Fokker–Planck and Kolmogorov backward equations for continuous time random walk scaling limits

Baeumer, Boris; Straka, Peter

doi:10.1090/proc/13203

Fokker–Planck and Kolmogorov backward equations for continuous time random walk scaling limits
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by Boris Baeumer and Peter Straka

Proc. Amer. Math. Soc. 145 (2017), 399-412

DOI: https://doi.org/10.1090/proc/13203

Published electronically: July 12, 2016

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Abstract:

It is proved that the distributions of scaling limits of Continuous Time Random Walks (CTRWs) solve integro-differential equations akin to Fokker–Planck equations for diffusion processes. In contrast to previous such results, it is not assumed that the underlying process has absolutely continuous laws. Moreover, governing equations in the backward variables are derived. Three examples of anomalous diffusion processes illustrate the theory.

References

David Applebaum, Lévy processes and stochastic calculus, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 116, Cambridge University Press, Cambridge, 2009. MR 2512800, DOI 10.1017/CBO9780511809781
Boris Baeumer and Mark M. Meerschaert, Stochastic solutions for fractional Cauchy problems, Fract. Calc. Appl. Anal. 4 (2001), no. 4, 481–500. MR 1874479
Boris Baeumer, Mark M. Meerschaert, and Jeff Mortensen, Space-time fractional derivative operators, Proc. Amer. Math. Soc. 133 (2005), no. 8, 2273–2282. MR 2138870, DOI 10.1090/S0002-9939-05-07949-9
Emilia Grigorova Bajlekova, Fractional evolution equations in Banach spaces, Eindhoven University of Technology, Eindhoven, 2001. Dissertation, Technische Universiteit Eindhoven, Eindhoven, 2001. MR 1868564
E. Barkai, R. Metzler, and J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E (3) 61 (2000), no. 1, 132–138. MR 1736459, DOI 10.1103/PhysRevE.61.132
R. F. Bass, Uniqueness in law for pure jump Markov processes, Probab. Theory Related Fields 79 (1988), no. 2, 271–287. MR 958291, DOI 10.1007/BF00320922
Peter Becker-Kern, Mark M. Meerschaert, and Hans-Peter Scheffler, Limit theorems for coupled continuous time random walks, Ann. Probab. 32 (2004), no. 1B, 730–756. MR 2039941, DOI 10.1214/aop/1079021462
B. Berkowitz, A. Cortis, M. Dentz, and H. Scher, Modeling non-Fickian transport in geological formations as a continuous time random walk, Rev. Geophys. 44 (2):RG2003, 2006. ISSN 8755-1209. DOI 10.1029/2005RG000178.
Jean Bertoin, Subordinators: examples and applications, Lectures on probability theory and statistics (Saint-Flour, 1997) Lecture Notes in Math., vol. 1717, Springer, Berlin, 1999, pp. 1–91. MR 1746300, DOI 10.1007/978-3-540-48115-7_{1}
Shai Carmi, Lior Turgeman, and Eli Barkai, On distributions of functionals of anomalous diffusion paths, J. Stat. Phys. 141 (2010), no. 6, 1071–1092. MR 2740404, DOI 10.1007/s10955-010-0086-6
A. V. Chechkin, R. Gorenflo, and I. M. Sokolov, Fractional diffusion in inhomogeneous media, J. Phys. A 38 (2005), no. 42, L679–L684. MR 2186196, DOI 10.1088/0305-4470/38/42/L03
S. Fedotov and S. Falconer, Subdiffusive master equation with space-dependent anomalous exponent and structural instability, Phys. Rev. E 85(3):031132, March 2012. DOI 10.1103/PhysRevE.85.031132.
S. Fedotov and A. Iomin, Migration and proliferation dichotomy in tumor-cell invasion, Phys. Rev. Lett. 98:118101, 2007.
Marjorie Hahn, Kei Kobayashi, and Sabir Umarov, SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations, J. Theoret. Probab. 25 (2012), no. 1, 262–279. MR 2886388, DOI 10.1007/s10959-010-0289-4
B. I. Henry and S. L. Wearne, Fractional reaction-diffusion, Phys. A 276 (2000), no. 3-4, 448–455. MR 1780373, DOI 10.1016/S0378-4371(99)00469-0
B. Henry, T. Langlands, and P. Straka, Fractional Fokker-Planck equations for subdiffusion with space- and time-dependent forces, Phys. Rev. Lett. 105\penalty0 (17):\penalty0 170602, 2010. DOI 10.1103/PhysRevLett.105.170602.
Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 1987. MR 959133, DOI 10.1007/978-3-662-02514-7
A. Jurlewicz, P. Kern, M. M. Meerschaert, and H.-P. Scheffler, Fractional governing equations for coupled random walks, Comput. Math. Appl. 64 (2012), no. 10, 3021–3036. MR 2989331, DOI 10.1016/j.camwa.2011.10.010
