aa r X i v : . [ m a t h . P R ] A ug fractional brownian flows ∗ Sreekar Vadlamani † November 3, 2018
Abstract
We consider stochastic flow on R n driven by fractional Brownian motion with Hurstparameter H ∈ ( , R n as they evolve under the flow.The main result is a bound on the rate of (global) growth in terms of the (local)H¨older norm of the flow. Our main objective is to study the global geometric properties of a manifold embedded inEuclidean space, as it evolves under a stochastic flow of diffeomorphisms driven by a nondiffusive process. This follows on from our previous paper [19] in which we obtained preciseestimates for the rate of growth of the
Lipschitz-Killing curvatures of smooth, ( n − R n , as they evolve under an isotropic Brownian flow.In this paper, however, we turn to the non-Markovian, non-diffusive situation in which theflow is driven by fractional Brownian motions.Although extensive literature is available for stochastic flows driven by standard Brownianmotion (see [5, 13]), very little is known when the driver of the flow is changed to a non-Markovian, non-diffusive process, such as fractional Brownian motion. For instance, someof the very basic results concerning the tangent flow are yet to be unearthed in the casewhen the flow is driven by fractional Brownian motion. Here we intend to target preciselythis aspect of the flow on our way to the main result of this paper.Recall that a fractional Brownian motion { B H ( t ) , t ≥ } with Hurst parameter H ∈ (0 , E [ B H ( s ) B H ( t )] = 12 (cid:0) t H + s H − | t − s | H (cid:1) . (1) ∗ AMS Subject Classifications: Primary 60G99, 60H10, 60J60; Secondary 53A05, 28A75.Keywords and phrases: Stochastic flows, fractional Brownian motion, manifolds † Research supported in part by the US-Israel Binational Science Foundation, grant 2004064. For a detailed exposition on Lipschitz-Killing curvatures, we refer the reader to [1]. H = 1 / B H is the standard Brownian motion, which is a Markov process and also amartingale. However for H = 1 / B H is neither a Markov process, nor a semi-martingale.In order to construct a non-diffusive flows, we start with a collection of independent fractionalBrownian motions, { B Hγ } γ ∈ N , a collection { U γ } γ ∈ N of deterministic vector fields on R n , anddefine, for some fixed but generic set I ⊂ N with | I | < ∞ , where | I | denotes the cardinalityof I , U I ( x, t ) = X γ ∈ I U γ ( x ) B Hγ ( t ) . (2)The flow of diffeomorphisms Φ t : R n → R n , or, equivalently, the stochastic flow driven by afractional Brownian motion, can then be defined pointwise by settingΦ t ( x ) = x + X γ ∈ I “ Z t U γ (Φ s ( x )) dB Hγ ( s )” . (3)Clearly, we shall need to place conditions on the vector fields for the result to give a diffeo-morphism, but, prior to that, we need to make sense of the stochastic integrals here.For H = , the integral can be interpreted in either the Itˆo or a Stratonovich sense. When H = the standard semimartingale arguments cease to work and we have to make achoice of definition. There is a plethora of literature on various ways to define an integral R ba f ( s ) dB H ( s ), where f is random and B H the fractional Brownian motion. See, forinstance, [2, 6, 8, 10, 14, 20].We shall adopt the pathwise definition given by Z¨ahle [20, 21], based on which Nualartand R˘a¸scanu ([16]) proved existence and uniqueness of the solutions of multidimensionalstochastic differential equations of the form X t = X + Z t σ ( s, X s ) dB H ( s ) + Z t b ( s, X s ) ds, for H > . Using this, Decreusefond and Nualart in [9] established the existence of ahomeomorphic stochastic flow driven by fractional Brownian motion, and so our flows arewell defined. We note here that stochastic integrals can also be defined for H < / m -dimensional, ( m < n ) C manifold embedded in R n , and consider itsimage under Φ, setting M t = Φ t ( M ) = { x ∈ R n : x = Φ t ( y ) for some y ∈ M } . Our interest is how M t behaves as a function of t .2lthough in [19] we were able to obtain information on all the Lipschitz-Killing curvatures of M t , in the current, non-diffusion scenario everything is much harder, and so we shall sufficeby studying only the size of M t , as measured through its m -dimensional Hausdorff measure, H m ( M t ), which basically measures the m -dimensional Lebesgue measure of the set M t . Ourmain result is Theorem 3.5, however one can see the main flavour of the result already fora flow driven by a single fractional Brownian motion. In this case we have Theorem 1.1
In the notation above, assuming that | I | = 1 in (3) , and under conditions(A1)–(A3) of Section 2 on the vector field U , for every β < H and H > there existconstants c and C , such that sup t ∈ [0 ,T ] H m ( M t ) ≤ c H m ( M ) 2 C T k B H k /ββ,T , where k B H k β,T is the β -H¨older norm of B H (cf. (10) ). It is not hard to see that the H¨older norm k B H k /ββ,T grows no faster than O ( T ǫ ) for any ǫ >
0, so that the overall rate of growth of H m ( M t ) given by Theorem 1.1 is O (2 CT ǫ ).One should hope for something that was smaller, and H -dependent, but current techniquesfail to establish this.Similarly, recent results of Baudoin and Coutin [4] seem to indicate that correct growthrate should be O (2 CT H ). These results, however, are based on the rough path approach of[3, 15]. While Hairer and Ohashi in [12] have proved existence of a stationary solution of(3), under conditions on the vector fields U γ and assuming that for | I | < ∞ , an approachvia rough paths also seems unable to reach a better growth rate.Our proof of Theorem 1.1 and the more general Theorem 3.5 will be based on the approachof Hu and Nualart [17], who obtained growth estimates on the solution of (3). The detailsfollow in the remaining two sections.In Section 2, apart from being more formal about setting up notation, we shall recall somebasic formulae from the fractional calculus required for our main analysis. Estimates on thetangent flow and the flow itself, together with the proof of the main results, will form thebulk of Section 3. We start by listing some of the basic formulae required from the deterministic fractionalcalculus, and the fractional spaces associated with them. (See [20, 21] for a complete accountof fractional calculus.)For a, b ∈ R , a < b , let L p ( a, b ), p ≥
1, be the space of Lebesgue measurable functions f : [ a, b ] → R with k f k L p ( a,b ) < ∞ , where k f k L p ( a,b ) = (cid:26) ( R ba | f ( x ) | p dx ) p , if 1 ≤ p < ∞ ess sup | f ( x ) | : x ∈ [ a, b ] , if p = ∞ . f ∈ L ( a, b ) of order α > I αa + f ( x ) = 1Γ( α ) Z xa ( x − y ) α − f ( y ) dy, for almost all x ∈ ( a, b ), where Γ( α ) is the standard Euler function. Similarly, the rightsided fractional integral is defined, for almost all x ∈ ( a, b ), as I αb − f ( x ) = ( − − α Γ( α ) Z bx ( y − x ) α − f ( y ) dy, where ( − − α = e − iπα . If we consider the fractional integral I αa + (or I αb − ) as an operatorwith domain L p ( a, b ), then its range is denoted by I αa + ( L p ( a, b )) (or I αb − ( L p ( a, b ))). Clearly,for α = 1, I αa + is the standard left integral operator, and a simple calculation yields thatlim α → ( I αa + f )( x ) = f ( x − ) = lim ε ↓ f ( x − ε ), for each x ∈ ( a, b ). An immediate consequenceof the definition of the fractional integral is that I αa + ( I βa + f ) = I α + βa + f, (4)for all α, β >
