B. Tsirelson

Scaling limit, noise, stability

Recent works

Boris Tsirelson,
"Scaling limit, noise, stability."
In: Lecture Notes in Mathematics 1840, Springer, pp. 1-106 (2004).
Available online (free of charge) from e-print archive (USA):
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A course of St. Flour summer school; 108 pages, 42 (small) figures, bibl. 27 refs.

Linear functions of many independent random variables lead to classical noises (white, Poisson, and their combinations) in the scaling limit. Some singular stochastic flows and some models of oriented percolation involve very nonlinear functions and lead to nonclassical noises. Two examples are examined, Warren's `noise made by a Poisson snake' and the author's `Brownian web as a black noise'. Classical noises are stable, nonclassical are not. A new framework for the scaling limit is proposed. Old and new results are presented about noises, stability, and spectral measures.

  1. Introduction
  2. A first look.
  3. Abstract nonsense of the scaling limit.
  4. Scaling limit and independence.
  5. Example: The noise made by a Poisson snake.
  6. Stability.
  7. Generalizing Wiener chaos.
  8. Example: The Brownian web as a black noise.
  9. Miscellany.
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