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Florent Malrieu
Some simple but challenging Markov processes
Annales de la faculté des sciences de Toulouse Sér. 6, 24 no. 4: Numéro Spécial : Conférence “Talking Across Fields” du 24 au 28 mars 2014 à l’Institut de Mathématiques de Toulouse (2015), p. 857-883, doi: 10.5802/afst.1468
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Résumé - Abstract

In this note, we present few examples of Piecewise Deterministic Markov Processes and their long time behavior. They share two important features: they are related to concrete models (in biology, networks, chemistry,...) and they are mathematically rich. Their mathematical study relies on coupling method, spectral decomposition, PDE technics, functional inequalities. We also relate these simple examples to recent and open problems.

Bibliography

[1] Aldous (D.) and Diaconis (P.).— Strong uniform times and finite random walks, Adv. in Appl. Math. 8, no. 1, p. 69-97 (1987).  MR 876954 |  Zbl 0631.60065
[2] Bakhtin (Y.) and Hurth (T.).— Invariant densities for dynamical systems with random switching, Nonlinearity 25 no. 10, p. 2937-2952 (2012).  MR 2979976 |  Zbl 1251.93132
[3] Bakhtin (Y.), Hurth (T.), and Mattingly (J. C.).— Regularity of invariant densities for 1D-systems with random switching, arXiv:1406.5425, (2014).
[4] Balde (M.) and Boscain (U.).— , Stability of planar switched systems: the nondiagonalizable case, Commun. Pure Appl. Anal. 7, no. 1, p. 1-21 (2008).  MR 2358351 |  Zbl 1147.93038
[5] Balde (M.), Boscain (U.), and Mason (P.).— A note on stability conditions for planar switched systems, Internat. J. Control 82, no. 10, p. 1882-1888 (2009).  MR 2567235 |  Zbl 1178.93121
[6] Bardet (J.-B.), Christen (A.), Guillin (A.), Malrieu (A.), and Zitt (P.-A.).— Total variation estimates for the TCP process, Electron. J. Probab. 18, no. 10, p. 1-21 (2013).  MR 3035738 |  Zbl 1283.68067
[7] Benaïm (M.), Le Borgne (S.), Malrieu (F.), and Zitt (P.-A.).— Qualitative properties of certain piecewise deterministic Markov processes, Ann. Inst. Henri Poincar? Probab. Stat. 51, no. 3, p. 1040-1075 (2015).  MR 3365972
[8] Benaïm (M.), Le Borgne (S.), Malrieu (F.), and Zitt (P.-A.).— Quantitative ergodicity for some switched dynamical systems, Electron. Commun. Probab. 17, no. 56, p. 14 (2012).  MR 3005729
[9] Benaïm (M.), Le Borgne (S.), Malrieu (F.), and Zitt (P.-A.).— On the stability of planar randomly switched systems, Ann. Appl. Probab. 24, no. 1, p. 292-311 (2014).  MR 3161648 |  Zbl 1288.93090
[10] Berestycki (J.), Bertoin (J.), Haas (B.), and Miermont (G.).— Quelques aspects fractals des fragmentations aléatoires, Quelques interactions entre analyse, probabilités et fractals, Panor. Synthèses, vol. 32, Soc. Math. France, Paris, p. 191-243 (2010).  Zbl 1238.60091
[11] Bertail (P.), S. Clémençon (S.), and Tressou (J.).— A storage model with random release rate for modeling exposure to food contaminants, Math. Biosci. Eng. 5, no. 1, p. 35-60 (2008).  MR 2401278 |  Zbl 1143.92026
[12] Borkovec (M.), Dasgupta (A.), Resnick (S.), and Samorodnitsky (G.).— A single channel on/off model with TCP-like control, Stoch. Models 18, no. 3, p. 333-367 (2002).  MR 1928492 |  Zbl 1015.60087
[13] Boscain (U.),.— Stability of planar switched systems: the linear single input case, SIAM J. Control Optim. 41 no. 1, p. 89-112 (2002).  MR 1920158 |  Zbl 1012.93055
[14] Bouguet (F.).— Quantitative exponential rates of convergence for exposure to food contaminants, To appear in ESAIM PS, arXiv:1310.3948, (2013).
