Asymptotic Growth Of Spatial Derivatives Of Isotropic Flows

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.


Introduction
The evolution of the diameter of a bounded set under the evolution of a stochastic flow has been studied since the 1990's (see [5], [6], [11], [17], [12] and the survey article [16] to name just a few references). It is known to be linearly growing in time if the flow has a positive Lyapunov exponent. Of course the considered diameter links to the the supremum of |φ t (x)| ranging over x in a subset of d . In the following we will consider the case where the flow is replaced by its spatial derivative. We emphasize that we consider the asymptotics in time (the spatial asymptotics for a fixed time horizon have been considered in [8] in a very general setting and in [2] in the particular one treated here).
If the flow has a positive top exponent it is known that the growth is at least exponentially fast which is then true even for a singleton (this follows directly from the Multiplicative Ergodic Theorem). We will show in the case of an isotropic Brownian flow (IBF) or an isotropic Ornstein-Uhlenbeck flow (IOUF) that sup x | log Dx t | grows at most linearly in time t where the supremum is taken over x in a bounded subset of d no matter what the top Lyapunov exponent is. This shows that the growth of the norm of the derivative is indeed at most exponentially fast but it also gives some insight into the distance of Dφ t (x) to singularity by bounding sup 0≤t≤T t −1 inf x log Dx t from below in the lim inf sense. This excludes super-exponential decay to singularity which might be of interest especially if the top exponent is negative. Exponential bounds on the growth of spatial derivatives play a role in the proof of Pesin's formula for stochastic flows (see [13]). It has also been conjectured that this should yield a new proof of the fact that the diameter grows at most linearly in time (but there are much simpler proofs known for this -see the references given above). Despite the fact that we can come up with an upper bound for the exponential growth rate we make no claims about its optimality (and we conjecture that our bound is far from optimal).

Definition And Prerequisites
In this section we will recall the definition of an isotropic Brownian Flow (IBF) from [4] and an Isotropic Ornstein-Uhlenbeck Flow (IOUF) from [2]. We will keep the convention of speaking of an IOUF only if its drift c is not equal to zero. (see [2] or [3] for a discusion of this issue).

Definition 1.1 (IBF and IOUF).
Let c > 0 and F (t, x, ω) be an isotropic Brownian field with a C 4 -covariance tensor i.e. F i (·, x), F j (·, y) t = t b i j (x − y) where the function b(·) = b i j (·) : d → d×d is four times continuously differentiable with bounded derivatives up to order four and preserved by rigid motions (see [4] or [7]). We define the semimartingale field V (t, x, ω) := F (t, x, ω) − c x t and an IOUF to be the solution φ = φ s,t (x, ω) of the Kunita-type stochastic differential equation (SDE) Note that the definition of b and [9,Theorem 3.1.3] imply If one puts c = 0 in (1) one gets the definition of an IBF.
We call c the drift of φ and b the covariance tensor of φ. Note that we write x t for φ t (x) = φ 0,t (x) = φ 0,t (x, ω) and x 1 t , . . . , x d t for its components. We will write the spatial derivatives as and Dx t := Dφ t (x) . The same notations are used for y t = φ 0,t ( y, ω) etc.. The assumptions on the smoothness of b ensure that the local characteristics (b, c·) are smooth enough to guarantee the existence of a solution flow to (1) which can be shown to be of class C 3,δ for arbitrary 1 > δ > 0 (see [9,Theorem 3.4.1]). In fact one has to choose a modification to get the mentioned smoothness (which we do without change of notation). IOUFs have been studied in [7] and in [2] and we will recall some facts that we use later.

Lemma 1.2 (some finite-dimensional marginals).
Let φ be an IOUF with drift c and covariance tensor b and let x, y ∈ d . Then we have 1. φ is a Brownian flow (i.e. it has independent increments) and its law is invariant under orthogonal transformations.
for a standard Brownian motion (W t ) t≥0 . Therein B L and B N are the longitudinal and normal correlation functions of b respectively (see [4] or Lemma 1.3). There are constants λ > 0 and σ > 0 such that the following is true.
(a) There is a standard Brownian motion (W t ) t≥0 such that we have a.s. for all t ≥ 0 that |x t − y t | ≤ |x − y|eσ sup 0≤s≤t W s +λt .
(b) We have for each x, y ∈ d , T > 0 and q ≥ 1 that We will use the symbol k as a shorthand for d k=1 (also for multiple summation indeces). . We will use this convention throughout the whole paper. The following lemma gives some insight into the structure of b which is known since [18].