J. Klafter and I. M. Sokolov, First steps in random walks, Oxford University Press, Oxford, 2011. From tools to applications. MR 2894872, DOI 10.1093/acprof:oso/9780199234868.001.0001
Kei Kobayashi, Stochastic calculus for a time-changed semimartingale and the associated stochastic differential equations, J. Theoret. Probab. 24 (2011), no. 3, 789–820. MR 2822482, DOI 10.1007/s10959-010-0320-9
V. N. Kolokol′tsov, Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics, Teor. Veroyatn. Primen. 53 (2008), no. 4, 684–703 (Russian, with Russian summary); English transl., Theory Probab. Appl. 53 (2009), no. 4, 594–609. MR 2766141, DOI 10.1137/S0040585X97983857
Vassili N. Kolokoltsov, Markov processes, semigroups and generators, De Gruyter Studies in Mathematics, vol. 38, Walter de Gruyter & Co., Berlin, 2011. MR 2780345
T. A. M. Langlands and B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys. 205 (2005), no. 2, 719–736. MR 2135000, DOI 10.1016/j.jcp.2004.11.025
T. A. M. Langlands and B. I. Henry, Fractional chemotaxis diffusion equations, Phys. Rev. E (3) 81 (2010), no. 5, 051102, 12. MR 2736241, DOI 10.1103/PhysRevE.81.051102
Marcin Magdziarz, Janusz Gajda, and Tomasz Zorawik, Comment on fractional Fokker-Planck equation with space and time dependent drift and diffusion, J. Stat. Phys. 154 (2014), no. 5, 1241–1250. MR 3176407, DOI 10.1007/s10955-014-0919-9
Mark M. Meerschaert and Hans-Peter Scheffler, Limit theorems for continuous-time random walks with infinite mean waiting times, J. Appl. Probab. 41 (2004), no. 3, 623–638. MR 2074812, DOI 10.1239/jap/1091543414
Mark M. Meerschaert and Hans-Peter Scheffler, Triangular array limits for continuous time random walks, Stochastic Process. Appl. 118 (2008), no. 9, 1606–1633. MR 2442372, DOI 10.1016/j.spa.2007.10.005
Mark M. Meerschaert and Alla Sikorskii, Stochastic models for fractional calculus, De Gruyter Studies in Mathematics, vol. 43, Walter de Gruyter & Co., Berlin, 2012. MR 2884383
Mark M. Meerschaert and Peter Straka, Semi-Markov approach to continuous time random walk limit processes, Ann. Probab. 42 (2014), no. 4, 1699–1723. MR 3262490, DOI 10.1214/13-AOP905
Ralf Metzler and Joseph Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), no. 1, 77. MR 1809268, DOI 10.1016/S0370-1573(00)00070-3
Erkan Nane and Yinan Ni, Stochastic solution of fractional Fokker-Planck equations with space-time-dependent coefficients, J. Math. Anal. Appl. 442 (2016), no. 1, 103–116. MR 3498320, DOI 10.1016/j.jmaa.2016.03.033
R. M. Neupauer and J. L. Wilson, Adjoint method for obtaining backward-in-time location and travel time probabilities of a conservative groundwater contaminant, Water Resour. Res. 35\penalty0 (11):\penalty0 3389–3398, Nov. 1999. DOI 10.1029/1999WR900190.
R. S. Phillips, The adjoint semi-group, Pacific J. Math. 5 (1955), 269–283. MR 70976
Jan Prüss, Evolutionary integral equations and applications, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1993. [2012] reprint of the 1993 edition. MR 2964432, DOI 10.1007/978-3-0348-8570-6
M. Raberto, E. Scalas, and F. Mainardi, Waiting-times and returns in high-frequency financial data: an empirical study, Phys. A Stat. Mech. Appl. 314\penalty0 (1-4):\penalty0 749–755, Nov. 2002. DOI 10.1016/S0378-4371(02)01048-8.
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
R. Schumer, D. A. Benson, M. M. Meerschaert, and B. Baeumer, Fractal mobile/immobile solute transport, Water Resour. Res. 39\penalty0 (10), Oct. 2003. DOI 10.1029/2003WR002141.
I. M. Sokolov and J. Klafter, Field-induced dispersion in subdiffusion. Phys. Rev. Lett. 97\penalty0 (14):\penalty0 1–4, Oct. 2006. ISSN 0031-9007. DOI 10.1103/PhysRevLett.97.140602. URL http://link.aps.org/doi/10.1103/PhysRevLett.97.140602.
P. Straka and B. I. Henry, Lagging and leading coupled continuous time random walks, renewal times and their joint limits, Stochastic Process. Appl. 121 (2011), no. 2, 324–336. MR 2746178, DOI 10.1016/j.spa.2010.10.003
Sabir Umarov, Introduction to fractional pseudo-differential equations with singular symbols, Developments in Mathematics, vol. 41, Springer, Cham, 2015. MR 3381285, DOI 10.1007/978-3-319-20771-1
A. Weron and M. Magdziarz, Modeling of subdiffusion in space-time-dependent force fields beyond the fractional Fokker-Planck equation,Phys. Rev. E 77\penalty0 (3):\penalty0 1–6, Mar. 2008. ISSN 1539-3755. DOI 10.1103/PhysRevE.77.036704.