[15] Calvez (V.), Doumic Jauffret (M.), and Gabriel (P.).— Self-similarity in a general aggregation-fragmentation problem. Application to fitness analysis, J. Math. Pures Appl. (9) 98, no. 1, p. 1-27 (2012).  MR 2935368 |  Zbl 1259.35151
[16] Chafaï (D.), Malrieu (F.), and Paroux (K.).— On the long time behavior of the TCP window size process, Stochastic Process. Appl. 120, no. 8, p. 1518-1534 (2010).  MR 2653264 |  Zbl 1196.68028
[17] Cloez (B.) and Hairer (M.).— Exponential ergodicity for Markov processes with random switching, Bernoulli 21, no. 1, p. 505-536 (2015).  MR 3322329
[18] Comets (F.), Popov (S.), Schütz (M.), and Vachkovskaia (M.).— Billiards in a general domain with random reflections, Arch. Ration. Mech. Anal. 191, no. 3, p. 497-537 (2009).  MR 2481068 |  Zbl 1186.37049
[19] Crudu (A.), Debussche (A.), Muller (A.), and Radulescu (O.).— Convergence of stochastic gene networks to hybrid piecewise deterministic processes, Ann. Appl. Probab. 22, no. 5, p. 1822-1859 (2012).  MR 3025682 |  Zbl 1261.60073
[20] Davis (M. H. A.).— Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B 46, no. 3, p. 353-388, With discussion (1984).  MR 790622 |  Zbl 0565.60070
[21] Davis (M. H. A.).— Markov models and optimization, Monographs on Statistics and Applied Probability, vol. 49, Chapman & Hall, London, (1993).  MR 1283589 |  Zbl 0780.60002
[22] Diaconis (P.) and Freedman (P.).— Iterated random functions, SIAM Rev. 41, no. 1, p. 45-76 (1999).  MR 1669737 |  Zbl 0926.60056
[23] Doumic Jauffret (M.) and Gabriel (P.).— Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci. 20, no. 5, p. 757-783 (2010).  MR 2652618 |  Zbl 1201.35086
[24] Dumas (V.), Guillemin (F.), and Robert (Ph.).— A Markovian analysis of additive-increase multiplicative-decrease algorithms, Adv. in Appl. Probab. 34, no. 1, p. 85-111 (2002).  MR 1895332 |  Zbl 1002.60091
[25] Erban (R.) and Othmer (H. G.).— From individual to collective behavior in bacterial chemotaxis, SIAM J. Appl. Math. 65, no. 2, p. 361-391 (electronic) (2004/05).  MR 2123062 |  Zbl 1073.35116
[26] Erban (R.) and Othmer (H. G.).— From signal transduction to spatial pattern formation in E. coli: a paradigm for multiscale modeling in biology, Multiscale Model. Simul. 3, no. 2, p. 362-394 (electronic) (2005).  MR 2122993 |  Zbl 1073.35205
[27] Fontbona (J.), Guérin (H.), and Malrieu (F.).— Quantitative estimates for the long-time behavior of an ergodic variant of the telegraph process, Adv. in Appl. Probab. 44, no. 4, p. 977-994 (2012).  MR 3052846 |  Zbl 1274.60240
[28] Gadat (S.) and Miclo (L.).— Spectral decompositions and ${\mathbb{L}}^2$-operator norms of toy hypocoercive semi-groups, Kinet. Relat. Models 6, no. 2, p. 317-372 (2013).  MR 3030715 |  Zbl 1262.35134
[29] Graham (C.) and Robert (Ph.).— Interacting multi-class transmissions in large stochastic networks, Ann. Appl. Probab. 19, no. 6, p. 2334-236 (2009)1.  MR 2588247 |  Zbl 1179.60067
[30] Graham (C.) and Robert (Ph.).— Self-adaptive congestion control for multiclass intermittent connections in a communication network, Queueing Syst. 69, no. 3-4, p. 237-257 (2011).  MR 2886470 |  Zbl 1236.90022
