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Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures
On discrete-time semi-Markov processes
| 1. | Faculty of Computing, Engineering and Science, University of South Wales, UK |
| 2. | Department of Mathematics "G. Peano", University of Torino, Italy |
In the last years, several authors studied a class of continuous-time semi-Markov processes obtained by time-changing Markov processes by hitting times of independent subordinators. Such processes are governed by integro-differential convolution equations of generalized fractional type. The aim of this paper is to develop a discrete-time counterpart of such a theory and to show relationships and differences with respect to the continuous time case. We present a class of discrete-time semi-Markov chains which can be constructed as time-changed Markov chains and we obtain the related governing convolution type equations. Such processes converge weakly to those in continuous time under suitable scaling limits.
References:
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D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511809781. |
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V. S. Barbu and N. Limnios, Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications, Lecture Notes in Statistics, 191, Springer, New York, 2008.
doi: 10.1007/978-0-387-73173-5. |
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P. Becker-Kern, M. M. Meerschaert and H.-P. Scheffler,
Limit theorems for coupled continuous-time random walks, Ann. Probab., 32 (2004), 730-756.
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L. Beghin,
Fractional relaxation equations and Brownian crossing probabilities of a random boundary, Adv. in Appl. Probab., 44 (2012), 479-505.
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L. Beghin and E. Orsingher,
Fractional Poisson processes and related planar random motions, Electron. J. Probab., 14 (2009), 1790-1827.
doi: 10.1214/EJP.v14-675. |
| [6] |
L. Beghin and E. Orsingher,
Poisson-type processes governed by fractional and higher-order recursive differential equations, Electron. J. Probab., 15 (2010), 684-709.
doi: 10.1214/EJP.v15-762. |
| [7] |
L. Beghin and C. Ricciuti,
Time-inhomogeneous fractional Poisson processes defined by the multistable subordinator, Stoch. Anal. Appl., 37 (2019), 171-188.
doi: 10.1080/07362994.2018.1548970. |
| [8] |
J. Bertoin, Subordinators: Examples and Applications, in Lectures on Probability Theory and Statistics, Lectures Notes in Math., 1717, Springer, Berlin, 1999, 1–91.
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N. H. Bingham,
Limit theorems for occupation times of Markov processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 17 (1971), 1-22.
doi: 10.1007/BF00538470. |
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N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1987.
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A triptych of discrete distributions related to the stable law, Statist. Probab. Lett., 18 (1993), 349-351.
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Fractional relaxation with time-varying coefficient, Fract. Calc. Appl. Anal., 17 (2014), 424-439.
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Probabilistic solutions to nonlinear fractional differential equations of generalized Caputo and Riemann–Liouville type, Stochastics, 90 (2018), 224-255.
doi: 10.1080/17442508.2017.1334059. |
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M. E. Hernández-Hernández, V. N. Kolokoltsov and L. Toniazzi,
Generalized fractional evolutions equations of Caputo type, Chaos Solitons Fractals, 102 (2017), 184-196.
doi: 10.1016/j.chaos.2017.05.005. |
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J. Jacod,
Systèmes régenératifs and processus semi-markoviens, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 31 (1974/75), 1-23.
doi: 10.1007/BF00538712. |
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V. N. Kolokol'tsov,
Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics, Theory Probab. Appl., 53 (2009), 594-609.
doi: 10.1137/S0040585X97983857. |
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V. N. Kolokoltsov, Markov Processes, Semigroups and Generators, De Gruyter Studies in Mathematics, 38, Walter de Gruyter & Co., Berlin, 2011.
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V. Korolyuk and A. Swishchuk, Semi-Markov Random Evolutions, Mathematics and its Applications, 308, Kluwer Academic Publishers, Dordrecht, 1995.
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A. Kumar, E. Nane and P. Vellaisamy,
Time-changed Poisson processes, Statist. Probab. Lett., 81 (2011), 1899-1910.
