A free stochastic partial differential equation
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 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.

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.

DOI : 10.1214/13-AIHP548
Classification : 46L54, 60H15
Mots clés : 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},
     pages = {1404--1455},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {4},
     year = {2014},
     doi = {10.1214/13-AIHP548},
     mrnumber = {3270000},
     zbl = {06377560},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/13-AIHP548/}
}
TY  - JOUR
AU  - Dabrowski, Yoann
TI  - A free stochastic partial differential equation
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2014
SP  - 1404
EP  - 1455
VL  - 50
IS  - 4
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/13-AIHP548/
DO  - 10.1214/13-AIHP548
LA  - en
ID  - AIHPB_2014__50_4_1404_0
ER  - 
%0 Journal Article
%A Dabrowski, Yoann
%T A free stochastic partial differential equation
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2014
%P 1404-1455
%V 50
%N 4
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/13-AIHP548/
%R 10.1214/13-AIHP548
%G en
%F AIHPB_2014__50_4_1404_0
Dabrowski, Yoann. A free stochastic partial differential equation. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1404-1455. doi : 10.1214/13-AIHP548. http://archive.numdam.org/articles/10.1214/13-AIHP548/

[1] S. Avsec. Strong solidity of the q-Gaussian algebras for all -1<q<1. Available at arXiv:1110.4918.

[2] P. Biane, M. Capitaine and A. Guionnet. Large deviation bounds for matrix Brownian motion. Invent. Math. 152 (2003) 433-459. | MR | Zbl

[3] P. Biane and R. Speicher. Stochastic calculus with respect to free Brownian motion. Probab. Theory Related Fields 112 (1998) 373-409. | MR | Zbl

[4] P. Biane and D. Voiculescu. A free probability analogue of the Wasserstein metric on the trace-state space. Geom. Func. Anal. 11 (2001) 1125-1138. | MR | Zbl

[5] M. Bożejko. Ultracontractivity and strong Sobolev inequality for q-Ornstein-Uhlenbeck semigroup (-1<q<1). Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 (1998) 203-220. | MR | Zbl

[6] M. Bożejko, B. Kummerer and R. Speicher. q-Gaussian processes: Non-commutative and classical aspects. Commun. Math. Phys. 185 (1997) 129-154. | MR | Zbl

[7] M. Bożejko and R. Speicher. An example of a generalized Brownian motion. Commun. Math. Phys. 137 (1991) 519-531. | MR | Zbl

[8] A. M. Cheboratev and F. Fagnola. Sufficient conditions for conservativity of minimal quantum dynamical semigroups. J. Funct. Anal. 153 (1998) 382-404. | MR | Zbl

[9] F. Cipriani and J.-L. Sauvageot. Derivations as square roots of Dirichlet forms. J. Funct. Anal. 201 (2003) 78-120. | MR | Zbl

[10] A. Connes and D. Shlyakhtenko. L 2 -homology for von Neumann algebras. J. Reine Angew. Math. 586 (2005) 125-168. | MR | Zbl

[11] Y. Dabrowski. A note about proving non-Γ under a finite non-microstates free Fisher information Assumption. J. Funct. Anal. 258 (2008) 3662-3674. | MR | Zbl

[12] Y. Dabrowski. A non-commutative path space approach to stationary free stochastic differential equations. Preprint, 2010, available at arXiv:1006.4351. | MR

[13] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press, Cambridge, 1992. | MR

[14] E. B. Davis and J. M. Lindsay. Non-commutative symmetric Markov semigroups. Math. Z. 210 (1992) 379-411. | Zbl

[15] C. Donati-Martin. Stochastic integration with respect to q Brownian motion. Probab. Theory Related Fields 125 (2003) 77-95. | MR | Zbl

[16] K. Dykema and A. Nica. On the Fock representation of the q-commutation relations. J. Reine Angew. Math. 440 (1993) 201-212. | MR | Zbl

[17] F. Fagnola. H-P quantum stochastic differential equations. In Non-Commutativity, Infinite-Dimensionality, and Probability at the Crossroads. The Proceedings of the RIMS Workshop on Infinite Dimensional Analysis and Quantum Probability: Kyoto, Japan, 20-22 November, 2001 51-96. N. Obata, T. Matsui and A. Hora (Eds). World Scientific, River Edge, NJ, 2002. | MR | Zbl

[18] F. Fagnola and S. J. Wills. Mild solutions of quantum stochastic differential equations. Electron. Commun. Probab. 5 (2000) 158-171. | MR | Zbl

[19] A. Guionnet and D. Shlyakhtenko. Free diffusions and matrix models with strictly convex interaction. Geom. Func. Anal. 18 (2007) 1875-1916. | MR | Zbl

[20] T. Kato. Perturbation Theory for Linear Operators, 2nd edition. Springer-Verlag, Berlin, 1980. | Zbl

