1988, Prog. Probab. 21, 43-62. Springer, 1990. Math. Review 92c:60006
  20. K.D. Elworthy. Stochastic differential equations on manifolds, volume 70 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1982. Math. Review 84d:58080
  21. K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, 2000. Math. Review 1721989
  22. S.Ethier and T.Kurtz. Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1986. Characterization and convergence. Math. Review 838085
  23. E. Gine and M.B. Marcus. The central limit theorem for stochastic integrals with respect to Levy processes. Ann. Probab., 11(1):58--77, 1983. Math. Review 84d:60009
  24. G.G. Hamedani and V.Mandrekar. Levy-Khinchine representation and Banach spaces of type and cotype. Studia Math., 66(3):299--306, 1979/80. Math. Review 81g:60005
  25. N. Ikeda and S. Watanabe. Stochastic differential equations and diffusion processes, volume 24 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, second edition, 1989. Math. Review 90m:60069
  26. J. Jacod and A. Shiryaev. Limit theorems for stochastic processes, volume 288 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, second edition, 2003. Math. Review 1943877
  27. G. Kallianpur and J. Xiong. A nuclear space-valued stochastic differential equation driven by Poisson random measures. In Stochastic partial differential equations and their applications, volume 176 of Lect. Notes Control Inf. Sci., pages 135--143. Springer Verlag, 1987. Math. Review 1176779
  28. G. Kallianpur and J. Xiong. Stochastic differential equations in infinite dimensional spaces., volume 26 of Lecture Notes - Monograph Series. 1996. Math. Review 98h:60001
  29. C. Knoche. SPDEs in infinite dimension with Poisson noise. C. R. Math. Acad. Sci. Paris, 339(9):647--652, 2004. Math. Review 2103204
  30. A.U. Kussmaul. Stochastic integration and generalized martingales. Research Notes in Mathematics. 11. London etc.: Pitman Publishing, 1977. Math. Review 0488281
  31. W. Linde. Probability in Banach spaces - stable and infinitely divisible distributions. 2nd ed. A Wiley-Interscience Publication, 1986. Math. Review 87m:60018
  32. A. Lunardi. Interpolation Theory. Scuola Normale Superiore Pisa - Appunti, 1999. Math. Review number not available.
  33. C. Mueller. The heat equation with Levy noise. Stochastic Process. Appl., 74(1):67--82, 1998. Math. Review 99g:60107
  34. L. Mytnik. Stochastic partial differential equation driven by stable noise. Probab. Theory Related Fields, 123(2):157--201, 2002. Math. Review 2003e:60145
  35. A. L. Neidhardt. Stochastic Integrals in 2-uniformly smooth Banach Spaces. Technical report, University of Wisconsin, 1978. Habilitation.
  36. A. Pazy. Semigroups of linear operators and applications to partial differential equations, volume 44 of Applied Mathematical Sciences. Springer-Verlag, 1983. Math. Review 85g:47061
  37. G. Pisier. Martingales with values in uniformly convex spaces. Isr. J. Math., 20:326--350, 1975. Math. Review 0394135
  38. G. Pisier. Probabilistic methods in the geometry of Banach spaces. In Probability and analysis, Lect. Sess. C.I.M.E., Varenna/Italy 1985, Lect. Notes Math. 1206, 167-24. Springer, 1986. Math. Review 88d:46032
  39. P. Protter and D. Talay. The Euler scheme for Levy driven stochastic differential equations. Ann. Probab., 25(1):393--423, 1997. Math. Review 98c:60063
  40. J. Rosinski. Random integrals of Banach space valued functions. Studia Math., 78(1):15--38, 1984. Math. Review 766703
  41. B.Ruediger. Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces. Stoch. Stoch. Rep., 76(3):213--242, 2004. Math. Review 2072381
  42. T. Runst and W. Sickel. Sobolev spaces of fractional order, Nemytskij operators and nonlinear partial differential equations., volume 3 of de Gruyter Series in Nonlinear Analysis and Applications. Berlin: de Gruyter, 1996. Math. Review 98a:47071
  43. Dang Hung Thang. On Ito stochastic integral with respect to vector stable random measures. Acta Math. Vietnam., 21(2):171--181, 1996. Math. Review 98e:60082
  44. H. Triebel. Interpolation theory, function spaces, differential operators. 2nd rev. a. enl. ed. Leipzig: Barth., 1995. Math. Review 80i:46032a
  45. J. Walsh. A stochastic model of neural response. Adv. in Appl. Prob., 13:231--281, 1981. Math. Review 82f:92020
  46. J. Walsh. An introduction to stochastic partial differential equations. In D. Williams, editor, Ecole d'ete de probabilites de Saint-Flour XIV - 1984, volume 1180 of Lect. Notes Math., pages 265--437. Springer Verlag, 1986. Math. Review 88a:60114
  47. W. Woyczynski. Geometry and martingales in Banach spaces. In Probability---Winter School (Proc. Fourth Winter School, Karpacz, 1975), pages 229--275. Lecture Notes in Math., Vol. 472. Springer, Berlin, 1975. Math. Review 52:14936
  48. W. Woyczknski. Geometry and martingales in Banach spaces. II. Independent increments. In Probability on Banach spaces, volume 4 of Adv. Probab. Related Topics, pages 267--517. Dekker, New York, 1978. Math. Review 80d:46035
  49. E. Zeidler. Nonlinear functional analysis and its applications. I Springer-Verlag, New York, 1986. Fixed-point theorems, Translated from the German by Peter R. Wadsack. Math. Review 816732
















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Electronic Journal of Probability. ISSN: 1083-6489