0. With some obvious variations where needed, all hold also for right sidedfractional integrals; viz. ( I b − f )( x ) = ( − Z bx f ( y ) dy, lim α → ( I αb − f )( x ) = f ( x +) = lim ε ↓ f ( x + ε ) ,I αb − ( I βb − f ) = I α + βb − f, ∀ α, β > f in I αa + ( L p ( a, b )) there corresponds a φ ∈ L p ( a, b ), such that I αa + φ = f . This φ is uniquein L p ( a, b ) and agrees almost everywhere with the fractional derivative, known as the leftsided Riemann-Liouville or Weyl derivative, of α th -order and defined as D αa + f ( x ) = (cid:16) − α ) ddx Z xa f ( y )( x − y ) α dy (cid:17) ( a,b ) ( x )= 1Γ(1 − α ) (cid:16) f ( x )( x − a ) α + α Z xa f ( x ) − f ( y )( x − y ) α dy (cid:17) ( a,b ) ( x ) . (5)Equivalently, we can write D αa + f = D ( I − αa + f ) , where D is the standard derivative operator.Similarly, we can define the right sided Weyl derivative as D αb − f = D ( I − αb − f ), for which D αb − f ( x ) = (cid:16) ( − α − Γ(1 − α ) ddx Z bx f ( y )( y − x ) α dy (cid:17) ( a,b ) ( x )= ( − α Γ(1 − α ) (cid:16) f ( x )( b − x ) α + α Z bx f ( x ) − f ( y )( y − x ) α dy (cid:17) ( a,b ) ( x ) . (6)4s in the case of the integral operators, there is an analogue of the composition formula,given, for all α, β >
0, by D αa + ( D βa + f ) = D α + βa + f. (7)A similar formula also holds for the right sided derivatives, and is given by, D αb − ( D βb − f ) = D α + βb − f, (8)as long as all the fractional derivatives are well defined.We note that the linear spaces I αa + ( L p ( a, b )), for various choices of α and p , are Banachspaces equipped with the norms k f k I αa + ( L p ( a,b )) = k f k L p ( a,b ) + k D αa + f k L p ( a,b ) , and a similar norm is defined on the space I αb − ( L p ( a, b )).Let f ( a +) = lim ε ↓ f ( a + ε ) , and g ( b − ) = lim ε ↓ f ( b − ε ), whenever the limit exists and isfinite, and define f a + ( x ) = ( f ( x ) − f ( a +))1 ( a,b ) ( x ) ,g b − ( x ) = ( g ( x ) − g ( b − ))1 ( a,b ) ( x ) . Using the methods of fractional calculus (see [20]), an extension of the Stieltjes integral,called the generalized Stieltjes integral , of f with respect to g can be defined as Z ba f ( x ) dg ( x ) = ( − α Z ba D αa + f ( x ) D − αb − g b − ( x ) dx, (9)where f a + ∈ I αa + ( L p ( a, b )) and g b − ∈ I − αb − ( L q ( a, b )) for some p, q ≥ , /p + 1 /q ≤ , ≤ α ≤ , and αp < C λ ( a, b ; R d ), the space of λ -H¨older continuous functions, with λ ∈ (0 , R d valued functions for some fixed d ∈ N , the set of natural numbers, equippedwith the norm given by k f k λ,a,b := sup a ≤ c ≤ d ≤ b k f ( d ) − f ( c ) k | d − c | λ , (10)where k · k is the usual Euclidean norm in the appropriate dimension. (When a = 0, weshall write k f k λ,b for k f k λ, ,b .)In [20], Z¨ahle proved that the conditions of the definition (9) are met if f ∈ C λ (0 , T ; R )and g ∈ C µ (0 , T ; R ) for λ + µ >
1, in which case the integral defined in (9) coincides withthe Riemann-Stieltjes integral. Now we state the following well known result concerning theH¨older coefficient and exponent of fractional Brownian motion with Hurst parameter H . Lemma 2.1
For { B H ( t ) : t ∈ [0 , T ] } , a fractional Brownian motion with Hurst parameter H ∈ (0 , , there exists, for each < ε < H and T > , a positive random variable η ε,T ,such that E ( | η ε,T | p ) < ∞ for all p ∈ [1 , ∞ ) and, for all s, t ∈ [0 , T ] , | B H ( t ) − B H ( s ) | ≤ η ε,T | t − s | H − ε a.s.,where η ε,T = C H,ε T H − ε ξ T , with the L q (Ω) norm of ξ T bounded by c ε,q T ε for q ≥ ε . α < .Define W − α, ∞ T (0 , T ; R ) to be the space of measurable functions g : [0 , T ] → R , endowedwith and finite under the norm k g k − α, ∞ ,T := sup
0. Moreover, if g ∈ W − α, ∞ T (0 , T ; R ), then g | (0 ,t ) ∈ I − αt − ( L ∞ (0 , t )) for all t ∈ (0 , T ).Recalling now the vector fields introduced in (3), the time has come to demand a set ofregularity assumptions. Assume that there exist constants M γ , M (1) γ and M (2) γ for all γ ∈ N such that:( A | U iγ ( x ) | ≤ M γ , ∀ x ∈ R n and γ ∈ N , where U iγ denotes the i -th component of U γ .( A | U iγ ( x ) − U iγ ( y ) | ≤ M (1) γ k x − y k , ∀ x, y ∈ R n and γ ∈ N , where k · k denotes thestandard Euclidean norm in the appropriate dimension.( A | W iγ, j ( x ) − W iγ, j ( y ) | ≤ M (2) γ k x − y k , ∀ x, y ∈ R n and γ ∈ N , where W γ ( x ) denotesthe spatial derivative of U γ ( x ), and W iγ, j ( · ) denotes the ( i, j )-th element of the matrix W γ ( · ).( A M (1) = P α ∈ N M (1) α < ∞ , M (2) = P α ∈ N M (2) α < ∞ , and M (3) = P α ∈ N M (3) α < ∞ .Under conditions ( A − ( A C − α (0 , T ; R n ) is proven in [16] for | I | < ∞ .In fact, the existence and uniqueness of the solution can be proven under far weaker condi-tions, but without necessarily giving a solution which provids a diffeomorphism in R n (cf.[9] for details). Properties of the solution of the flow equation are also obtained in [16], andimproved on in [17]. We shall now adopt and adapt the approach developed in [17] to derive some estimates onsome of the basic geometric characteristics of the flow (3).6ith M , as usual, a C , m -dimensional manifold embedded in R n , we write T x M for itstangent space at x . Let v ∈ T x M . Then its push-forward under the flow Φ t is denoted by v t = D Φ t ( x ) v, where D Φ t ( x ) = ( ∂ Φ it ( x ) ∂x j ) ij denotes the matrix of spatial derivatives of the flow Φ t ( x ), and v t ∈ T x t M t . From now on we shall write x t for Φ t ( x ).We now prove the following technical result, which will form the basis for much of thesubsequent analysis. Theorem 3.1
Under assumptions ( A − ( A , and for α = 1 − H + δ , β = H − ε , suchthat (1 − H ) < α < / and δ > ε , there exist a constant c and a random variable C T , suchthat sup r ∈ [0 ,T ] k v r k ≤ sup r ∈ [0 ,T ] k v r k ≤ c C T T , where k v r k and k v r k denote the l and l norms, respectively, of the vector v r as an elementin R n . The random variable C T depends on α , β , n , I , and {k B Hγ k β,T , M γ , M (1) γ , M (2) γ } γ ∈ I .Furthermore, E [ C T ] β ≤ C · E [ k B H k β,T ] , where the constant C depends only on α , β , n , | I | and { M γ , M (1) γ , M (2) γ } γ ∈ I . Remark 3.2
For a better understanding of the results of Theorem 3.1, we note that for thecase | I | = 1 , this result simplifies to sup r ∈ [0 ,T ] k v r k ≤ c C T k B H k /ββ,T , for some constants c and C , dependent only on the various uniform bounds and the Lipschitzcoefficients corresponding to the vector field. Remark 3.3
