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On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension. (English) Zbl 1078.60048
Summary: Existence, uniqueness and regularity of the trajectories of mild solutions of one-dimensional nonlinear stochastic fractional partial differential equations of order $\alpha > 1$ containing derivatives of entire order and perturbed by space-time white noise are studied. The fractional derivative operator is defined by means of a generalized Riesz-Feller potential.

60H15Stochastic partial differential equations
26A33Fractional derivatives and integrals (real functions)
60G60Random fields
Full Text: DOI
[1] Angulo, J. M.; Ruiz-Medina, M. D.; Anh, V. V.; Grecksch, W.: Fractional diffusion and fractional heat equation. Adv. appl. Probab. 32, 1077-1099 (2000) · Zbl 0986.60077
[2] Bonaccorsi, S.; Tubaro, L.: Mittag-Leffler’s function and stochastic linear Volterra equations of convolution type. Stochast. anal. Appl. 21, No. 1, 61-78 (2003) · Zbl 1035.60067
[3] Dalang, R.; Mueller, C.: Some nonlinear s.p.d.e.’s that are second order in time. Electron. J. Probab. 8, No. 1, 1-21 (2003) · Zbl 1013.60044
[4] L. Debbi, Explicit solutions of some fractional equations via stable subordinators, preprint. · Zbl 1127.35316
[5] L. Debbi, On some properties of a high fractional differential operator which is not in general selfadjoint, preprint. · Zbl 1143.26004
[6] L. Debbi, L. Abbaoui, Explicit solution of some fractional heat equations via Lévy motion, Maghreb Math. Rev., to appear. · Zbl 1085.47015
[7] W. Feller, Generalization of Marcel Riesz’ potentials and the semi-groups generated by them, Meddelanden Fran Lunds Universitets Matematiska Seminarium Supplementband 1952, pp. 73 -- 81.
[8] Gorenflo, R.; Mainardi, F.: Random walk models for space-fractional diffusion processes. Fract. calculus appl. Anal. 1, No. 2, 167-191 (1998) · Zbl 0946.60039
[9] Komatsu, T.: On the martingale problem for generators of stable processes with perturbations. Osaka J. Math. 21, 113-132 (1984) · Zbl 0535.60063
[10] Existence, P. Kotelenez: Uniqueness and smoothness for a class of function valued stochastic partial differential equations. Stochast. stochast. Reports 41, 177-199 (1992) · Zbl 0766.60078
[11] Kylov, V. Yu.: Some properties of the distribution corresponding to the equation $\partial u/\partial $t=(-1)q+$1\partial 2qu/\partial t$2q. Soviet math. Dokl. 1, 260-263 (1960)
[12] A. Le Mehaute, T. Machado, J.C. Trigeassou, J. Sabatier, Fractional differentiation and its applications, FDA’04, Proceedings of the first IFAC Workshop, vol. 2004-1, International Federation of Automatic Control, ENSEIRB, Bordeaux, France, July 19 -- 21, 2004.
[13] X. Leoncini, G. Zaslavsky, Ets, stickiness, and anomalous transport, Phys. Rev. E 65 (2002) 046216 -- 1 -- 046216 -- 16. · Zbl 1244.76014
[14] E. Lukacs, Characteristic Functions, second ed., 1970, Griffin, 1960. · Zbl 0087.33605
[15] Mainardi, F.; Luchko, Y.; Pagnini, G.: The fundamental solution of the space -- time fractional diffusion equation. Fract. calculus appl. Anal. 4, No. 2, 153-192 (2001) · Zbl 1054.35156
[16] Mueller, C.: The heat equation with Lévy noise. Stochast. proc. Appl. 74, 67-82 (1998) · Zbl 0934.60056
[17] V.V. Uchaikin, V.M. Zolotarev, Chance and stability, stable distributions and their applications, Mod. Probab. Statist. VSP. 1999. · Zbl 0944.60006
[18] J.B. Walsh, An Introduction To Stochastic Partial Differential Equations, Lectures Notes in Mathematics, vol. 1180, Springer, Berlin, 1986, Ecole d’été de Probabilités de Saint-Flour XIV-1984. · Zbl 0608.60060
[19] J.L. Wu, Fractal Burgers equation with stable Lévy noise, International Conference SPDE and Applications -- VII, January 4 -- 10, 2004.
[20] J. Zabczyk, Symmetric solutions of semilinear stochastic equations, Lecture Notes in Mathematics, vol. 1390, Springer, Berlin, 1988, pp. 237 -- 256.