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We study some finite time transport properties of isotropic Brownian flows.Under a certain nondegeneracy condition on the potential spectral measure, weprove that uniform shrinking or expansion of balls under the flow over somebounded time interval can happen with positive probability. We also provide acontrol theorem for isotropic Brownian flows with drift. Finally, we apply theabove results to show that under the nondegeneracy condition the length of arectifiable curve evolving in an isotropic Brownian flow with strictly negativetop Lyapunov exponent converges to zero as $t\to \infty$ with positiveprobability.

Uniform shrinking and expansion under isotropic Brownian flows