The results listed in this section hold true for any I ⊂ N as long as thecardinality of the set satisfies | I | < ∞ . However, extensions of these results to the case I = N , though possible, require unnatural conditions on the summability of the constantsappearing in Assumptions ( A − ( A . For instance, extending Lemma 3.1 to the case I = N would require P γ ∈ N M (1) γ k B Hγ k β,T P γ ∈ N M (2) γ k B Hγ k β,T < ∞ . This, in turn would be implied by P γ ∈ N M (1) γ /M (2) γ < ∞ , which does not seem to have aclear meaning in terms of the vector fields U γ . , T ] into smaller units ofsize ∆, on which reasonable estimates of k v r k are possible, and then to glue the intervalstogether to obtain the required result. However, in the process, we shall need to derive anestimate on the flow, presented in the following lemma, the proof of which relies on someresults of [17]. Lemma 3.1
Let M , M (1) be constants as defined in Assumptions ( A − ( A , and ≤ s ≤ t ≤ T be such that ( t − s ) − β > n α (2 α + β − − α )(1 − α )( α + β − α )Γ(1 − α ) X γ ∈ I M (1) γ k B Hγ k β,T , where α = 1 − H + δ , β = H − ε , such that (1 − H ) < α < / and δ > ε . Then for x t defined in (3) there exists a positive random variable K ∗ s,t such that Z ts k x t − x r k ( t − r ) α dr ≤ K ∗ s,t ( t − s ) β − α . (13) Furthermore, K ∗ s,t can be bounded above by another random variable, independent of s and t , with finite moments of order greater than , as long as ( t − s ) is chosen sufficiently small. Remark 3.4
Note that under the aforementioned conditions concerning α and β , we have α + β > , and β > α . Proof:
Writing U iγ ( · ) for the i -th component of the vector U γ ( · ) and choosing { e i } ni =1 asthe canonical basis of R n , we have h ( x t − x s ) , e i i = X γ ∈ I Z ts U iγ ( x r ) dB Hγ ( r ) , which is true by linearity of the operation, and where h· , ·i denotes the standard Euclideaninner product. Hence for α ∈ (1 − H, ), using (9), we obtain |h ( x t − x s ) , e i i| = (cid:12)(cid:12)(cid:12) X γ ∈ I Z ts U iγ ( x r ) dB Hγ ( r ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) X γ ∈ I Z ts D αs + U iγ ( x r ) D − αt − B Hγ,t − ( r ) dr (cid:12)(cid:12)(cid:12) ≤ X γ ∈ I Z ts | D αs + U iγ ( x r ) | · | D − αt − B Hγ,t − ( r ) | dr To obtain a bound on the second term in the integrand, choose β < H , such that α + β > | D − αt − B Hγ,t − ( r ) | = (cid:12)(cid:12)(cid:12) ( − − α Γ( α ) (cid:16) B Hγ ( t ) − B Hγ ( r )( t − r ) − α + α Z tr B Hγ ( u ) − B Hγ ( r )( u − r ) − α du (cid:17)(cid:12)(cid:12)(cid:12) ≤ α ) (cid:16) | B Hγ ( t ) − B Hγ ( r ) || t − r | − α + α Z tr | B Hγ ( u ) − B Hγ ( r ) | ( u − r ) − α du (cid:17) = 1Γ( α ) (cid:16) | B Hγ ( t ) − B Hγ ( r ) | ( t − r ) β ( t − r ) β ( t − r ) − α + α Z tr | B Hγ ( u ) − B Hγ ( r ) | ( u − r ) β ( u − r ) α + β − du (cid:17) ≤ α ) (cid:16) k B Hγ k β,T ( t − r ) α + β − + α k B Hγ k β,T ( t − r ) α + β − α + β − (cid:17) = k ( α, β ) k B Hγ k β,T ( t − r ) α + β − , (14)where k ( α, β ) = (2 α + β − α + β − α ) .To bound the first term we use (5) and assumptions ( A − ( A
