Communications in Mathematical Physics

, Volume 185, Issue 1, pp 129–154

q-Gaussian Processes: Non-commutative and Classical Aspects

  • M. Bożejko
  • B. Kümmerer
  • R. Speicher

DOI: 10.1007/s002200050084

Cite this article as:
Bożejko, M., Kümmerer, B. & Speicher, R. Comm Math Phys (1997) 185: 129. doi:10.1007/s002200050084


We examine, for −1<q<1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) \(\)– where the at fulfill the q-commutation relations \(\) for some covariance function \(\)– equipped with the vacuum expectation state. We show that there is a q-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possesses a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB].

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • M. Bożejko
    • 1
  • B. Kümmerer
    • 2
  • R. Speicher
    • 3
  1. 1.Instytut Matematyczny, Uniwersytet Wrocławski, Plac Grunwaldzki 2/4, 50-384 Wrocław, Poland.¶E-mail: bozejko@math.uni.wroc.plPL
  2. 2.Mathematisches Institut A, Pfaffenwaldring 57, D-70569 Stuttgart, Germany.¶E-mail: DE
  3. 3.Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany.¶E-mail: roland.speicher@urz.uni-heidelberg.deDE