A Weak Limit Shape Theorem For Planar Isotropic Brownian Flows

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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 ε>0, arbitrary connected γ⊂⊂² consisting of at least two different points and arbitrarily large times T that (1-ε)TB⊂ ∪₀≤ ₜ≤ ₜ∪ₓ∈ᵧ Φ₀,ₜ(x)⊂(1+ε)TB.