Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T01:21:51.862Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized fractional kinetic equations: another point of view

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  01 July 2016

David Márquez-Carreras
David Márquez-Carreras*
Affiliation:
Universitat de Barcelona
*
∗ Postal address: Departament de Probabilitat, Lògica i Estadística, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007-Barcelona, Spain. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we deal with generalized fractional kinetic equations driven by a Gaussian noise, white in time and correlated in space, and where the diffusion operator is the composition of the Bessel and Riesz potentials for any fractional parameters. We give results on the existence and uniqueness of solutions by means of a weak formulation and study the Hölder continuity. Moreover, we prove the existence of a smooth density associated to the solution process and study the asymptotics of this density. Finally, when the diffusion coefficient is constant, we look for its Gaussian index.

Keywords

Stochastic fractional kinetic and heat equationsBessel and Riesz potentialsGaussian processesMalliavin calculus

MSC classification

Primary: 60G60: Random fields 60H15: Stochastic partial differential equations 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60G10: Stationary processes 60G15: Gaussian processes 60H07: Stochastic calculus of variations and the Malliavin calculus
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 3 , September 2009 , pp. 893 - 910
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Partially supported by the grant MTM 2006-01351.

References

Adler, R. J. (1981). The Geometry of Random Fields. John Wiley, Chichester.Google Scholar
Angulo, J. M., Anh, V. V., McVinish, R. and Ruiz-Medina, M. D. (2005). Fractional kinetic equations driven by Gaussian or infinitely divisible noise. Adv. Appl. Prob. 37, 366–392.Google Scholar
Angulo, J. M., Ruiz-Medina, M. D., Anh, V. V. and Grecksch, W. (2000). Fractional diffusion and fractional heat equation. Adv. Appl. Prob. 32, 1077–1099.Google Scholar
Anh, V. V. and Leonenko, N. N. (2001). Spectral analysis of fractional kinetic equations with random data. J. Statist. Phys. 104, 1349–1387.Google Scholar
Anh, V. V. and Leonenko, N. N. (2002). Renormalization and homogenization of fractional diffusion equations with random data. Prob. Theory Relat. Fields 124, 381–408.CrossRefGoogle Scholar
Anh, V. V., Angulo, J. M. and Ruiz-Medina, M. D. (1999). Possible long-range dependence in fractional random fields. J. Statist. Planning Infer. 80, 95–110.Google Scholar
Dalang, R. C. (1999). Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e's. Electron. J. Prob. 4, 29pp.CrossRefGoogle Scholar
Dalang, R. C. and Frangos, N. E. (1998). The stochastic wave equation in two spatial dimensions. Ann. Prob. 26, 187–212.Google Scholar
Dautray, R. and Lions, J.-L. (1988). Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2. Springer, Berlin.Google Scholar
Florit, C. and Márquez-Carreras, D. (2007). The generalized heat equation. Submitted.Google Scholar
Kunita, H. (1988). Stochastic differential equations and stochastic flow of diffeomorphisms. In École d'Été de Probabilités de Saint-Flour XII-1982 (Lecture Notes Math. 1097), Springer, Berlin, pp. 143–303.Google Scholar
Márquez-Carreras, D. (2006). On the asymptotics of the density in perturbed SPDE's with spatially correlated noise. Infin. Dimens. Anal. Quantum Prob. Relat. Top. 9, 271–285.CrossRefGoogle Scholar
Márquez-Carreras, D. and Sanz-Solé, M. (1998). Taylor expansion of the density in a stochastic heat equation. Collect. Math. 49, 399–415.Google Scholar
Márquez-Carreras, D. and Sarrà, M. (2003). Behaviour of the density in perturbed SPDE's with spatially correlated noise. Bull. Sci. Math. 127, 348–367.Google Scholar
Márquez-Carreras, D., Mellouk, M. and Sarrà, M. (2001). On stochastic partial differential equations with spatially correlated noise: smoothness of the law. Stoch. Process. Appl. 93, 269–284.Google Scholar
Nualart, D. (1998). Analysis on Wiener space and anticipating stochastic calculus. In École d'Été de Probabilités de Saint-Flour XXV-1995 (Lecture Notes Math. 1690), Springer, New York, pp. 123–227.Google Scholar
Nualart, D. (1998). Malliavin Calculus and Related Topics. Springer, New York.Google Scholar
Ruiz-Medina, M. D., Angulo, J. M. and Anh, V. V. (2001). Scaling limit solution of a fractional Burgers equation. Stoch. Process. Appl. 93, 285–300.Google Scholar
Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1987). Fractional Integrals and Derivatives. Gordon and Breach, New York.Google Scholar
Schwartz, L. (1966). Théorie des Distributions. Hermann, Paris.Google Scholar
Stein, E. M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton University Press.Google Scholar
Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École d'Été de Probabilites de Saint-Flour XIV-1984 (Lecture Notes Math. 1180), Springer, Berlin, pp. 265–439.Google Scholar