2013, 2(1): 173-192. doi: 10.3934/eect.2013.2.173

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

1. 

Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, United States

2. 

Department of Mathematics and Statistics, University of Maryland Baltimore County (UMBC), Baltimore, MD 21250

Received  November 2012 Revised  December 2012 Published  January 2013

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.
Citation: Stephen Thompson, Thomas I. Seidman. Approximation of a semigroup model of anomalous diffusion in a bounded set. Evolution Equations & Control Theory, 2013, 2 (1) : 173-192. doi: 10.3934/eect.2013.2.173
References:
[1]

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

[2]

D. Brockmann, Human mobility and spatial disease dynamics,, in, 2 (2009), 1.

[3]

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

[4]

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

[5]

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.

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

L. Lorenzi, A. Lundardi, G. Metafune and D. Pallara, "Analytic Semigroups and Reaction-Diffusion Problems,", unpublished Lecture Notes, ().

[17]

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

[18]

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.

[19]

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

[20]

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

show all references

References:
[1]

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

[2]

D. Brockmann, Human mobility and spatial disease dynamics,, in, 2 (2009), 1.

[3]

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

[4]

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

[5]

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.

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

L. Lorenzi, A. Lundardi, G. Metafune and D. Pallara, "Analytic Semigroups and Reaction-Diffusion Problems,", unpublished Lecture Notes, ().

[17]

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

[18]

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.

[19]

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

[20]

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

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