[31] Guillemin (F.), Robert (Ph.), and Zwart (B.).— AIMD algorithms and exponential functionals, Ann. Appl. Probab. 14, no. 1, p. 90-117 (2004).  MR 2023017 |  Zbl 1041.60072
[32] Herrmann (S.) and Vallois (P.).— From persistent random walk to the telegraph noise, Stoch. Dyn. 10, no. 2, p. 161-196 (2010).  MR 2652885 |  Zbl 1196.60082
[33] Hespanha (J. P.).— A model for stochastic hybrid systems with application to communication networks, Nonlinear Anal. 62, no. 8, p. 1353-1383 (2005).  MR 2164929 |  Zbl 1131.90322
[34] Jacobsen (M.).— Point process theory and applications, Probability and its Applications, Birkh?user Boston Inc., Boston, MA, Marked point and piecewise deterministic processes (2006).  MR 2189574 |  Zbl 1093.60002
[35] Kac (M.).— A stochastic model related to the telegrapherÕs equation, Rocky Mountain J. Math. 4, p. 497-509 (1974).  MR 510166 |  Zbl 0314.60052
[36] Karmakar (R.) and Bose (I.).— Graded and binary responses in stochastic gene expression, Physical Biology 197 no. 1, p. 197-214 (2004).
[37] Lamberton (D.) and Pagès (G.).— A penalized bandit algorithm, Electron. J. Probab. 13, no. 13, p. 341-373 (2008).  MR 2386736 |  Zbl 1206.62139
[38] Lawley (S. D.), Mattingly (J. C.), and Reed (M. C.).— Sensitivity to switching rates in stochastically switched ODEs, Commun. Math. Sci. 12, no. 7, p. 1343-1352 (2014).  MR 3210750
[39] van Leeuwaarden (J. S. H.) and Löpker (A. H.).— Transient moments of the TCP window size process, J. Appl. Probab. 45, no. 1, p. 163-175 (2008).  MR 2409318 |  Zbl 1142.60049
[40] van Leeuwaarden (J. S. H.), Löpker (A. H.), and Ott (T. J.).— TCP and iso-stationary transformations, Queueing Syst. 63, no. 1-4, p. 459-475 (2009).  MR 2576022 |  Zbl 1196.60136
[41] Levin (D. A.), Peres (Y.), and Wilmer (E. L.).— Markov chains and mixing times, American Mathematical Society, Providence, RI, With a chapter by James G. Propp and David B. Wilson (2009).  MR 2466937 |  Zbl 1160.60001
[42] Mackey (M. C.), Tyran-Kaminska (M.), and Yvinec (R.).— Dynamic behavior of stochastic gene expression models in the presence of bursting, SIAM J. Appl. Math. 73, no. 5, p. 1830-1852 (2013).  MR 3097042 |  Zbl 1279.92029
[43] Maulik (K.) and Zwart (B.).— Tail asymptotics for exponential functionals of Lévy processes, Stochastic Process. Appl. 116, no. 2, p. 156-177 (2006).  MR 2197972 |  Zbl 1090.60046
[44] Maulik (K.) and Zwart (B.).— An extension of the square root law of TCP, Ann. Oper. Res. 170, p. 217-232 (2009).  MR 2506283 |  Zbl 1174.94002
[45] Miclo (L.) and Monmarché (P.).— , ?tude spectrale minutieuse de processus moins indécis que les autres, Séminaire de Probabilités XLV, Lecture Notes in Math., vol. 2078, Springer, Cham, p. 459-481 (2013).  MR 3185926
[46] Mischler (S.) and Scher (J.).— Spectral analysis of semigroups and growth-fragmentation equations, arXiv:1310.7773, (2013).
[47] Monmarché (P.).— Hypocoercive relaxation to equilibrium for some kinetic models, Kinet. Relat. Models 7, no. 2, p. 341-360 (2014).  MR 3195078
[48] Monmarché (P.).— On $H^1$ and entropic convergence for contractive PDMP, arXiv:1404.4220 (2014).
