A free stochastic partial differential equation

Dabrowski, Yoann

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014), p. 1404-1455 / Harvested from Numdam

Résumé

Nous construisons des solutions stationnaires de certaines équations différentielles stochastiques libres à coefficients opérateurs non-bornés. Comme application, nous montrons l’égalité des dimensions entropiques libres microcanonique et non-microcanonique sous l’hypothèse d’une variable conjuguée Lipschitz pour les générateurs $X_{1},\ldots ,X_{N}$ d’un espace de probabilité non-commutatif inscriptible dans une ultrapuissance $R^{\omega }$ du facteur hyperfini. Cette hypothèse de variable conjuguée Lipschitz inclut le cas de $N$ variables aléatories $q$-Gaussiennes pour de petits $q$ par exemple $|q|N\le 0.13$.

Abstract

We get stationary solutions of a free stochastic partial differential equation. As an application, we prove equality of non-microstate and microstate free entropy dimensions under a Lipschitz like condition on conjugate variables, assuming also the von Neumann algebra $R^{\omega }$ embeddable. This includes an $N$-tuple of $q$-Gaussian random variables e.g. for $|q|N\le 0.13$.

Détail

Publié le : 2014-01-01

MR 3270000 | Zbl 06377560

DOI : https://doi.org/10.1214/13-AIHP548
Classification:  46L54,  60H15

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