Asymptotic Growth of Spatial Derivatives of Isotropic Flows

Abstract

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.

A Weak Limit Shape Theorem for Planar Isotropic Brownian Flows

“…Now we turn this into an estimate for f (d t ). Abbreviate ρ t := ρ x (1) x (2) t and consider for K > 0…”

“…There is a constant R > 0 such that for any m ∈ N there is κ (2) m > 0 (not depending on γ, P or r) such that for β > 1 we have P τ R (γ, P ) > κ (2) m βr ≤ κ…”

“…Isotropic Brownian flows (IBFs) are a fairly natural class of stochastic flows and have been studied by various authors in different directions, e.g. [1], [12], [7] and [2] -just to name a few references. [6] and [14] study the evolution of the diameter of a bounded and non-trivial set under the evolution of such a flow giving upper and lower bounds for the linear growth rate.…”

“…(1) t , x(2) t , y(1) t , y(2) t ∈ γ t with |x (1) t − x (2) t | = d t , |x (i) t − y (i) t | = r ( ) 10 : i = 1, 2 and define z (1) :=x (1) t + x (1) t − x (2) t |x (1) t − x (2) t |r , z (2) := x (2) t + x (2) t − x (1) t |x (2) t − x (1) t |r , r (i) 1 :=|x (i) t − z (i) | ∧ |y (i) t − z (i) | =r : i = 1, 2 and r (i) 2 :=|x (i) t+1 − z (i) | ∧ |y (i) t+1 − z (i) | : i = 1, 2 (seeFig. 1for the geometry at time t).…”

Exponential Growth Rate for Derivatives of Stochastic Flows

“…Our bound depends on the box dimension of the set. In the case of IBFs such a result has been obtained in [13] with a different (and more technical) proof. We will be much more general with our set-up but, contrary to [13], will not derive lower bounds for the growth rates.…”