A Weak Limit Shape Theorem for Planar Isotropic Brownian Flows

Abstract

It has been shown by various authors under different assumptions that the diameter of a bounded non-trivial set γ 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 2-dimensional isotropic Brownian flow Φ with a positive Lyapunov exponent that there exists a non-random set B such that we have for > 0, arbitrary connected γ ⊂⊂ R 2 consisting of at least two different points and arbitrarily large times T thatDefinition 1.2 A function b : R d → R d×d is an isotropic covariance tensor if x → b(x) is C ∞ and all derivatives of any order are bounded, b(0) = E d (the d-dimensional unit matrix), x → b(x) is not constant and b(x) = O * b(Ox)O holds for any x ∈ R d and any orthogonal matrix O.

Asymptotic support theorem for planar isotropic Brownian flows

“…This, together with (17), yields convergence of (16) to 1. Combining (15) and (16) via (14) implies the assertion.…”

“…Moreover we have v(x) ≡ 0. In this paper we will assume b ∈ C ∞ , since we will use results of [15], where smoothness of b has to be assumed. In this case ϕ s,t (·) ∈ C ∞ (R d ) are diffeomorphims.…”

“…By this remark, and since we would like to use results from [15], we always will assume a strictly positive top-Lyapunov exponent. But we conjecture that the results in [15], and hence our main result, are also true for µ 1 = 0. For more details on Lyapunov exponents for random dynamical systems we refer to [1].…”

“…where β L and β N are as in Section 2.1, and · denotes the spectral norm on R 2×2 . According to [15], Lemma 1.6, x is an eigenvector of b(x) to the eigenvalue B L (|x|), and any vector x ⊥ = 0 perpendicular to x is an eigenvector of b(x) to the eigenvalue B N (|x|). Since the matrix A(t, x, y) is symmetric, we have…”

“…In the case of planar periodic stochastic flows (stochastic flows on the torus) Dolgopyat, Kaloshin and Koralov [7] used a new approach based on the so-called stable norm, to identify the precise deterministic linear growth rate of such flows. By this approach, van Bargen [15] identified the precise deterministic growth rate for planar IBFs, which have a strictly positive top-Lyapunov exponent.…”