Advanced Search
Article Contents
Article Contents

Approximation of a semigroup model of anomalous diffusion in a bounded set

Abstract Related Papers Cited by
  • The convergence is established for a sequence of operator semigroups, where the limiting object is the transition semigroup for a reflected stable processes. For semilinear equations involving the generators of these transition semigroups, an approximation method is developed as well. This makes it possible to derive an a priori bound for solutions to these equations, and therefore prove global existence of solutions. An application to epidemiology is also given.
    Mathematics Subject Classification: Primary: 47D06, 60J35; Secondary: 60J75, 45L05, 58D25, 35K57, 92D30.


    \begin{equation} \\ \end{equation}
  • [1]

    K. Bogdan, K. Burdzy and Z. Chen, Censored stable processes, Probab. Theory Relat. Fields, 19 (2003), 89-152.doi: 10.1007/s00440-003-0275-1.


    D. Brockmann, Human mobility and spatial disease dynamics, in "Reviews of Nonlinear Dynamics and Complexity, 2" (ed. H. G. Schuster), Wiley-VCH, (2009), 1-24.


    D. Brockmann, L. Hufnagel and T. Geisel, The scaling laws of human travel, Nature, 439 (2006), 462-465.


    Z. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on d-sets, Stoch. Process. Appl., 108 (2003), 27-62.doi: 10.1016/S0304-4149(03)00105-4.


    Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.


    K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer, New York, 1995.


    M. Fukushima, T. Oshima and M. Takeda, "Dirichlet Forms and Symmetric Markov Processes," Walter de Gruyter, Berlin, 1994.doi: 10.1515/9783110889741.


    P. Grisvard, Caractérisation de quelques espaces d'interpolation, Arch. Rational Mech. Anal., 26 (1967), 431-458.


    Q. Guan and Z. Ma, Reflected symmetric $\alpha$-stable processes and regional fractional laplacian, Probab. Theory Relat. Fields, 134 (2006), 649-694.doi: 10.1007/s00440-005-0438-3.


    K. Gustafson and G. Lumer, Multiplicative perturbation of semigroup generators, Pac. J. Math., 41 (1972), 731-742.


    E. Hanert, Front dynamics in a two-species competition model driven by Lévy flights, J. Theor. Biol., 300 (2012), 134-142.doi: 10.1016/j.jtbi.2012.01.022.


    E. Hanert, E. Schumacher and E. Eleersnijder, Front dynamics in fractional-order epidemic models, J. Theor. Biol., 279 (2011), 9-16.


    K. Ito and F. Kappel, The trotter kato theorem and approximation of PDEs, Math. Comput., 67 (1998), 21-44.doi: 10.1090/S0025-5718-98-00915-6.


    P. Kim, Weak convergence of censored and reflected stable processes, Stoch. Process. Appl., 116 (2006), 1792-1814.doi: 10.1016/j.spa.2006.04.006.


    R. Klages, G. Radons and I. M. Sokolov, "Anomalous Transport," Wiley-VCH, Weinheim 2008.


    L. Lorenzi, A. Lundardi, G. Metafune and D. Pallara, "Analytic Semigroups and Reaction-Diffusion Problems," unpublished Lecture Notes, http://www.math.unipr.it/~lunardi/LectureNotes/I-Sem2005.pdf.


    T. Lux and M. Marchesi, Scaling and criticality in a stochastic multi-agent model of a financial market, Nature, 397 (1999), 498-500.


    G. M. Viswanathan, S. V. Buldyrev, S. Havlin, M. G. E. da Luz, E. P. Raposo and H. E. Stanley, Optimizing the success of random searches, Nature, 401 (1999), 911-914.


    J. Wloka, "Partial Differential Equations," Cambridge University Press, London 1987.


    M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries: Analysis, Differential Calculus, and Optimization," Society for Industrial and Applied Mathematics, Philadelphia 2001.

  • 加载中

Article Metrics

HTML views() PDF downloads(82) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint