
Mathematical Biosciences and Engineering, 2019, 16(6): 63866405. doi: 10.3934/mbe.2019319.
Research article
Export file:
Format
 RIS(for EndNote,Reference Manager,ProCite)
 BibTex
 Text
Content
 Citation Only
 Citation and Abstract
Probabilistic analysis of systems alternating for statedependent dichotomous noise
Dipartimento di Matematica, Università degli Studi di Salerno, Via Giovanni Paolo II n. 132, 84084 Fisciano (SA), Italy
Received: , Accepted: , Published:
Keywords: telegraph process; random motion; intensity function; interarrival times; gamma distribution; generalized MittagLeffler function; firstpassagetime problem, constant boundaries
Citation: Antonio Di Crescenzo, Fabio Travaglino. Probabilistic analysis of systems alternating for statedependent dichotomous noise. Mathematical Biosciences and Engineering, 2019, 16(6): 63866405. doi: 10.3934/mbe.2019319
References:
 1. I. Bena, Dichotomous Markov noise: exact results for outofequilibrium systems, Int. J. Modern Phys. B, 20 (2006), 2825–2888.
 2. P. Li, L. R. Nie, C. Z. Shu, et al., Effect of correlated dichotomous noises on stochastic resonance in a linear system, J. Stat. Phys., 146 (2012), 1184–1202.
 3. Y. Jin, Z. Ma and S. Xiao, Coherence and stochastic resonance in a periodic potential driven by multiplicative dichotomous and additive white noise, Chaos Solit. Fract., 103 (2017), 470–475.
 4. F. MüllerHansen, F. Droste and B. Lindner, Statistics of a neuron model driven by asymmetric colored noise, Phys. Rev. E, 91 (2015), 022718.
 5. F. Droste and B. Lindner, Integrateandfire neurons driven by asymmetric dichotomous noise, Biol. Cybernet., 108 (2014), 825–843.
 6. R. Mankin and N. Lumi, Statistics of a leaky integrateandfire model of neurons driven by dichotomous noise, Phys. Rev. E, 93 (2016), 052143.
 7. S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation, Quart. J. Mech. Appl. Math., 4 (1951), 129–156.
 8. M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mountain J. Math., 4 (1974), 497–509.
 9. A. D. Kolesnik and N. Ratanov, Telegraph Processes and Option Pricing, Springer Briefs in Statistics, Springer, Heidelberg, 2013.
 10. E. Orsingher, Probability law, flow function, maximum distribution of wavegoverned random motions and their connections with Kirchoff's laws, Stoch. Process. Appl., 34 (1990), 49–66.
 11. L. Beghin, L. Nieddu and E. Orsingher, Probabilistic analysis of the telegrapher's process with drift by means of relativistic transformations, J. Appl. Math. Stochastic Anal., 14 (2001), 11–25.
 12. O. López and N. Ratanov, On the asymmetric telegraph processes, J. Appl. Probab., 51 (2014), 569–589.
 13. D. Bshouty, A. Di Crescenzo, B. Martinucci, et al., Generalized telegraph process with random delays, J. Appl. Probab., 49 (2012), 850–865.
 14. A. Di Crescenzo and B. Martinucci, A damped telegraph random process with logistic stationary distribution, J. Appl. Probab., 47 (2010), 84–96.
 15. A. De Gregorio and C. Macci, Large deviations for a damped telegraph process, in Modern problems in insurance mathematics (eds. D. Silvestrov and A. MartinLöf), EAA Ser., Springer, Cham, (2014), 275–289.
 16. A.D. Kolesnik, Probability distribution function for the Euclidean distance between two telegraph processes, Adv. Appl. Probab., 46 (2014), 1172–1193.
 17. A. D. Kolesnik, The explicit probability distribution of the sum of two telegraph processes, Stoch. Dyn., 15 (2015), 1550013.
 18. B. Martinucci and A. Meoli, Certain functionals of squared telegraph processes, Stoch. Dyn., (2019) Online Ready.
 19. A. Di Crescenzo, B. Martinucci and S. Zacks, Telegraph process with elastic boundary at the origin, Methodol. Comput. Appl. Probab., 20 (2018), 333–352.
 20. A. A. Pogorui, R. M. Rodr´ ıguezDagnino and T. Kolomiets, The first passage time and estimation of the number of levelcrossings for a telegraph process, Ukrainian Math. J., 67 (2015), 998–1007.
 21. N. Ratanov, Telegraph processes with random jumps and complete market models, Methodol. Comput. Appl. Probab., 17 (2015), 677–695.
 22. A. Di Crescenzo and S. Zacks, Probability law and flow function of Brownian motion driven by a generalized telegraph process, Methodol. Comput. Appl. Probab., 17 (2015), 761–780.
 23. F. Travaglino, A. Di Crescenzo, B. Martinucci, et al., A new model of Campi Flegrei inflation and deflation episodes based on Brownian motion driven by the telegraph process, Math. Geosci., 50 (2018), 961–975.
 24. J. Bierkens, P. Fearnhead and G. Roberts, The zigzag process and superefficient sampling for Bayesian analysis of big data, arXiv:1607.03188v2.
 25. S. M. Iacus, Statistical analysis of the inhomogeneous telegrapher's process, Statist. Probab. Lett., 55 (2001), 83–88.
 26. R. Garra and E. Orsingher, Random motions with spacevarying velocities, in Modern problems of stochastic analysis and statistics (ed. V. Panov), Springer Proc. Math. Stat., 208, Springer, Cham, (2017), 25–39.
 27. G. D'Onofrio, P. Lansky and E. Pirozzi, On two diffusion neuronal models with multiplicative noise: The mean firstpassage time properties, Chaos, 28 (2018), 043103.
 28. A. Di Crescenzo and B. Martinucci, Analysis of a stochastic neuronal model with excitatory inputs and statedependent effects, Math. Biosci., 209 (2007), 547–563.
 29. A. Di Crescenzo, On random motions with velocities alternating at Erlangdistributed random times, Adv. Appl. Probab., 33 (2001), 690–701.
 30. W. Alharbi and S. Petrovskii, Critical domain problem for the reactiontelegraph equation model of population dynamics, Mathematics, 6 (2018), 59.
 31. L. Angelani and R. Garra, Probability distributions for the runandtumble models with variable speed and tumbling rate, Mod. Stoch. Theory Appl., 6 (2019), 3–12.
 32. R. Garcia, F. Moss, A. Nihongi, et al., Optimal foraging by zooplankton within patches: the case of Daphnia, Math. Biosci., 207 (2007), 165–188.
 33. A. Di Crescenzo and B. Martinucci, Random motion with gammadistributed alternating velocities in biological modeling, in Computer Aided Systems Theory  Eurocast 2007, Lecture Notes in Computer Science (eds. R. MorenoDiaz, F. Pichler and A. QuesadaArencibia), SpringerVerlag, Berlin, (2007), 163–170.
 34. G. Le Caër, A new family of solvable PearsonDirichlet random walks, J. Stat. Phys., 144 (2011), 23–45.
 35. A. A. Pogorui and R.M. RodríguezDagnino, Random motion with gamma steps in higher dimensions, Statist. Probab. Lett., 83 (2013), 638–1643.
 36. R. Gorenflo, A. A. Kilbas, F. Mainardi, et al., The twoparametric MittagLeffler function, in MittagLeffler functions, related topics and applications, Springer Monographs in Mathematics. Springer, Berlin, Heidelberg, (2014), 55–96.
 37. H. J. Haubold, A. M. Mathai and R. K. Saxena, MittagLeffler functions and their applications, J. Appl. Math., (2011), Art. ID 298628, 51 pp.
 38. M. Alipour, L. Beghin and D. Rostamy, Generalized fractional nonlinear birth processes, Methodol. Comput. Appl. Probab., 17 (2015), 525–540.
 39. A. Di Crescenzo, B. Martinucci and A. Meoli, A fractional counting process and its connection with the Poisson process, ALEA Lat. Am. J. Probab. Math. Stat., 13 (2016), 291–307.
 40. E. Orsingher and F. Polito, On a fractional linear birthdeath process, Bernoulli, 17 (2011), 114–137.
 41. P. Lansky, L. Sacerdote and C. Zucca, Optimum signal in a diffusion leaky integrateandfire neuronal model, Math. Biosci., 207 (2007), 261–274.
Reader Comments
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)
Associated material
Metrics
Other articles by authors
Related pages
Tools
your name: * your email: *