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T. G. Kurtz,
Comparison of semi-Markov and Markov processes, Ann. Math. Statist., 42 (1971), 991-1002.
doi: 10.1214/aoms/1177693327. |
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N. N. Leonenko, M. M. Meerschaert, R. L. Shilling and A. Sikorskii, Correlation structure of time-changed Lévy processes, Commun. Appl. Ind. Math., 6 (2014), 22pp.
doi: 10.1685/journal.caim.483. |
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N. N. Leonenko, E. Scalas and M. Trinh,
The fractional non-homogeneous Poisson process, Statist. Probab. Lett., 120 (2017), 147-156.
doi: 10.1016/j.spl.2016.09.024. |
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F. Mainardi, R. Gorenflo and E. Scalas,
A fractional generalization of the Poisson process, Vietnam J. Math., 32 (2007), 53-64.
|
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M. M. Meerschaert and H.-P. Scheffler,
Limit theorems for continuous-time random walks with infinite mean waiting times, J. Appl. Probab., 41 (2004), 623-638.
doi: 10.1239/jap/1091543414. |
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M. M. Meerschaert and H.-P. Scheffler,
Triangular array limits for continuous time random walks, Stochastic Process. Appl., 118 (2008), 1606-1633.
doi: 10.1016/j.spa.2007.10.005. |
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M. M. Meerschaert, E. Nane and P. Vellaisamy,
The fractional Poisson process and the inverse stable subordinator, Electron. J. Probab., 16 (2011), 1600-1620.
doi: 10.1214/EJP.v16-920. |
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M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, 43, Walter de Gruyter & Co., Berlin, 2012.
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Semi-Markov approach to continuous time random walk limit processes, Ann. Probab., 42 (2014), 1699-1723.
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Fractional pure birth processes, Bernoulli, 16 (2010), 858-881.
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On a fractional linear birth-death process, Bernoulli, 17 (2011), 114-137.
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Fractional non-linear, linear and sub-linear death processes, J. Stat. Phys., 141 (2010), 68-93.
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Time-inhomogeneous jump processes and variable order operators, Potential Anal., 45 (2016), 435-461.
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E. Orsingher, C. Ricciuti and B. Toaldo,
On semi-Markov processes and their Kolmogorov's integro-differential equations, J. Funct. Anal., 275 (2018), 830-868.
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show all references
References:
| [1] |
D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009.
doi: 10.1017/CBO9780511809781. |
| [2] |
V. S. Barbu and N. Limnios, Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications, Lecture Notes in Statistics, 191, Springer, New York, 2008.
doi: 10.1007/978-0-387-73173-5. |
| [3] |
P. Becker-Kern, M. M. Meerschaert and H.-P. Scheffler,
Limit theorems for coupled continuous-time random walks, Ann. Probab., 32 (2004), 730-756.
doi: 10.1214/aop/1079021462. |
| [4] |
L. Beghin,
Fractional relaxation equations and Brownian crossing probabilities of a random boundary, Adv. in Appl. Probab., 44 (2012), 479-505.
doi: 10.1017/S0001867800005693. |
| [5] |
L. Beghin and E. Orsingher,
Fractional Poisson processes and related planar random motions, Electron. J. Probab., 14 (2009), 1790-1827.
doi: 10.1214/EJP.v14-675. |
| [6] |
L. Beghin and E. Orsingher,
Poisson-type processes governed by fractional and higher-order recursive differential equations, Electron. J. Probab., 15 (2010), 684-709.
doi: 10.1214/EJP.v15-762. |
| [7] |
L. Beghin and C. Ricciuti,
Time-inhomogeneous fractional Poisson processes defined by the multistable subordinator, Stoch. Anal. Appl., 37 (2019), 171-188.
doi: 10.1080/07362994.2018.1548970. |
| [8] |
J. Bertoin, Subordinators: Examples and Applications, in Lectures on Probability Theory and Statistics, Lectures Notes in Math., 1717, Springer, Berlin, 1999, 1–91.
doi: 10.1007/978-3-540-48115-7_1. |
| [9] |
P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. |
| [10] |
N. H. Bingham,
Limit theorems for occupation times of Markov processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 17 (1971), 1-22.
doi: 10.1007/BF00538470. |
| [11] |
N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1987.