2) to see that | D αs + U iγ ( x r ) | = 1Γ(1 − α ) (cid:12)(cid:12)(cid:12) U iγ ( x r )( r − s ) α + α Z rs ( U iγ ( x r ) − U iγ ( x θ ))( r − θ ) α dθ (cid:12)(cid:12)(cid:12) ≤ − α ) (cid:16) | U iγ ( x r ) | ( r − s ) α + α Z rs | U iγ ( x r ) − U iγ ( x θ ) | ( r − θ ) α dθ (cid:17) ≤ c α (cid:16) M γ ( r − s ) α + α Z rs M (1) γ k x r − x θ k ( r − θ ) α dθ (cid:17) ≤ c α (cid:16) M γ ( r − s ) − α + M (1) γ,α k x k s,r, − α ( r − s ) − α (cid:17) , (15)where c α = Γ(1 − α ) − , M (1) γ,α = αM (1) γ (1 − α ) and k x k s,r, − α is the H¨older norm as defined in (10).Therefore, combining the above two estimates , we find |h ( x t − x s ) , e i i| ≤ c α k ( α, β ) X γ ∈ I k B Hγ k β,T Z ts (cid:16) M γ ( r − s ) − α ( t − r ) α + β − + M (1) γ,α k x k s,r, − α ( r − s ) − α ( t − r ) α + β − (cid:17) dr ≤ c α k ( α, β ) X γ ∈ I k B Hγ k β,T ( t − s ) α + β − Z ts (cid:16) M γ ( r − s ) − α + M (1) γ,α k x k s,r, − α ( r − s ) − α (cid:17) dr ≤ c α k ( α, β ) X γ ∈ I k B Hγ k β,T (cid:16) M γ ( t − s ) β (1 − α ) − + M (1) γ,α k x k s,t, − α ( t − s ) − α + β (2 − α ) − (cid:17) . M α = (1 − α ) − X γ ∈ I M γ k B Hγ k β,T , (16)and ˜ M (1) α = (2 − α ) − X γ ∈ I M (1) γ,α k B Hγ k β,T . (17)Then k ( x t − x s ) k = n X i =1 |h ( x t − x s ) , e i i|≤ c α nk ( α, β ) (cid:16) M α ( t − s ) β + ˜ M (1) α k x k s,t, − α ( t − s ) − α + β (cid:17) . Equivalently, k ( x t − x s ) k ( t − s ) − α ≤ c α nk ( α, β ) (cid:16) M α ( t − s ) α + β − + ˜ M (1) α k x k s,t, − α ( t − s ) β (cid:17) . (18)(Recall that α + β > k · k is bounded above by k · k , we have k x k s,t, − α = sup s ≤ u ≤ v ≤ t k ( x v − x u ) k ( v − u ) − α ≤ sup s ≤ u ≤ v ≤ t k ( x v − x u ) k ( v − u ) − α ≤ sup s ≤ u ≤ v ≤ t c α nk ( α, β ) (cid:16) M α ( v − u ) α + β − + ˜ M (1) α k x k u,v, − α ( v − u ) β (cid:17) ≤ c α nk ( α, β ) (cid:16) M α ( t − s ) α + β − + ˜ M (1) α k x k s,t, − α ( t − s ) β (cid:17) . (19)Now choosing s, t such that ( t − s ) − β > c α nk ( α, β ) ˜ M (1) α , (20)(19) can be rewritten as k x k s,t, − α ≤ c α nk ( α, β ) M α ( t − s ) α + β − − c α nk ( α, β ) ˜ M (1) α ( t − s ) β = K s,t ( t − s ) α + β − , (21)10here K s,t = c α nk ( α,β ) M α − c α nk ( α,β ) ˜ M (1) α ( t − s ) β .Therefore, Z ts k x t − x r k ( t − r ) α dr = Z ts k x t − x r k ( t − r ) − α ( t − r ) − α dr ≤ k x k s,t, − α Z ts ( t − r ) − α dr ≤ K s,t ( t − s ) β − α (1 − α )= K ∗ s,t ( t − s ) β − α , where K ∗ s,t = K s,t (1 − α ) , thus establishing (13). The final claim, that K ∗ s,t can be bounded bya random variable independent of s and t , will be proven later. Proof of Theorem 3.1:
Taking the space derivative of (3), the existence of which isensured by Theorem 3.2 in [17], we have D Φ t ( x ) = I + X γ ∈ I Z t W γ (Φ s ( x )) D Φ s ( x ) dB Hγ ( s ) , where the matrix W γ ( · ) = ( W iγ,j ( · )) i,j denotes the spatial derivative of the vector field U .Now using the definition of the pushforward of a vector, we can write the evolution equationof the tangent vector as follows v t = v + X γ ∈ I Z t W γ ( x s ) v s dB Hγ ( s ) . Recall that k v t k = P ni =1 |h v t , e i i| , where h· , ·i is the standard Euclidean inner product, and { e i } ni =1 denotes the canonical basis of R n . Since, h v t , e i i = x + X γ ∈ I Z t h W γ ( x r ) v r , e i i dB Hγ ( r ) , we have |h v t , e i i − h v s , e i i| = (cid:12)(cid:12)(cid:12) X γ ∈ I Z ts h W γ ( x r ) v r , e i i dB Hγ ( r ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) X γ ∈ I Z ts D αs + h W γ ( x r ) v r , e i i D − αt − B Hγ,t − ( r ) dr (cid:12)(cid:12)(cid:12) ≤ X γ ∈ I Z ts | D αs + h W γ ( x r ) v r , e i i| · | D − αt − B Hγ,t − ( r ) | dr. s and t , but we are interested in pairs for which( t − s ) is sufficiently small. To this end, note first that from (14) we can bound the secondintegrand by | D − αt − B Hγ,t − ( r ) | ≤ k ( α, β ) k B Hγ k β,T ( t − r ) α + β − . Now using (5) and Assumptions (A2)–(A4), the first integrand can be bounded