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PDE's for the Dyson, Airy and Sine processes

Mark Adler (2005)

Annales de l’institut Fourier

In 1962, Dyson showed that the spectrum of a n × n random Hermitian matrix, whose entries (real and imaginary) diffuse according to n 2 independent Ornstein-Uhlenbeck processes, evolves as n non-colliding Brownian particles held together by a drift term. When n , the largest eigenvalue, with time and space properly rescaled, tends to the so-called Airy process, which is a non-markovian continuous stationary process. Similarly the eigenvalues in the bulk, with a different time and space rescaling, tend...

Poincaré inequalities and hitting times

Patrick Cattiaux, Arnaud Guillin, Pierre André Zitt (2013)

Annales de l'I.H.P. Probabilités et statistiques

Equivalence of the spectral gap, exponential integrability of hitting times and Lyapunov conditions is well known. We give here the correspondence (with quantitative results) for reversible diffusion processes. As a consequence, we generalize results of Bobkov in the one dimensional case on the value of the Poincaré constant for log-concave measures to superlinear potentials. Finally, we study various functional inequalities under different hitting times integrability conditions (polynomial,…)....

Probabilistic properties of a Markov-switching periodic G A R C H process

Billel Aliat, Fayçal Hamdi (2019)

Kybernetika

In this paper, we propose an extension of a periodic G A R C H ( P G A R C H ) model to a Markov-switching periodic G A R C H ( M S - P G A R C H ), and provide some probabilistic properties of this class of models. In particular, we address the question of strictly periodically...

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