Nonclassical stochastic flows and continuous products

Abstract

Contrary to the classical wisdom, processes with independent values (defined properly) are much more diverse than white noise 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.

A survey, 126 pages. Version 3 (final): former Question 9d4 is solved; 8a1 reformulated. Ref [41] added. For readability, sections are reordered (123456..->142536..). Cosmetic changes, mostly in 1b, 2a, 3d, (4a7) (v3 numbers) and Introduction

References in corpus (11)

• Flows, coalescence and noise
• The Brownian web: Characterization and convergence
• From random sets to continuous tensor products: answers to three questions of W. Arveson
• Unitary Brownian motions are linearizable
• On automorphisms of type II Arveson systems (probabilistic approach)
• Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces
• Noise sensitivity on continuous products: an answer to an old question of J. Feldman
• Splitting: Tanaka's SDE revisited
• Spectral densities describing off-white noises
• On Product Systems arising from Sum Systems
• The automorphism group of the Gaussian measure cannot act pointwise

Cited by in corpus (20)

• On the scaling limits of planar percolation
• Turbulent superfluid as continuous vortex mixture
• Consistent families of Brownian motions and stochastic flows of kernels
• On automorphisms of type II Arveson systems (probabilistic approach)
• Weak convergence of the localized disturbance flow to the coalescing Brownian flow
• Planar aggregation and the coalescing Brownian flow
• KPZ equation limit of sticky Brownian motion
• Stochastic processes in turbulent transport
• Noise as a Boolean algebra of $\sigma$-fields
• Noncommutative continuous Bernoulli shifts
• Percolation, boundary, noise: an experiment
• The Brownian web is a two-dimensional black noise
• Kolmogorov complexity and the geometry of Brownian motion
• Brownian local minima and other random dense countable sets
• Is the Universe Noise-Sensitive?
• A potpourri of results in the general theory of stochastic noises
• Simulation Example of a Black Noise
• Some applications of the Tsirelson spectral measures to noise filtering problems
• Product systems arising from Lévy processe
• Itô-Wiener expansion for functionals of the Arratia's flow $n$-point motion