by | D αs + h W γ ( x r ) v r , e i i| ≤ − α ) h |h W γ ( x r ) v r , e i i| ( r − s ) α + α Z rs |h W γ ( x r ) v r , e i i − h W γ ( x θ ) v θ , e i i| ( r − θ ) α dθ i ≤ − α ) h P nj =1 | W iγ,j ( x r ) h v r , e j i| ( r − s ) α + α Z rs |h W γ ( x r ) v r , e i i − h W γ ( x θ ) v θ , e i i| ( r − θ ) α dθ i ≤ − α ) h M (1) γ n X j =1 |h v r , e j i| ( r − s ) α + α Z rs |h W γ ( x r ) v r , e i i − h W γ ( x θ ) v r , e i i| ( r − θ ) α dθ + α Z rs |h W γ ( x θ ) v r , e i i − h W γ ( x θ ) v θ , e i i| ( r − θ ) α dθ i = 1Γ(1 − α ) h M (1) γ n X j =1 |h v r , e j i| ( r − s ) α + α Z rs | P nj =1 ( W iγ,j ( x r ) h v r , e j i − W iγ,j ( x θ ) h v r , e j i ) | ( r − θ ) α dθ + α Z rs | P nj =1 ( W iγ,j ( x θ ) h v r , e j i − W iγ,j ( x θ ) h v θ , e i i ) | ( r − θ ) α dθ i ≤ − α ) h M (1) γ n X j =1 |h v r , e j i| ( r − s ) α + α Z rs P nj =1 | W iγ,j ( x r ) − W iγ,j ( x θ ) | · |h v r , e j i| ( r − θ ) α dθ + α Z rs P nj =1 | W iγ,j ( x θ ) | · |h v r , e j i − h v θ , e i i| ( r − θ ) α dθ i ≤ − α ) h M (1) γ n X j =1 |h v r , e j i| ( r − s ) α + αM (2) γ n X j =1 |h v r , e j i| Z rs k x r − x θ k ( r − θ ) α dθ + αM (1) γ n X j =1 Z rs |h v r , e j i − h v θ , e i i| ( r − θ ) α dθ i . Now using the result proven in Lemma 3.1, for r such that s < r < t , with ( t − s ) satisfying(20), we have Z rs k x r − x θ k ( r − θ ) α dθ ≤ K ∗ s,r ( r − s ) β − α . | D αs + h W γ ( x r ) v r , e i i| ≤ n X j =1 h |h v r , e j i| ( r − s ) α (cid:16) M (1) γ + αM (2) γ K ∗ s,r ( r − s ) β Γ(1 − α ) (cid:17) + αM (1) γ Γ(1 − α ) Z rs |h v r , e j i − h v θ , e j i| ( r − θ ) α dθ i = n X j =1 h a γ,s,r, |h v r , e j i| ( r − s ) α + αM (1) γ Γ(1 − α ) Z rs |h v r , e j i − h v θ , e j i| ( r − θ ) α dθ i ≤ n X j =1 h a γ,s,r, |h v r , e j i| ( r − s ) − α + b γ, kh v · , e j ik s,t,β ( r − s ) β − α i , where a γ,s,r, = M (1) γ + αM (2) γ K ∗ s,r ( r − s ) β Γ(1 − α ) , (22)and b γ, = αM (1) γ ( β − α )Γ(1 − α ) . (23)Note that a γ,s,r, ≤ a γ,s,t, , for s ≤ r ≤ t .Writing a s,r, = P γ ∈ I a γ,s,r, k B Hγ k β,T and b = P γ ∈ I b γ, k B Hγ k β,T , and using the above13stimates for the integrands, together with (14) and Remark 3.4, we have |h ( v t − v s ) , e i i| ≤ k ( α, β ) Z ts n X j =1 (cid:16) a s,r, |h v r , e j i| ( r − s ) − α ( t − r ) α + β − + b kh v · , e j ik s,t,β ( r − s ) β − α ( t − r ) α + β − (cid:17) dr ≤ k ( α, β )( t − s ) α + β − Z ts n X j =1 (cid:16) a s,r, |h v r , e j i| ( r − s ) − α + b kh v · , e j ik s,t,β ( r − s ) β − α (cid:17) dr ≤ k ( α, β )( t − s ) α + β − Z ts n X j =1 (cid:16) a s,r, kh v · , e j ik s,t, ∞ ( r − s ) − α + b kh v · , e j ik s,t,β ( r − s ) β − α (cid:17) dr ≤ k ( α, β )( t − s ) α + β − n X j =1 (cid:16) a s,t, kh v · , e j ik s,t, ∞ ( t − s ) − α − α + b kh v · , e j ik s,t,β ( t − s ) β − α β − α (cid:17) = k ( α, β ) n X j =1 (cid:16) a s,t, kh v · , e j ik s,t, ∞ ( t − s ) β + b kh v · , e j ik s,t,β ( t − s ) β (cid:17) , where a s,t, = a s,t, (1 − α ) − and b = b (1 − α + β ) − .Therefore, kh v · , e i ik s,t,β = sup s ≤ r ≤ θ ≤ t |h ( v θ − v r ) , e i i| ( θ − r ) β ≤ k ( α, β ) n X j =1 sup s ≤ r ≤ θ ≤ t (cid:16) a r,θ, kh v · , e j ik r,θ, ∞ + b kh v · , e j ik r,θ,β ( θ − r ) β (cid:17) ≤ k ( α, β ) n X j =1 (cid:16) a s,t, kh v · , e j ik s,t, ∞ + b kh v · , e j ik s,t,β ( t − s ) β (cid:17) . As a consequence of the above estimate we have n X i =1 kh v · , e i ik s,t,β ≤ n k ( α, β ) n X j =1 (cid:16) a s,t, kh v · , e j ik s,t, ∞ + b kh v · , e j