[49] Morris (C.) and Lecar (H.).— Voltage oscillations in the barnacle giant muscle fiber, Biophys. J. 35, no. 1, p. 193-213 (1981).
[50] Othmer (H. G.), Dunbar (S. R.), and Alt (W.).— Models of dispersal in biological systems, J. Math. Biol. 26, no. 3, p. 263-298 (1988).  MR 949094 |  Zbl 0713.92018
[51] Ott (T. J.).— Rate of convergence for the Ôsquare root formulaÕ in the internet transmission control protocol, Adv. in Appl. Probab. 38, no. 4, p. 1132-1154 (2006).  MR 2285697 |  Zbl 1107.60065
[52] Ott (T. J.) and Kemperman (J. H. B.).— Transient behavior of processes in the TCP paradigm, Probab. Engrg. Inform. Sci. 22, no. 3, p. 431-471 (2008).  MR 2426601 |  Zbl 1151.68328
[53] Ott (T. J.), Kemperman (J. H. B.), and Mathis (M.).— The stationary behavior of ideal TCP congestion avoidance, unpublished manuscript available at http://www.teunisott.com/ (1996).
[54] Ott (T. J.) and Swanson (J.).— Asymptotic behavior of a generalized TCP congestion avoidance algorithm, J. Appl. Probab. 44, no. 3, p. 618-635 (2007).  MR 2355580 |  Zbl 1233.60016
[55] Pakdaman (K.), Thieullen (M.), and Wainrib (G.).— Fluid limit theorems for stochastic hybrid systems with application to neuron models, Adv. in Appl. Probab. 42, no. 3, p. 761-794 (2010).  MR 2779558 |  Zbl 1232.60019
[56] Perthame (B.).— Transport equations in biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel (2007).  MR 2270822 |  Zbl 1185.92006
[57] Perthame (B.) and Ryzhik (L.).— Exponential decay for the fragmentation or cell-division equation, J. Differential Equations 210, no. 1, p. 155-177 (2005).  MR 2114128 |  Zbl 1072.35195
[58] Radulescu (O.), Muller (A.), and Crudu (A.).— Théorèmes limites pour des processus de Markov à sauts. Synthèse des résultats et applications en biologie moléculaire, Technique et Science Informatiques 26, no. 3-4, p. 443-469 (2007).
[59] Roberts (G. O.) and Tweedie (R. L.).— Rates of convergence of stochastically monotone and continuous time Markov models, J. Appl. Probab. 37, no. 2, p. 359-373 (2000).  MR 1780996 |  Zbl 0979.60060
[60] Rousset (M.) and Samaey (G.).— Individual-based models for bacterial chemotaxis in the diffusion asymptotics, Math. Models Methods Appl. Sci. 23, no. 11, p. 2005-2037 (2013).  MR 3084742
[61] Shao (J.).— Ergodicity of regime-switching diffusions in Wasserstein distances, Stochastic Process. Appl. 125, no. 2, p. 739-758 (2015).  MR 3293301
[62] Villani (C.).— Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI (2003).  MR 1964483 |  Zbl 1106.90001
[63] Wainrib (G.), Thieullen (M.), and Pakdaman (K.).— Intrinsic variability of latency to first-spike, Biol. Cybernet. 103, no. 1, p. 43-56 (2010).  MR 2658681 |  Zbl 1266.92024
[64] Yin (G. G.) and Zhu (C.).— Hybrid switching diffusions, Stochastic Modelling and Applied Probability, vol. 63, Springer, New York, Properties and applications (2010).  MR 2559912 |  Zbl 1279.60007
[65] Yvinec (R.), Zhuge (C.), Lei (J.), and Mackey (M. C.).— Adiabatic reduction of a model of stochastic gene expression with jump Markov process, J. Math. Biol. 68, no. 5, p. 1051-1070 (2014).  MR 3175198 |  Zbl 1284.92031
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