A Semigroup Approach to Fractional Poisson Processes

C. Lizama, R. Rebolledo

Research output: Contribution to journalArticle

Abstract

It is well-known that fractional Poisson processes (FPP) constitute an important example of a non-Markovian structure. That is, the FPP has no Markov semigroup associated via the customary Chapman–Kolmogorov equation. This is physically interpreted as the existence of a memory effect. Here, solving a difference-differential equation, we construct a family of contraction semigroups (Tα)α∈]0,1], Tα=(Tα(t))t≥0. If C([ 0 , ∞[ , B(X)) denotes the Banach space of continuous maps from [ 0 , ∞[ into the Banach space of endomorphisms of a Banach space X, it holds that Tα∈ C([ 0 , ∞[ , B(X)) and α↦ Tα is a continuous map from ]0, 1] into C([ 0 , ∞[ , B(X)). Moreover, T1 becomes the Markov semigroup of a Poisson process. © 2018, Springer International Publishing AG, part of Springer Nature.
LanguageEnglish
Pages777-785
Number of pages9
JournalComplex Analysis and Operator Theory
Volume12
Issue number3
DOIs
Publication statusPublished - 2018

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Banach spaces
Poisson process
Markov Semigroups
Fractional
Semigroup
Banach space
Continuous Map
Difference-differential Equations
Contraction Semigroup
Kolmogorov Equation
Memory Effect
Endomorphisms
Differential equations
Denote
Data storage equipment

Keywords

  • Chapman–Kolmogorov equation
  • Fractional Poisson process
  • Markov semigroup

Cite this

A Semigroup Approach to Fractional Poisson Processes. / Lizama, C.; Rebolledo, R.

In: Complex Analysis and Operator Theory, Vol. 12, No. 3, 2018, p. 777-785.

Research output: Contribution to journalArticle

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title = "A Semigroup Approach to Fractional Poisson Processes",
abstract = "It is well-known that fractional Poisson processes (FPP) constitute an important example of a non-Markovian structure. That is, the FPP has no Markov semigroup associated via the customary Chapman–Kolmogorov equation. This is physically interpreted as the existence of a memory effect. Here, solving a difference-differential equation, we construct a family of contraction semigroups (Tα)α∈]0,1], Tα=(Tα(t))t≥0. If C([ 0 , ∞[ , B(X)) denotes the Banach space of continuous maps from [ 0 , ∞[ into the Banach space of endomorphisms of a Banach space X, it holds that Tα∈ C([ 0 , ∞[ , B(X)) and α↦ Tα is a continuous map from ]0, 1] into C([ 0 , ∞[ , B(X)). Moreover, T1 becomes the Markov semigroup of a Poisson process. {\circledC} 2018, Springer International Publishing AG, part of Springer Nature.",
keywords = "Chapman–Kolmogorov equation, Fractional Poisson process, Markov semigroup",
author = "C. Lizama and R. Rebolledo",
note = "Export Date: 6 April 2018 Correspondence Address: Lizama, C.; Departamento de Matem{\'a}tica y Ciencia de la Computaci{\'o}n, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307, Correo 2, Chile; email: carlos.lizama@usach.cl References: Arendt, W., Batty, C., Hieber, M., Neubrander, F., (2001) Vector-Valued Laplace Transforms and Cauchy Problems, 96. , Monographs in mathematics, Birkh{\"a}user, Basel; Barrachina, X., Peris, A., Distributionally chaotic translation semigroups (2012) J. Differ. Equ. Appl., 18 (4), pp. 751-761; Beghin, L., Orsingher, E., Fractional Poisson processes and related random motions (2009) Electron. J. Probab., 14, pp. 1790-1826; Beghin, L., Orsingher, E., Poisson-type processes governed by fractional and higher-order recursive differential equations (2010) Electron. J. Probab., 15, pp. 684-709; Ethier, S.N., Kurtz, T.G., (1986) Markov Processes: Characterization and Convergence, , Wiley, New York; Gorenflo, R., Mainardi, F., Fractional Calculus: Integral and Differential Equations of Fractional Order (2008) CISM lecture notes, 805, p. 3823. , http://arxiv.org/abs/0805.3823; Grosse-Erdmann, K.G., Peris, A., (2011) Linear Chaos, , Universitext, Springer, Berlin; Jumarie, G., Fractional master equation: non-standard analysis and Liouville–Riemann derivative (2001) Chaos Solitons Fractals, 12, pp. 2577-2587; Kilbas, A., Srivastava, H., Trujillo, J., (2006) Theory and Applications of Fractional Differential Equations, 204. , North-Holland mathematics studies, Elsevier, Amsterdam; Laskin, N., Fractional Poisson process (2003) Commun. Nonlinear Sci. Numer. Simul., 8, pp. 201-213; Laskin, N., Some applications of the fractional Poisson probability distribution (2009) J. Math. Phys, 50, p. 113513; Lizama, C., The Poisson distribution, abstract fractional difference equations, and stability (2017) Proc. Am. Math. Soc., 145 (9), pp. 3809-3827; Mainardi, F., Gorenflo, R., Scalas, E., A fractional generalization of the Poisson processes (2004) Vietnam J. Math., 32, pp. 53-64; Mainardi, F., Gorenflo, R., Vivoli, A., Beyond the Poisson renewal process: a tutorial survey (2007) J. Comput. Appl. Math., 205, pp. 725-735; Meerschaert, M.M., Nane, E., Vellaisamy, P., The fractional Poisson process and the inverse stable subordinator (2011) Electron. J. Probab., 16, pp. 1600-1620; Pazy, A., (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations, 44. , Applied mathematical sciences, Springer, New York; Podlubny, I., (1999) Fractional Differential Equations, 198. , Mathematics in science and engineering, Academic Press Inc., San Diego; Pr{\"u}ss, J., (1993) Evolutionary Integral Equations and Applications, , Birkh{\"a}user, Basel; Repin, O.N., Saichev, A.I., Fractional Poisson law (2000) Radiophys. Quantum Electron., 43, pp. 738-741; Samko, S.-G., Kilbas, A., Marichev, O., (1993) Fractional Integrals and Derivatives, , Gordon and Breach Science Publishers, Yverdon; Uchaikin, V.V., Cahoy, D.O., Sibatov, R.T., Fractional processes: from Poisson to branching one (2008) Int. J. Bifur. Chaos Appl. Sci. Eng., 18, pp. 2717-2725",
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PY - 2018

