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摘要
It has been shown by various authors under different assumptions that the diameter of a bounded nontrivial set gamma under the action of a stochastic flow grows linearly in time. We show that the asymptotic linear expansion speed if properly defined is deterministic, i.e. we show for a two-dimensional isotropic Brownian flow Phi with a positive Lyapunov exponent that there exists a non-random set B such that we have for epsilon > 0, arbitrary connected gamma subset of subset of R(2) consisting of at least two different points and arbitrarily large times T that
(1 - epsilon)TB subset of boolean OR boolean OR(0 <= t <= Tx is an element of gamma) Phi(0,t)(x) subset of (1 + epsilon)TB.
- 出版日期2011-12
