It is known from the multiplicative ergodic theorem that the norm of the derivative of certain stochastic flows at a previously fixed point grows exponentially fast in time as the flows evolves. We prove that this is also true if one takes the supremum over a bounded set of initial points. We give an explicit bound for the exponential growth rate which is far different from the lower bound coming from the Multiplicative Ergodic Theorem.
"Asymptotic Growth of Spatial Derivatives of Isotropic Flows." Electron. J. Probab. 14 2328 - 2351, 2009. https://doi.org/10.1214/EJP.v14-704