Abstract
The article deals with SPDEs driven by Poisson random measure with non Lipschitz coefficients. Let A:E→E be a generator of an analytic semigroup on E, E being a certain Banach space. Let
be a stochastic basis carrying an E-valued Poisson random measure η with characteristic measure ν and compensator γ. Let 1≤p≤2. Our point of interest is the existence of solutions to SPDE's of e.g.the following type
where g:E→L(E,E
0) is some mapping satisfying ∫
E
|g(x,z)−g(y,z)|p
ν(dz)≤C|x−y|rp, x,y ∈ E, where 0<r<1 satisfy certain condition specified later and
is again a certain Banach space.
References
Albeverio, S., Wu, J., Zhang, T.: Parabolic SPDEs driven by Poisson white noise. Stochastic Processes Appl. 74, 21–36 (1998)
Applebaum, D., Wu, J.L.: Stochastic partial differential equations driven by Lévy space-time white noise. Random Oper. Stochastic Equations 8 (3), 245–259 (2000)
Bié, E.: Étude d'une EDPS conduite par un bruit poissonnien. (Study of a stochastic partial differential equation driven by a Poisson noise). Probab. Theory Relat. Fields 111, 287–321 (1998)
Dettweiler, E.: Banach space valued processes with independent increments and stochastic integration. In: Probability in Banach spaces IV, Proc. Semin., Oberwolfach 1982, Lect. Notes Math. pp. 54–83. Springer, 1983
Elworthy, K.D.: Stochastic differential equations on manifolds, London Mathematical Society Lecture Note Series, vol. 70. Cambridge University Press, Cambridge, 1982
Ethier, S., Kurtz, T.: Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York 1986. Characterization and convergence
Granas, A., Dugundji, J.: Fixed point theory. Springer Monographs in Mathematics. Springer-Verlag, New York, 2003
Gutman, S.: Compact perturbations of m-accretive operators in general Banach spaces. SIAM J. Math. Anal. 13 (5), 789–800 (1982)
Hausenblas, E.: Existence, Uniqueness and Regularity of Parabolic SPDEs driven by Poisson random measure. Electron. J. Probab. 10, 1496–1546, (electronic) (2005)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, vol. 24, second edn. North-Holland Publishing Co., Amsterdam, 1989
Kallenberg, O.: Foundations of modern probability. second edn. Probability and its Applications (New York). Springer-Verlag, New York, 2002
Kallianpur, G., Xiong, J.: A nuclear space-valued stochastic differential equation driven by Poisson random measures. In: Stochastic partial differential equations and their applications, Lect. Notes Control Inf. Sci., vol. 176, pp. 135–143. Springer Verlag, 1987
Kallianpur, G., Xiong, J.: Stochastic differential equations in infinite dimensional spaces, Lecture Notes - Monograph Series, vol. 26. Institute of Mathematical Statistics, 1996
Knoche, C.: SPDEs in infinite dimension with Poisson noise. C. R. Math. Acad. Sci. Paris 339 (9), 647–652 (2004)
Linde, W.: Probability in Banach spaces - stable and infinitely divisible distributions. 2nd ed. A Wiley-Interscience Publication, 1986
Mueller, C.: The heat equation with Lévy noise. Stochastic Process. Appl. 74 (1), 67–82 (1998)
Mueller, C., Mytnik, L., Stan, A.: The heat equation with time-independent multiplicaive stable Lévy noise. Stochastic Process. Appl. 116 (1), 70–100 (2006)
Mytnik, L.: Stochastic partial differential equation driven by stable noise. Probab. Theory Related Fields 123 (2), 157–201 (2002)
Parthasarathy, K.R.: Probability measures on metric spaces. Probability and Mathematical Statistics, No. 3. Academic Press Inc., New York, 1967
Pazy, A.: Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44. New York etc.: Springer-Verlag, 1983
Pisier, G.: 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, 167–241 . Springer, 1986
Reed, M., Simon, B.: Methods of modern mathematical physics. I. Functional analysis. Academic Press, New York, 1972
Runst, T., Sickel, W.: Sobolev spaces of fractional order, Nemytskij operators and nonlinear partial differential equations. de Gruyter Series in Nonlinear Analysis and Applications, vol. 3. Berlin: de Gruyter, 1996
Triebel, H.: Interpolation theory, function spaces, differential operators. 2nd rev. a. enl. ed. Leipzig: Barth. 1995
Walsh, J.B.: A stochastic model of neural response. Adv. Appl. Probab. 13, 231–281 (1981)
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This work was supported by the Austrian Academy of Science, APART 700 and FWF-Project P17273-N12
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Hausenblas, E. SPDEs driven by Poisson random measure with non Lipschitz coefficients: existence results. Probab. Theory Relat. Fields 137, 161–200 (2007). https://doi.org/10.1007/s00440-006-0501-8
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DOI: https://doi.org/10.1007/s00440-006-0501-8
Mathematics Subject Classification (2000)
- 60H15
Key words or phrases
- Stochastic partial differential equations
- Poisson random measure