B. Tsirelson

Nonclassical stochastic flows and continuous products

Recent works

Boris Tsirelson,
"Nonclassical stochastic flows and continuous products."
Probability Surveys 1, 173-298 (2004).
Available online (free of charge):
via DOI: 10.1214/154957804100000042
or from Project Euclid site:
or the journal site:
See also arXiv.org/abs/math.PR/0402431/

A survey; 126 pages, bibl. 51 refs.

Contrary to the classical wisdom, processes with independent values (defined properly) are much more diverse than white noises combined with Poisson point processes, and product systems are much more diverse than Fock spaces.

This text is a survey of recent progress in constructing and investigating nonclassical stochastic flows and continuous products of probability spaces and Hilbert spaces.

  1. Introduction
  2. Singularity concentrated in time (toy models).
  3. Singularity concentrated in space (examples).
  4. From convolution semigroups to continuous products of probability spaces.
  5. Stability and sensitivity.
  6. Continuous products: from probability spaces to Hilbert spaces.
  7. Classical part of a continuous product.
  8. Distributed singularity, black noise (according to Le Jan and Raimond).
  9. Unitary flows are classical.
  10. Random sets as degrees of nonclassicality.
  11. Continuous products of measure classes.
  12. Beyond the one-dimensional time.
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