[21] T. Kotelenez. Stochastic Ordinary and Stochastic Partial Differential Equations: Transition from Microscopic to Macroscopic Equations. Springer, Berlin, 2007. | MR | Zbl

[22] N. V. Krylov. An analytic approach to SPDEs. In Stochastic Partial Differential Equations: Six Perspectives. R. A. Carmona and B. L. Rozovskii (Eds). Mathematical Surveys and Monographs 64. American Mathetical Society, Providence, 1999. | MR | Zbl

[23] N. V. Krylov and B. L. Rozovskii. Stochastic evolution equations (in Russian). Current Problems in Mathematics 14 (1979) 71-147. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow. | MR | Zbl

[24] T. M. Liggett. Continuous Time Markov Processes: An Introduction. American Mathetical Society, Providence, 2010. | MR | Zbl

[25] W. Lück. L 2 -Invariants: Theory and Applications to Geometry and K-Theory. Springer, Berlin, 2002. | Zbl

[26] Z. M. Ma and M. Röckner. Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Universitext. Springer, Berlin, 1992. | Zbl

[27] I. Mineyev and D. Shlyakhtenko. Non-microstate free entropy dimension for groups. Geom. Func. Anal. 15 (2005) 476-490. | MR | Zbl

[28] A. Nou. Non injectivity of the q-deformed von Neumann algebra. Math. Ann. 330 (2004) 17-38. | MR | Zbl

[29] F. Otto and C. Villani. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000) 361-400. | MR | Zbl

[30] E. Pardoux. Sur des équations aux dérivés partielles stochastiques monotones. C. R. Acad. Sci. Paris Sér. A-B 275 (1972) A101-A103. | MR | Zbl

[31] E. Pardoux. Équations aux dérivées partielles stochastiques de type monotone. In Séminaire sur les Équations aux Dérivées Partielles (1974-1975), III Exp. No. 2 1-10. Collège de France, Paris, 1975. | MR | Zbl

[32] J. Peterson. A 1-cohomology characterisation of Property (T) in von Neumann algebras. Pacific J. Math. 243 (2009) 181-199. | MR | Zbl

[33] J. Peterson. L 2 -rigidity in von Neumann algebras. Invent. Math. 175 (2009) 417-433. | MR | Zbl

[34] C. Prévôt and M. Röckner. A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics 1905. Springer, Berlin, 2007. | MR | Zbl

[35] B. L. Rozovskii. Stochastic Evolution Systems. Kluwer Academic, Dordrecht, 1990. | MR

[36] J.-L. Sauvageot. Strong Feller semigroups on C * -algebras. J. Operator Theory 42 (1999) 83-102. | MR | Zbl

[37] D. Shlyakhtenko. Some estimates for non-microstate free entropy dimension with applications to q-semicircular families. Int. Math. Res. Not. 51 (2004) 2757-2772. | MR | Zbl

[38] D. Shlyakhtenko. Remarks on free entropy dimension. In Operator Algebras Abel Symposia, Volume 1 249-257. Springer, Berlin, 2006. | MR | Zbl

[39] D. Shlyakhtenko. Lower estimates on microstate free entropy dimension. Anal. PDE 2 (2009) 119-146. | MR | Zbl

[40] P. Śniady. Gaussian random matrix models for q-deformed Gaussian variables. Comm. Math. Phys. 216 (2001) 515-537. | Zbl

[41] A. S. Ustunel. On the regularity of the solutions of stochastic partial differential equations. In Stochastic Differential Systems Filtering and Control. Lecture Notes in Control and Information Sciences 69 71-75. Springer, Berlin, 1985. | MR | Zbl

[42] D. Voiculescu. The analogs of entropy and of Fisher's information measure in free probability theory, II. Invent. Math. 118 (1994) 411-440. | MR | Zbl

[43] D. Voiculescu. The analogs of entropy and of Fisher's information measure in free probability theory, V: Non commutative Hilbert Transforms. Invent. Math. 132 (1998) 189-227. | MR | Zbl

[44] D. Voiculescu. The analogs of entropy and of Fisher's information measure in free probability theory, VI: Liberation and mutual free information. Adv. Math. 146 (1999) 101-166. | MR | Zbl

[45] D. Voiculescu. Free entropy. Bull. Lond. Math. Soc. 34 (2002) 257-278. | MR | Zbl

[46] D. Voiculescu. A note on cyclic gradients. Indiana Univ. Math. J. 49 (2000) 837-841. | MR | Zbl

[47] J. B. Walsh. An introduction to stochastic partial differential equations. In École d'été de probabilités de Saint-Flour, XIV-1984. Lecture Notes in Math. 1180 265-439. Springer, Berlin, 1986. | MR | Zbl

[48] N. Weaver. Lipschitz algebras and derivations of von Neumann algebras. J. Funct. Anal. 139 (1996) 261-300. | MR | Zbl

[49] D. Zagier. Realizability of a model in infinite statistics. Commun. Math. Phys. 147 (1992) 199-210. | MR | Zbl

Cité par Sources :