doi: 10.1017/CBO9780511721434. |
| [12] |
D. Buraczewski, E. Damek and T. Mikosch, Stochastic Models with Power-Law Tails, Springer Series in Operations Research and Financial Engineering, Springer, Cham, 2016.
doi: 10.1007/978-3-319-29679-1. |
| [13] |
E. Çinlar, Markov additive processes and semi-regeneration, in Proceedings of the Fifth Conference on Probability Theory, Editura Acad. R. S. R., Bucharest, 1977, 33-49. |
| [14] |
D. R. Cox, Renewal Theory, Methuen & Co. Ltd., London; John Wiley & Sons, Inc., New York, 1962. |
| [15] |
L. Devroye,
A triptych of discrete distributions related to the stable law, Statist. Probab. Lett., 18 (1993), 349-351.
doi: 10.1016/0167-7152(93)90027-G. |
| [16] |
M. D'Ovidio,
Wright functions governed by fractional directional derivatives and fractional advection diffusion equations, Methods Appl. Anal., 22 (2015), 1-36.
doi: 10.4310/MAA.2015.v22.n1.a1. |
| [17] |
W. Feller,
On semi-Markov processes, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 653-659.
doi: 10.1073/pnas.51.4.653. |
| [18] |
R. Garra, E. Orsingher and F. Polito,
State-dependent fractional point processes, J. Appl. Probab., 52 (2015), 18-36.
doi: 10.1239/jap/1429282604. |
| [19] |
R. Garra, A. Giusti, F. Mainardi and G. Pagnini,
Fractional relaxation with time-varying coefficient, Fract. Calc. Appl. Anal., 17 (2014), 424-439.
doi: 10.2478/s13540-014-0178-0. |
| [20] |
N. Georgiou, I. Z. Kiss and E. Scalas, Solvable non-Markovian dynamic network, Phys. Rev. E (3), 92 (2015), 9pp.
doi: 10.1103/PhysRevE.92.042801. |
| [21] |
B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley Educational Publishers Inc., Cambridge, Mass., 1954. |
| [22] |
I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes. II, der Mathematischen Wissenschaften, Band 218, Springer-Verlag, New York-Heidelberg, 1975. |
| [23] |
C. W. J. Granger and R. Joyeux,
An introduction to long-memory time series models and fractional differencing, J. Time Ser. Anal., 1 (1980), 15-29.
doi: 10.1111/j.1467-9892.1980.tb00297.x. |
| [24] |
A. Gut, Probability: A Graduate Course, Springer Texts in Statistics, 75, Springer, New York, 2013.
doi: 10.1007/978-1-4614-4708-5. |
| [25] |
M. E. Hernández-Hernández and V. N. Kolokoltsov,
Probabilistic solutions to nonlinear fractional differential equations of generalized Caputo and Riemann–Liouville type, Stochastics, 90 (2018), 224-255.
doi: 10.1080/17442508.2017.1334059. |
| [26] |
M. E. Hernández-Hernández, V. N. Kolokoltsov and L. Toniazzi,
Generalized fractional evolutions equations of Caputo type, Chaos Solitons Fractals, 102 (2017), 184-196.
doi: 10.1016/j.chaos.2017.05.005. |
| [27] |
J. Jacod,
Systèmes régenératifs and processus semi-markoviens, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 31 (1974/75), 1-23.
doi: 10.1007/BF00538712. |
| [28] |
V. N. Kolokol'tsov,
Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics, Theory Probab. Appl., 53 (2009), 594-609.
doi: 10.1137/S0040585X97983857. |
| [29] |
V. N. Kolokoltsov, Markov Processes, Semigroups and Generators, De Gruyter Studies in Mathematics, 38, Walter de Gruyter & Co., Berlin, 2011.
doi: 10.1515/9783110250114. |
| [30] |
V. Korolyuk and A. Swishchuk, Semi-Markov Random Evolutions, Mathematics and its Applications, 308, Kluwer Academic Publishers, Dordrecht, 1995.
doi: 10.1007/978-94-011-1010-5. |
| [31] |
A. Kumar, E. Nane and P. Vellaisamy,
Time-changed Poisson processes, Statist. Probab. Lett., 81 (2011), 1899-1910.