Lemma 1.3 (local properties of b).
We have for i, j, k, l = 1, . . . , d that: x i x j |x|2 : Observe that our definition assumes the rigid-motion-part of b to vanish (which is different from the definition in [10] where one has to assume α = 1 to be consistent with our notation).

The partial derivatives of b at
3. There is 0 <r < 1 and C > 0 such that for x ∈ d with |x| <r we have Proof: [7, Proposition 1.2.2].

Lemma 1.4 (a lemma on real functions).
. Let further f be convex and g be concave. If we have for some t > 0 that f (t) ≥ g(t) and f ′ (t) ≥ g ′ (t) then we have for all s ≥ t that f (s) ≥ g(s).
Proof: easy undergraduate exercise The following result is the main tool that allows for the estimation of suprema of the derivatives. Observe that r + = r ∨ 0 denotes the positive part of r ∈ .
2. There exist Λ ≥ 0, σ > 0, q 0 ≥ 1 andc > 0 such that for each x, y ∈ d , T > 0 and even q ≥ q 0 Proof: A complete proof can be found in [16] as Theorem 5.1. Observe that the change of "q ≥ 1" to "even q ≥ q 0 " does not alter the statement at all, because the assumptions above guarantee the same assumptions for q ≥ 1 with a change only in the value ofc. The fact that we allowc to depend on q does not play any role because the proof in [16] is perfectly valid with q-dependingc. We will only briefly indicate what is involved in it. Choosing ε > 0 and a sufficiently small r 0 > 0 one can cover Ξ with at most e γT (∆+ε) subsets of d of diameter at most e −γT < r 0 . Denote these subsets by Ξ 1 , . . . , Ξ e γT (∆+ε) For fixed T the assumption sup 0≤t≤T sup x∈Ξ 1 T |ψ t (x)| > κ implies one of the following to occur up to time T : one of the Ξ i gets diameter at least one or the center of a Ξ i reaches a distance of κT − 1 from its original position. Hence we have for δ > 0 that The chaining based result [16, Theorem 3.1] now shows that the two-point condition is sufficient to control S 2 and the one-point-condition covers S 1 . In this way one obtains that the sum over T ∈ of the right hand side of (6) is finite for some γ and κ which completes the proof via the Borel-Cantelli-Lemma. The formula for the constant K is obtained via an optimization over γ and κ.

The Main Result
We are now ready to state the main result.

Theorem 2.1 (exponential growth of spatial derivatives).
Let φ be an IOUF or an IBF with b and c as above and let Ξ be a compact subset of d with box dimension ∆ > 0. Then where K := The Λ i and σ i depend only on b and d and will be specified later.
Proof: This follows directly from Theorem 1.5 applied to ψ : [0, ∞) × d × Ω → ; ψ t (x, ω) := log Dx t if we can verify the following lemmas. We will only use ψ in the meaning given above from now on. Note that we choose the matrix norm to be the Frobenius norm ||(a i, j ) 1≤i, j≤d || := ( i, j a 2 i, j ) 1/2 for its computational simplicity although the special choice of a norm is irrelevant because of to their equivalence.

Lemma 2.2 (condition on the one-point motion).
We have for each bounded S ⊂ d that for A and B as given in Theorem 2.1.

Lemma 2.3 (condition on the two-point motion).
We have for each x, y ∈ d , T > 0 and even q ≥ q 0 : for Λ and σ as given in Theorem 2.1 andc :=c 1 +c 2 +c 6 . Thec i are constants that depend on b and d and will be specified later.
The proofs of these lemmas will be given in the next sections. Observe thatc does not enter into the constant K, so we do not need to pay attention to get a small value for it.

Proof Of Lemma 2.2: The One-Point Condition
Before proving Lemma 2.2 we will need to establish some facts on Dx t 2 .
s. for all t > 0 and hence a true martingale.

We have the SDE Dx t
Proof: Lemma 1.2 and Itô's formula imply for Dx t By Lemma 1.2 this is equal to Since we have by Lemma 1.
This together with 2. proves 3. and 1. also follows from this and the next proposition.

