Chasing Balls through Martingale Fields

• Published in: The Annals of probability Vol. 30; no. 4; pp. 2046 - 2080
• Main Authors: Scheutzow, Michael, Steinsaltz, David
• Format: Journal Article
• Language: English
• Published: Hayward, CA Institute of Mathematical Statistics 01-10-2002; The Institute of Mathematical Statistics
• Subjects: 60H20; Ballistics; Continuous functions; dichotomy; Ellipticity; Exact sciences and technology; exceptional points; Homeomorphism; Hyperplanes; Integers; Law of large numbers; linear expansion; Martingales; Mathematics; Probability and statistics; Probability theory and stochastic processes; Random variables; Sciences and techniques of general use; Stochastic analysis; Stochastic flows; Stochastic processes; Stopping distances; submartingale inequalities; supermartingale inequalities; Brownian motion; Speed; Stochastic analysis; Filtration; Ellipticity condition; Martingale; One dimensional model; Lipschitz constant; Stochastic process; Stochastic flow; Flow; Cauchy Schwartz inequality
• ISSN: 0091-1798; 2168-894X
• DOI: 10.1214/aop/1039548381

We consider the way sets are dispersed by the action of stochastic flows derived from martingale fields. Under fairly general continuity and ellipticity conditions, the following dichotomy result is shown: any nontrivial connected set χ either contracts to a point under the action of the flow, or its diameter grows linearly in time, with speed at least a positive deterministic constant Λ. The linear growth may further be identified (again, almost surely), with a much stronger behavior, which we call "ball-chasing": if ψ is any path with Lipschitz constant smaller than Λ, the ball of radius ε around ψ(t) contains points of the image of χ for an asymptotically positive fraction of times t. If the ball grows as the logarithm of time, there are individual points in χ whose images eventually remain in the ball.