Concentración y desigualdades de tipo Poincaré para un proceso de Markov puro con salto degenerado


Estudiamos la concentración de Talagrand y las desigualdades de tipo Poincaré para procesos de Markov de salto puro no acotado. En particular, nos centramos en los procesos con saltos degenerados que dependen del pasado de todo el sistema, basado en el modelo introducido por Galves y Löcherbach, para describir la actividad de una red neuronal biológica. Como resultado obtenemos algunas propiedades de concentración.



Aida, S., Masuda, T., y Shigekawa, I. (1994). Logarithmic sobolev inequalities and exponential integrability. Journal of Functional Analysis, 126(1), 83–101.

Aida, S., y Stroock, D. (1994). Moment estimates derived from poincaré and logarithmic sobolev inequalities. Mathematical Research Letters, 1(1), 75–86.

André, M. (2019). A result of metastability for an infinite system of spiking neurons. Journal of Statistical Physics, 177(5), 984–1008.

André, M., y Planche, L. (2021). The effect of graph connectivity on metastability in a stochastic system of spiking neurons. Stochastic Processes and their Applications, 131, 292–310.

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

Azais, R., Bardet, J.-B., Génadot, A., Krell, N., y Zitt, P.-A. (2014). Piecewise deterministic markov process recent results. En Esaim: Proceedings (Vol. 44, pp. 276–290).

Bakry, D., Cattiaux, P., y Guillin, A. (2008). Rate of convergence for ergodic continuous markov processes: Lyapunov versus poincaré. Journal of Functional Analysis, 254(3), 727–759.

Bakry, D., Gentil, I., Ledoux, M., y cols. (2014). Analysis and geometry of markov diffusion operators (Vol. 103). Springer.

Barthe, F., y Roberto, C. (2008). Modified logarithmic sobolev inequalities on \R. Potential Analysis, 29(2), 167–193.

Bobkov, S., y Ledoux, M. (1997). Poincaré's inequalities and talagrand’s concentration phenomenon for the exponential distribution. Probability Theory and Related Fields, 107(3), 383–400.

Cattiaux, P., Guillin, A., Wang, F.-Y., y Wu, L. (2009). Lyapunov conditions for super poincaré inequalities. Journal of Functional Analysis, 256(6), 1821–1841.

Chafai, D. (2004). Entropies, convexity, and functional inequalities, on \ϕ − entropies and\ϕ − Sobolev inequalities. Journal of Mathematics of Kyoto University, 44(2), 325–363.

Chevallier, J. (2017). Mean-field limit of generalized hawkes processes. Stochastic Processes and their Applications, 127(12), 3870–3912.

Crudu, A., Debussche, A., Muller, A., y Radulescu, O. (2012). Convergence of stochastic gene networks to hybrid piecewise deterministic processes. The Annals of Applied Probability, 22(5), 1822–1859.

Davis, M. H. (1984). Piecewise-deterministic markov processes: A general class of non-diffusion stochastic models. Journal of the Royal Statistical Society: Series B (Methodological), 46(3), 353–376.

Davis, M. H. (2018). Markov models and optimization. Routledge.

Diaconis, P., y Saloff-Coste, L. (1996). Logarithmic sobolev inequalities for finite markov chains. The Annals of Applied Probability, 6(3), 695–750.

Duarte, A., Löcherbach, E., y Ost, G. (2019). Stability, convergence to equilibrium and simulation of non-linear hawkes processes with memory kernels given by the sum of erlang kernels. ESAIM: Probability and Statistics, 23, 770–796.

Duarte, A., y Ost, G. (2014). A model for neural activity in the absence of external stimuli. arXiv preprint arXiv:1410.6086.

Galves, A., y Löcherbach, E. (2013). Infinite systems of interacting chains with memory of variable length—a stochastic model for biological neural nets. Journal of Statistical Physics, 151(5), 896–921.

Gentil, I., Guillin, A., y Miclo, L. (2005). Modified logarithmic sobolev inequalities and transportation inequalities. Probability theory and related fields, 133(3), 409–436.

Gross, L., y Rothaus, O. (1998). Herbst inequalities for supercontractive semi- groups. Journal of Mathematics of Kyoto University, 38(2), 295–318.

Guionnet, A., y Zegarlinksi, B. (2003). Lectures on logarithmic sobolev inequalities. En Séminaire de probabilités xxxvi (pp. 1–134). Springer.

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

Hodara, P., Krell, N., y Löcherbach, E. (2018). Non-parametric estimation of the spiking rate in systems of interacting neurons. Statistical Inference for Stochastic Processes, 21(1), 81–111.

Hodara, P., y Löcherbach, E. (2017). Hawkes processes with variable length memory and an infinite number of components. Advances in Applied Probability, 49(1), 84–107.

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

Inglis, J., y Papageorgiou, I. (2014). Log-sobolev inequalities for infinitedimensional gibbs measures with non-quadratic interactions. arXiv preprint arXiv:1401.3219.

Ledoux, M. (1999). Concentration of measure and logarithmic sobolev inequalities. En Seminaire de probabilites xxxiii (pp. 120–216). Springer.

Ledoux, M. (2001). The concentration of measure phenomenon (n.o 89). American Mathematical Soc.

Locherbach, E. (2018). Absolute continuity of the invariant measure in piecewise deterministic markov processes having degenerate jumps. Stochastic Proces- ses and their Applications, 128(6), 1797–1829.

Pakdaman, K., Thieullen, M., y Wainrib, G. (2010). Fluid limit theorems for stochastic hybrid systems with application to neuron models. Advances in Applied Probability, 42(3), 761–794.

Papageorgiou, I. (2011). A note on the modified log-sobolev inequality. Potential Analysis, 35(3), 275–286.

Papageorgiou, I. (2020). Modified log-sobolev inequality for a compact pure jump markov process with degenerate jumps. Journal of Statistical Physics, 178(6),1293–1318.

Saloff, L. (1996). Coste. lectures on finite markov chains, volume 1665 of lectures on probability theory and statistics. Lecture Notes in Math. Springer Verlag, Berlin.

Talagrand, M. (1991). A new isoperimetric inequality and the concentration of measure phenomenon. En Geometric aspects of functional analysis (pp. 94–124). Springer.

Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Publications Mathématiques de l’Institut des Hautes Etudes Scientifiques, 81(1), 73–205.

Wang, F.-Y., y Yuan, C. (2010). Poincaré inequality on the path space of poisson point processes. Journal of Theoretical Probability, 23(3), 824–833.