Proposition 3.2 (a simple estimate).
We have Proof: The proof is just the combination of the triangle inequality with Schwarz' inequality. We leave the details to the reader. The reason why we state this fact as a proposition of its own is that we will use it again. Now we can turn to the proof of Lemma 2.2. Since we can write M t = W 〈M 〉 t for a standard Brownian Motion (W t ) t≥0 we get with Lemma 3.1 that where the latter means equality in distribution. Therefore we get for any k > 0 that Here we distinguish between two cases to treat (14). If (d −1)β N +β L −2c ≤ 0 then we immediately get (using 1 − Φ(t) ≤ exp(−1/2t 2 ) ) which gives lim sup T →∞ We now only have to exclude the possibility that the modulus of the logarithm in ψ t (x) might become large due to a very small Dx t . Observe, that (13) implies ψ t (x) ≥ 1/2((d−1)β N −β L −2c)t+1/2W 〈M 〉 t . Distinguishing between the signs of (d−1)β N −β L −2c we get with a completely analogous computation lim sup T →∞ . This completes the proof of Lemma 2.2.

General Estimates And Preparation
We now turn to the proof of Lemma 2.3 and start with a sketch of proof since it is rather technical. It consists of four steps.
3. Obtaining estimates of the type required in Lemma 2.3 for X t . This is done for small |x − y| estimating the probability for very fast increase in the beginning and with a Grönwall type argument in the case this increase does not occur. 4. Obtaining the estimate for ψ t (x) − ψ t ( y) as integrated versions of the ones on the X (i j) using the Burkholder-Davies-Gundy inequality.
Observe that we have by Lemma 3.1 To further analyze the latter we first prove the following lemma.

Lemma 4.1 (general estimates for A t andM t ).
WithM t and A t defined as above the following holds.

Introducing the abbreviationb
Using

Since (again by Lemma 1.3) we have
This completes the proof of Lemma 4.1.
This Lemma shows that we have to control terms like . We postpone this until we will have derived the following estimate from Lemma 4.1.

3.
Let r > 0 be fixed. We have for any x, y ∈ d with |x − y| ≥ r, T > 0 and q ≥ 1 that 2πc and Λ 1 and σ 1 as before.
Proof: Once again observe that by the triangle inequality Since we can writeM t = W 〈M〉 t and M t ≤c t a.s. we get using sup 0≤t≤TM that Stirling's formula for the Gamma function implies Thus 1. follows. For the proof of 2. it is sufficient to observe that since for any t ≥ 0 we have t ≤ e t/e and t ≤ 1/4 + t we also get This proofs 2. and 3. follows from this by using |x− y| r ≤ 1. Since we can now control the moments of ψ t (x) − ψ t ( y) provided x and y are not too close to each other we introduce the following stopping time. Letr be chosen according to Lemma 1.3 and r ≤r to be specified later. Remember that we assumedr ≤ 1. We now define for x, y ∈ d with |x − y| ≤ r the stopping time and assume r <r in the following (which ensures τ > 0 a.s.).

Derivation Of Formula H
We now proceed to work on

Proposition 4.3 (SDE for the direction of the derivative ).
We have the SDE Proof: Since by Itô's formula we only have to note that by Lemmas 1.2 and 3.1 which combined with Lemmas 1.2 and 3.1 and put into (24) yields This proves Proposition 4.3.
Of course the latter implies Letting we have to compute the cross variations to apply Itô's formula for powers to (27).

Proposition 4.4 (formula H).
We have for q ≥ q 0 that Proof: There is nothing left to show. Proposition 4.4 will be useful to estimate the expectation in Lemma 2.3 on the event {T ≤ τ} but since this requires some additional preparations we will first consider the reversed case in the following intermezzo.

Treating Small |x − y| And Large T
Since we obviously have from Schwarz' inequality that it seems reasonable to compute some useful estimate for the tails of τ. We will also immediatly specify conditions on r andr. Assume first with Lemma 1.3 thatr <r is small enough to ensure that for any 0 ≤ r ≤r we have

Lemma 4.5 (tails of τ ).
If we assume q ≥ q 0 then we have for T ≤ . Then we may start with (see (3) and (35)) for T ≤ logr |x− y| 16β L q (remember |x − y| < r <r ). Let now 2(d − 1)β N − 2β L − c > 0. In this case we can The proof is complete.
Thus the proof of Lemma 4.6 is complete.

Evaluation Of
Another application of Fubini's theorem now yields that the latter is less or equal to (remember that we chose even q ≥ 3)