A free stochastic partial differential equation
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 4, p. 1404-1455

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 ω embeddable. This includes an N-tuple of q-Gaussian random variables e.g. for |q|N0.13.

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 ω 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|N0.13.

DOI : https://doi.org/10.1214/13-AIHP548
Classification:  46L54,  60H15
Keywords: free stochastic partial differential equations, lower bounds on microstate free entropy dimension, free probability, q-gaussian variables
@article{AIHPB_2014__50_4_1404_0,
     author = {Dabrowski, Yoann},
     title = {A free stochastic partial differential equation},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {4},
     year = {2014},
     pages = {1404-1455},
     doi = {10.1214/13-AIHP548},
     zbl = {06377560},
     mrnumber = {3270000},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_4_1404_0}
}
Dabrowski, Yoann. A free stochastic partial differential equation. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 4, pp. 1404-1455. doi : 10.1214/13-AIHP548. http://www.numdam.org/item/AIHPB_2014__50_4_1404_0/

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