A free stochastic partial differential equation

Dabrowski, Yoann

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1404-1455.

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 $X_1,...,X_N$ 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 $\left|q\right|N\le 0.13$.

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 $\left|q\right|N\le 0.13$.

Classification : 46L54, 60H15
Mots clés : free stochastic partial differential equations, lower bounds on microstate free entropy dimension, free probability, $q$-gaussian variables

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