Scaling Limit, Noise, Stability

Abstract

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.

A summer school course, 108 pages, 42 small figs

References in corpus (8)

• Flows, coalescence and noise
• The Brownian Web
• Unitary Brownian motions are linearizable
• Splitting: Tanaka's SDE revisited
• Noise sensitivity on continuous products: an answer to an old question of J. Feldman
• Fourier-Walsh coefficients for a coalescing flow (discrete time)
• The noise of a Brownian sticky flow is black
• Scaling limit of Fourier-Walsh coefficients (a framework)

Cited by in corpus (14)

• Flows, coalescence and noise
• The Brownian web: Characterization and convergence
• On the scaling limits of planar percolation
• Fock factorizations, and decompositions of the $L^2$ spaces over general Levy processes
• Geometric hydrodynamics and infinite-dimensional Newton's equations
• Balls-in-boxes duality for coalescing random walks and coalescing Brownian motions
• Noise as a Boolean algebra of $\sigma$-fields
• The noise of a Brownian sticky flow is black
• Oded Schramm's contributions to noise sensitivity
• Marking (1,2) Points of the Brownian Web and Applications
• Is the Universe Noise-Sensitive?
• Simulation Example of a Black Noise
• Some applications of the Tsirelson spectral measures to noise filtering problems
• A potpourri of results in the general theory of stochastic noises