doi: 10.1016/j.spl.2011.08.002. |
| [32] |
T. G. Kurtz,
Comparison of semi-Markov and Markov processes, Ann. Math. Statist., 42 (1971), 991-1002.
doi: 10.1214/aoms/1177693327. |
| [33] |
N. N. Leonenko, M. M. Meerschaert, R. L. Shilling and A. Sikorskii, Correlation structure of time-changed Lévy processes, Commun. Appl. Ind. Math., 6 (2014), 22pp.
doi: 10.1685/journal.caim.483. |
| [34] |
N. N. Leonenko, E. Scalas and M. Trinh,
The fractional non-homogeneous Poisson process, Statist. Probab. Lett., 120 (2017), 147-156.
doi: 10.1016/j.spl.2016.09.024. |
| [35] |
P. Levy, Processus semi-markoviens, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, Erven P. Noordhoff N. V., Groningen; North-Holland Publishing Co., Amsterdam, 1956,416–426. |
| [36] |
F. Mainardi, R. Gorenflo and E. Scalas,
A fractional generalization of the Poisson process, Vietnam J. Math., 32 (2007), 53-64.
|
| [37] |
M. M. Meerschaert and H.-P. Scheffler,
Limit theorems for continuous-time random walks with infinite mean waiting times, J. Appl. Probab., 41 (2004), 623-638.
doi: 10.1239/jap/1091543414. |
| [38] |
M. M. Meerschaert and H.-P. Scheffler,
Triangular array limits for continuous time random walks, Stochastic Process. Appl., 118 (2008), 1606-1633.
doi: 10.1016/j.spa.2007.10.005. |
| [39] |
M. M. Meerschaert, E. Nane and P. Vellaisamy,
The fractional Poisson process and the inverse stable subordinator, Electron. J. Probab., 16 (2011), 1600-1620.
doi: 10.1214/EJP.v16-920. |
| [40] |
M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, 43, Walter de Gruyter & Co., Berlin, 2012.
doi: 10.1515/9783110258165. |
| [41] |
M. M. Meerschaert and P. Straka,
Semi-Markov approach to continuous time random walk limit processes, Ann. Probab., 42 (2014), 1699-1723.
doi: 10.1214/13-AOP905. |
| [42] |
M. M. Meerschaert and B. Toaldo,
Relaxation patterns and semi-Markov dynamics, Stochastic Process. Appl., 129 (2019), 2850-2879.
doi: 10.1016/j.spa.2018.08.004. |
| [43] |
T. Michelitsch, G. Maugin, S. Derogar and A. Nowakowski, Sur une généralisation de l'opérateur fractionnaire, preprint, arXiv: 1111.1898v1. |
| [44] |
J. R. Norris, Markov Chains, Cambridge Series in Statistical and Probabilistic Mathematics, 2, Cambridge University Press, Cambridge, 1998.
doi: 10.1017/CBO9780511810633. |
| [45] |
E. Orsingher and F. Polito,
Fractional pure birth processes, Bernoulli, 16 (2010), 858-881.
doi: 10.3150/09-BEJ235. |
| [46] |
E. Orsingher and F. Polito,
On a fractional linear birth-death process, Bernoulli, 17 (2011), 114-137.
doi: 10.3150/10-BEJ263. |
| [47] |
E. Orsingher, F. Polito and L. Sakhno,
Fractional non-linear, linear and sub-linear death processes, J. Stat. Phys., 141 (2010), 68-93.
doi: 10.1007/s10955-010-0045-2. |
| [48] |
E. Orsingher, C. Ricciuti and B. Toaldo,
Time-inhomogeneous jump processes and variable order operators, Potential Anal., 45 (2016), 435-461.
doi: 10.1007/s11118-016-9551-4. |
| [49] |
E. Orsingher, C. Ricciuti and B. Toaldo,
On semi-Markov processes and their Kolmogorov's integro-differential equations, J. Funct. Anal., 275 (2018), 830-868.
doi: 10.1016/j.jfa.2018.02.011. |
| [50] |
A. Pachon, F. Polito and L. Sacerdote,
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