A Weak Limit Shape Theorem For Planar Isotropic Brownian Flows

It has been shown by various authors under different assumptions that the diameter of a bounded non-trivial 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 $2$-dimensional isotropic Brownian flow $\mathrm{\Phi }$ with a positive Lyapunov exponent that there exists a non-random set $\mathcal{B}$ such that we have for $ϵ>0$, arbitrary connected $\gamma \subset \subset {\mathbb{R}}^2$ consisting of at least two different points and arbitrarily large times $T$ that $$(1-ϵ)T\mathcal{B}\subset {\cup }_{0\le t\le T}{\cup }_{x\in \gamma }{\mathrm{\Phi }}_{0,t}(x)\subset (1+ϵ)T\mathcal{B}\mathrm{.}$$