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Modified Log-Sobolev Inequality for a Compact Pure Jump Markov Process with Degenerate Jumps

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We study the modified log-Sobolev inequality for a class of pure jump Markov processes that describe the interactions between brain neurons. As a result, we obtain concentration properties for empirical approximations of the process. In particular, we focus on a finite and compact process with degenerate jumps inspired by the model introduced by Galves and Löcherbach (J Stat Phys 151:896–921, 2013).

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  1. 1.

    André, M.: A result of metastability for an infinite system of spiking neurons. J. Stat. Phys. 177, 984–1008 (2019)

  2. 2.

    Ane, C., Ledoux, M.: Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. Probab. Theory Relat. Fields 116, 573–602 (2000)

  3. 3.

    Azaïs, R., Bardet, J.B., Genadot, A., Krell, N., Zitt, P.A.: Piecewise deterministic Markov process (pdmps): recent results. ESIAM Proc. 44, 276–290 (2014)

  4. 4.

    Bakry, D.: L’hypercontructivité et son utilisation en théorie des semigroupes. Ecole d’Eté de Probabilités de St-Flour. Lecture Notes in Mathematics, vol. 1581, pp. 1–114. Springer, Berlin (1994)

  5. 5.

    Bakry, D.: On Sobolev and Logarithmic Sobolev Inequalities for Markov Semigroups. New Trends in Stochastic Analysis, pp. 43–75. World Scientific, Singapore (1997)

  6. 6.

    Bobkov, S., Ledoux, M.: Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution. Probab. Theory Relat. Fields 107, 383–400 (1997)

  7. 7.

    Bolley, F., Guillin, A., Villani, C.: Quantitative concentration inequalities for empirical measures on non-compact spaces. Probab. Theory Relat. Fields 137, 541–593 (2007)

  8. 8.

    Chafai, D.: Entropies, convexity, and functional inequalities. J. Math. Kyoto Univ. 44(2), 325–363 (2004)

  9. 9.

    Chevalier, J.: Mean-field limit of generalized Hawkes processes. Stoch. Process. Appl. 127(12), 3870–3912 (2017)

  10. 10.

    Crudu, A., Debussche, A., Muller, A., Radulescu, O.: Convergence of stochastic gene networks to hybrid piecewise deterministic processes. Ann. Appl. Probab. 22, 1822–1859 (2012)

  11. 11.

    Davis, M.H.A.: Piecewise-derministic Markov processes: a general class off nondiffusion stochastic models. J. R. Stat. Soc. Ser. B 46(3), 353–388 (1984)

  12. 12.

    Davis, M.H.A.: Markov Models and Optimization Monographs on Statistics and Applied Probability, vol. 49. Chapman & Hall, London (1993)

  13. 13.

    Diaconis, P., Saloff-Coste, L.: Logarithmic Sobolev inequalities for finite Markov Chanis. Ann. Appl. Probab. 6, 695–750 (1996)

  14. 14.

    Duarte, A., Ost, G.: A model for neural activity in the absence of external stimuli. Markov Process. Relat. Fields 22, 37–52 (2016)

  15. 15.

    Duarte, A., Löcherbach, E., Ost, G.: Stability, convergence to equilibrium and simulation of non-linear Hawkes Processes with memory kernels given by the sum of Erlang kernels. ESAIM: PS 23, 770–796 (2019)

  16. 16.

    Galves, A., Löcherbach, E.: Infinite systems of interacting chains with memory of variable length: a stochastic model for biological neural nets. J. Stat. Phys. 151, 896–921 (2013)

  17. 17.

    Guionnet, A., Zegarlinski, B.: Lectures on Logarithmic Sobolev Inequalities, IHP Course 98. In: Seminare de Probabilite XXVI. Lecture Notes in Mathematics, vol. 1801, pp. 1–134. Springer, Berlin (2003)

  18. 18.

    Hansen, N., Reynaud-Bouret, P., Rivoirard, V.: Lasso and probabilistic inequalities for multivariate point processes. Bernoulli 21(1), 83–143 (2015)

  19. 19.

    Hodara, P., Löcherbach, E.: Hawkes processes with variable length memory and an infinite number of components. Adv. Appl. Probab. 49, 84–107 (2017)

  20. 20.

    Hodara, P., Papageorgiou, I.: Poincaré type inequalities for compact degenerate pure jump Markov processes. Mathematics 7(6), 518 (2019)

  21. 21.

    Hodara, P., Krell, N., Löcherbach, E.: Non-parametric estimation of the spiking rate in systems of interacting neurons. Stat. Inference Stoch. Process. 21, 81–111 (2018)

  22. 22.

    Ledoux, M.: Concentration of measure and logarithmic Sobolev inequalities. In: Seminaire de Probabilites XXXV. Lecture Notes in Mathematics, vol. 1709, pp. 120-216. Springer, Berlin (1999)

  23. 23.

    Ledoux, M.: The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, vol. 89. AMS, Providence (2001)

  24. 24.

    Löcherbach, E.: Absolute continuity of the invariant measure in piecewise deterministic Markov Processes having degenerate jumps. Stoch. Process. Their Appl. 128, 1797–1829 (2018)

  25. 25.

    Malrieu, F.: Logarithmic Sobolev inequalities for some nonlinear PDE’s. Stoch. Process. Appl. 95, 109–132 (2001)

  26. 26.

    Norris, J.R.: Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 2. Cambridge University Press, Cambridge (1998)

  27. 27.

    Pakdaman, K., Thieulen, M., Wainrib, G.: Fluid limit theorems for stochastic hybrid systems with application to neuron models. Adv. Appl. Probab. 42, 761–794 (2010)

  28. 28.

    Saloff-Coste, L.: Lectures on finite Markov chains. IHP Course 98, Ecole d’ Ete de Probabilites de Saint-Flour XXVI, Lecture Notes in Mathematics, vol. 1665, pp. 301–413. Springer, Berlin (1996)

  29. 29.

    Talagrand, M.: A new isoperimetric inequality and concentration of measure phenomenon. In: Lindenstrauss, J., Milman, V.D. (eds.) Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics, vol. 1469, pp. 94–124. Springer, Berlin (1991)

  30. 30.

    Talagrand, M.: Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. IHES 81, 73–205 (1995)

  31. 31.

    Wang, F.-Y., Yuan, C.: Poincaré inequality on the path space of Poisson point processes. J. Theor. Probab. 23(3), 824–833 (2010)

  32. 32.

    Yosida, K.: Functional Analysis. Springer, Berlin (1980)

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Correspondence to Ioannis Papageorgiou.

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This article was produced as part of the activities of FAPESP Research, Innovation and Dissemination Center for Neuromathematics (Grant 2013/ 07699-0 , S. Paulo Research Foundation); This article is supported by FAPESP Grant (2017/15587-8)

Communicated by Eric A. Carlen.

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Papageorgiou, I. Modified Log-Sobolev Inequality for a Compact Pure Jump Markov Process with Degenerate Jumps. J Stat Phys (2020).

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  • Brain neuron networks
  • Pure jump Markov processes
  • Modified log-Sobolev inequality
  • Concentration
  • Empirical approximations

Mathematics Subject Classification

  • 60K35
  • 26D10
  • 60G99