Y1 - 2018

N2 - It is well-known that fractional Poisson processes (FPP) constitute an important example of a non-Markovian structure. That is, the FPP has no Markov semigroup associated via the customary Chapman–Kolmogorov equation. This is physically interpreted as the existence of a memory effect. Here, solving a difference-differential equation, we construct a family of contraction semigroups (Tα)α∈]0,1], Tα=(Tα(t))t≥0. If C([ 0 , ∞[ , B(X)) denotes the Banach space of continuous maps from [ 0 , ∞[ into the Banach space of endomorphisms of a Banach space X, it holds that Tα∈ C([ 0 , ∞[ , B(X)) and α↦ Tα is a continuous map from ]0, 1] into C([ 0 , ∞[ , B(X)). Moreover, T1 becomes the Markov semigroup of a Poisson process. © 2018, Springer International Publishing AG, part of Springer Nature.

AB - It is well-known that fractional Poisson processes (FPP) constitute an important example of a non-Markovian structure. That is, the FPP has no Markov semigroup associated via the customary Chapman–Kolmogorov equation. This is physically interpreted as the existence of a memory effect. Here, solving a difference-differential equation, we construct a family of contraction semigroups (Tα)α∈]0,1], Tα=(Tα(t))t≥0. If C([ 0 , ∞[ , B(X)) denotes the Banach space of continuous maps from [ 0 , ∞[ into the Banach space of endomorphisms of a Banach space X, it holds that Tα∈ C([ 0 , ∞[ , B(X)) and α↦ Tα is a continuous map from ]0, 1] into C([ 0 , ∞[ , B(X)). Moreover, T1 becomes the Markov semigroup of a Poisson process. © 2018, Springer International Publishing AG, part of Springer Nature.

KW - Chapman–Kolmogorov equation

KW - Fractional Poisson process

KW - Markov semigroup

U2 - 10.1007/s11785-018-0763-z

DO - 10.1007/s11785-018-0763-z

M3 - Article

VL - 12

SP - 777

EP - 785

JO - Complex Analysis and Operator Theory

T2 - Complex Analysis and Operator Theory

JF - Complex Analysis and Operator Theory

SN - 1661-8254

IS - 3

ER -