A stochastic differential equation model with jumps for fractional advection and dispersion

Journal of Statistical Physics

Volumn 136, Issue 3, 2009, Pages 527-551

  Chakraborty, Paramita ^a  

^a MICHIGAN STATE UNIVERSITY   (United States)

Author keywords

Forward equation; Infinitesimal generator; Jump L vy diffusion; Pure jump L vy process; Stochastic differential equation

Indexed keywords

EID: 70349306436     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-009-9794-1     Document Type: Article

References (23)

1. Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004)
2. Baeumer, B., Meerschaert, M.M.: Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4(4) (2001)
3. Bear, J.: Dynamics of Fluids in Porous Media. Dover, New York (1972)
4. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional order governing equation of Lévy motion. Water Resour. Res. 36(6) (2000)
5. Bhattacharya, R.N., Gupta, V.K., Sposito, G.: On the stochastic foundation of the theory of water flow through unsaturated soil. Water Resour. Res. 12(3) (1976)
6. Bhattacharya, R.N., Gupta, V.K.: A theoretical explanation of solute dispersion in saturated porous media at the Darcy scale. Water Resour. Res. 19(4), 938-944 (1983)
7. Bottcher, B., Schilling, R.L.: Approximation of feller processes by Markov chains with Lévy increments. Stoch. Dyn. 9(1), 71-80 (2009)
8. Cartea, A., del-Castillo-Negrete, D.: Fluid limit of the continuous-time random walk with general Lévy jump distribution functions. Phys. Rev. E 76, 041105 (2007)
9. Chakraborty, P., Meerschaert, M.M., Lim, C.Y.: Parameter estimation for fractional transport: a particle tracking approach. Water Resour. Res. (2009). doi:10.1029/2008WR007577
10. Fetter, C.W.: Applied Hydrogeology, 4th edn. Prentice Hall, New York (2001), 598 pp.
11. Gupta, V.K., Sposito, G., Bhattacharya, R.N.: Toward an analytical theory of water flow through inhomogeneous porous media. Water Resour. Res. 13(1), 208-210 (1977)
12. Hoh, W.A.: A symbolic calculus for pseudo differential operators generating feller semigroups. Osaka J. Math. 35, 798-820 (1998)
13. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Process. North-Holland, Amsterdam (1981)
14. Jacob, N.: Pseudo-Differential Operators and Markov Process vols. I and II. Imperial College Press, London (2002)
15. Jacob, N., Schilling, R.L.: Lévy-type processes and pseudo-differential operators. In: Resnick, S., Barndorff-Nielsen, O., Mikosch, Th. (eds.) Lévy Processes-Theory and Applications, pp. 139-168. Birkhäuser, Basel (2001)
16. Rosiński, J.: Tempering stable processes. Stoch. Process. Appl. 117, 677-707 (2007)
17. Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance. Chapman & Hall, New York (1994)
18. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, London (1993)
19. Sato, K.I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (2004)
20. Skorokhod, A.V.: Studies in the Theory of Random Processes. Addison-Wesley, Reading (1964)
21. Zhang, Y., Benson, D., Meerschaert, M.M., Scheffler, H.P.: On using random walks to solve the space-fractional advection-dispersion equation. J. Stat. Phys. 123(1) (2006)
22. Zhang, Y., Benson, D., Meerschaert, M.M., LaBolle, E., Scheffler, H.P.: Random walk approximation of fractional-order multiscaling anomalous diffusion. Phys. Rev. E 74(2), 706-715 (2006)
23. Zhang, Y., Benson, D., Meerschaert, M.M., LaBolle, E.: Space-fractional advection-dispersion equations with variable parameters: diverse formulas, numerical solutions, and application to the MADE-site data. Water Resour. Res. 43, W05439 (2007). doi:10.1029/2006WR004912