From random times to fractional kinetics

In this paper we study the effect of the subordination by a general random time-change to the solution of a model on spatial ecology in terms of its evolution density. In particular on traveling waves for a non-local spatial logistic equation. We study the Cesaro limit of the subordinated dynamics in a number of particular cases related to the considered fractional derivative making use of the Karamata-Tauberian theorem.

[1]  Joseph Lauber,et al.  1946 , 2019, The George Bell-Gerhard Leibholz Correspondence.

[2]  M. Manhart,et al.  Markov Processes , 2018, Introduction to Stochastic Processes and Simulation.

[3]  D. Finkelshtein,et al.  Doubly nonlocal Fisher-KPP equation: Existence and properties of traveling waves , 2018, 1804.10258.

[4]  Jian Wang,et al.  Heat kernel estimates for time fractional equations , 2017, Forum Mathematicum.

[5]  Y. Kondratiev,et al.  FRACTIONAL KINETICS IN A SPATIAL ECOLOGY MODEL , 2018 .

[6]  A. Kochubei,et al.  Fractional kinetic hierarchies and intermittency , 2016, 1604.03807.

[7]  E. Bazhlekova.  Subordination Principle for a Class of Fractional Order Differential Equations , 2015 .

[8]  Bruno Toaldo.  Convolution-Type Derivatives, Hitting-Times of Subordinators and Time-Changed C0-semigroups , 2013, 1308.1327.

[9]  Georg Prokert.  Evolution Equations , 2015 .

[10]  R. Gorenflo,et al.  Mittag-Leffler Functions, Related Topics and Applications , 2014, Springer Monographs in Mathematics.

[11]  Yuri Kozitsky,et al.  The statistical dynamics of a spatial logistic model and the related kinetic equation , 2014 .

[12]  佐藤 健一.  Lévy processes and infinitely divisible distributions , 2013 .

[13]  Dmitri Finkelshtein,et al.  Semigroup approach to birth-and-death stochastic dynamics in continuum , 2011, 1109.5094.

[14]  Anatoly N. Kochubei,et al.  General Fractional Calculus, Evolution Equations, and Renewal Processes , 2011, 1105.1239.

[15]  V. Kolokoltsov.  Markov Processes, Semigroups and Generators , 2011 .

[16]  F. Mainardi.  Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , 2010 .

[17]  Dmitri Finkelshtein,et al.  Vlasov Scaling for Stochastic Dynamics of Continuous Systems , 2010 .

[18]  René L. Schilling,et al.  Bernstein Functions: Theory and Applications , 2010 .

[19]  Dmitri Finkelshtein,et al.  Individual Based Model with Competition in Spatial Ecology , 2008, SIAM J. Math. Anal..

[20]  Sachin Bhalekar,et al.  Boundary value problems for multi-term fractional differential equations , 2008 .

[21]  Mark M. Meerschaert,et al.  Triangular array limits for continuous time random walks , 2008 .

[22]  Murad S. Taqqu,et al.  Non-Markovian diffusion equations and processes: Analysis and simulations , 2007, 0712.0240.

[23]  A. Kochubei.  Distributed order calculus and equations of ultraslow diffusion , 2007, math-ph/0703046.

[24]  Обобщенные случайные блуждания в непрерывном времени (CTRW), субординация временами достижения и дробная динамика@@@Generalized Continuous-Time Random Walks, Subordination by Hitting Times, and Fractional Dynamics , 2008 .

[25]  Anatoly N. Kochubei,et al.  Distributed-order calculus: An operator-theoretic interpretation , 2007, 0710.1710.

[26]  Vassili N. Kolokoltsov,et al.  Generalized Continuous-Time Random Walks (CTRW), Subordination by Hitting Times and Fractional Dynamics , 2007, 0706.1928.

[27]  A. Hanyga,et al.  Anomalous diffusion without scale invariance , 2007 .

[28]  M. Meerschaert,et al.  Stochastic model for ultraslow diffusion , 2006 .

[29]  Oleksandr Kutoviy,et al.  On the metrical properties of the configuration space , 2006 .

[30]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[31]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies) , 2006 .

[32]  Thomas Mikosch,et al.  Regularly varying functions , 2006 .

[33]  Rudolf Gorenflo,et al.  Cauchy and Nonlocal Multi-Point Problems for Distributed Order Pseudo-Differential Equations, Part One , 2005 .

[34]  A. Kochubei,et al.  Analytic Methods In The Theory Of Differential And Pseudo-Differential Equations Of Parabolic Type , 2004 .

[35]  Nicolas Fournier,et al.  A microscopic probabilistic description of a locally regulated population and macroscopic approximations , 2004, math/0503546.

[36]  Mark M. Meerschaert,et al.  Limit theorems for continuous-time random walks with infinite mean waiting times , 2004, Journal of Applied Probability.

[37]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[38]  Tobias Kuna,et al.  HARMONIC ANALYSIS ON CONFIGURATION SPACE I: GENERAL THEORY , 2002 .

[39]  J. López,et al.  Asymptotic Expansions of Generalized Stieltjes Transforms of Algebraically Decaying Functions , 2002 .

[40]  Roderick Wong,et al.  Asymptotic approximations of integrals , 1989, Classics in applied mathematics.

[41]  Emilia Bazhlekova,et al.  Fractional evolution equations in Banach spaces , 2001 .

[42]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[43]  D. Sornette.  Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools , 2000 .

[44]  E. Bazhlekova.  Subordination principle for fractional evolution equations , 1999 .

[45]  B. Bolker,et al.  Using Moment Equations to Understand Stochastically Driven Spatial Pattern Formation in Ecological Systems , 1997, Theoretical population biology.

[46]  L. Bondesson,et al.  Infinite divisibility of random variables and their integer parts , 1996 .

[47]  J. Prüss.  Evolutionary Integral Equations And Applications , 1993 .

[48]  M. Meerschaert.  Regular Variation in R k , 1988 .

[49]  G. Gripenberg.  Volterra integro-differential equations with accretive nonlinearity , 1985 .

[50]  Yuji Kasahara,et al.  Limit Theorems of Occupation Times for Markov Processes , 1976 .

[51]  N. Bleistein,et al.  Asymptotic Expansions of Integrals , 1975 .

[52]  N. N. Bogolyubov,et al.  Problems of a Dynamical Theory in Statistical Physics , 1959 .

[53]  T. Teichmann,et al.  Harmonic Analysis and the Theory of Probability , 1957, The Mathematical Gazette.

[54]  J. Karamata,et al.  Sur un mode de croissance régulière. Théorèmes fondamentaux , 1933 .