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 $\Phi$ with a positive Lyapunov exponent that there exists a non-random set $\mathcal{B}$ such that we have for $\epsilon>0$, arbitrary connected $\gamma\subset\subset\R^2$ consisting of at least two different points and arbitrarily large times $T$ that $$(1-\epsilon)T\mathcal{B}\subset \cup_{0\leq t\leq T}\cup_{x\in\gamma} \Phi_{0,t}(x)\subset(1+\epsilon)T\mathcal{B}.$$