Probability Theory and Related Fields

, Volume 112, Issue 3, pp 373–409

Stochastic calculus with respect to free Brownian motion and analysis on Wigner space

  • Philippe Biane
  • Roland Speicher

DOI: 10.1007/s004400050194

Cite this article as:
Biane, P. & Speicher, R. Probab Theory Relat Fields (1998) 112: 373. doi:10.1007/s004400050194


We define stochastic integrals with respect to free Brownian motion, and show that they satisfy Burkholder-Gundy type inequalities in operator norm. We prove also a version of Itô's predictable representation theorem, as well as product form and functional form of Itô's formula. Finally we develop stochastic analysis on the free Fock space, in analogy with stochastic analysis on the Wiener space.

Mathematics Subject Classification (1991): 60H05 46L50 81S25 

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Philippe Biane
    • 1
  • Roland Speicher
    • 2
  1. 1.CNRS, DMI, Ecole Normale Supérieure, 45, rue d'Ulm, 75005 Paris, France. e-mail: biane@dmi.ens,frFR
  2. 2.Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany. e-mail: roland.Speicher@urz.uni-heidelberg.deDE

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