ik s,t,β ( t − s ) β (cid:17) . (24)14or further analysis we shall require that( t − s ) − β > nk ( α, β ) b . (25)Thereby, for ( t − s ) satisfying conditions (20) and (25), we can rewrite (24) as n X i =1 kh v · , e i ik s,t,β ≤ n k ( α, β ) a s,t, n X i =1 kh v · , e i ik s,t, ∞ (1 − n k ( α, β ) b ( t − s ) β ) . Hence, n X i =1 |h v t , e i i| ≤ n X i =1 (cid:16) |h v s , e i i| + |h v t , e i i − h v s , e i i| (cid:17) ≤ n X i =1 (cid:16) |h v s , e i i| + kh v · , e i ik s,t,β ( t − s ) β (cid:17) ≤ n X i =1 (cid:16) |h v s , e i i| + n k ( α, β ) a s,t, kh v · , e i ik s,t, ∞ ( t − s ) β (1 − n k ( α, β ) b ( t − s ) β ) (cid:17) Clearly, for any r ∈ [ s, t ] we have n X i =1 |h v r , e i i| ≤ n X i =1 (cid:16) |h v s , e i i| + n k ( α, β ) a s,r, kh v · , e i ik s,r, ∞ ( r − s ) β (1 − n k ( α, β ) b ( r − s ) β ) (cid:17) . Now using the fact that s < r < t , so that kh v · , e i ik s,r, ∞ ≤ kh v · , e i ik s,t, ∞ and a s,r, ≤ a s,t, ,we have n X i =1 kh v · , e i ik s,t, ∞ ≤ n X i =1 (cid:16) |h v s , e i i| + n k ( α, β ) a s,t, kh v · , e i ik s,t, ∞ ( t − s ) β (1 − n k ( α, β ) b ( t − s ) β ) (cid:17) . (26)Finally, we shall require ( t − s ) to satisfy( t − s ) − β > n k ( α, β ) [ a s,t, + b ] , (27)to allow us to rewrite (26) as n X i =1 kh v · , e i ik s,t, ∞ h − n k ( α, β ) a s,t, ( t − s ) β (1 − n k ( α, β ) b ( t − s ) β ) i ≤ n X i =1 |h v s , e i i| . We shall note that for ( t − s ) sufficiently small, the inequality (27) does hold true, as a s,t, is a decreasing function of ( t − s ). 15his, in turn implies, n X i =1 sup ≤ r ≤ t |h v r , e i i| = n X i =1 max { sup ≤ r ≤ s |h v r , e i i| , kh v · , e i ik s,t, ∞ }≤ n X i =1 max { sup ≤ r ≤ s |h v r , e i i| , |h v s , e i i| h − nk ( α,β ) a s,t, ( t − s ) β (1 − nk ( α,β ) b ( t − s ) β ) i }≤ n X i =1 max { sup ≤ r ≤ s |h v r , e i i| , sup ≤ r ≤ s |h v r , e i i| h − nk ( α,β ) a s,t, ( t − s ) β (1 − nk ( α,β ) b ( t − s ) β ) i } = n X i =1 sup ≤ r ≤ s |h v r , e i i| h − nk ( α,β ) a s,t, ( t − s ) β (1 − nk ( α,β ) b ( t − s ) β ) i = S n X i =1 sup ≤ r ≤ s |h v r , e i i| , (28)where S = h − nk ( αβ ) a s,t, ( t − s ) β (1 − nk ( α,β ) b ( t − s ) β ) i − . Next we divide the interval [0 , T ] into p pieces of size ∆ = ( t − s ), with ∆ being smallenough, so that none of the above estimates fail, and write a ∆ , for a s,t, , as a s,t, dependson s, t only through the difference ( t − s ) = ∆.More precisely, in view of (20), (25) and (27), we require ∆ to satisfy∆ − β > n k ( α, β ) · max[ c α ˜ M (1) α , b , ( a ∆ , + b )]= n k ( α, β ) · max[ c α ˜ M (1) α , ( a ∆ , + b )] . For example, we can choose∆ − β = 3 n k ( α, β ) · max[ c α ˜ M (1) α , ( a ∆ , + b )] , (29)and thus, for this specific choice of ∆, we have S ≤ . To ensure the existence of such a ∆, we start with∆ − β = 3 n k ( α, β ) c α ˜ M (1) α . Then, if ∆ − β ≥ n k ( α, β ) ( a ∆ , + b ) , (30)we choose ∆ = ∆ . Otherwise we solve the equation∆ − β = 3 n k ( α, β ) ( a ∆ , + b ) , in the range ∆ ≤ ∆ . It is easy to see that the solution to this equation is ensured since theleft side increases to infinity as ∆ →
0, whereas the right side, which is larger than the leftside at ∆ = ∆ , decreases as ∆ decreases to zero.16sing the above notation, and repeatedly applying the technique used in (28), we can writesup t ∈ [0 ,T ] k v t k = sup t ∈ [0 , p ∆] [ n X i =1 |h v t , e i i| ] ≤ n X i =1 sup t ∈ [0 , p ∆] |h v t , e i i|≤ S p n X i =1 |h v, e i i| , where p = T ∆ = T (cid:16) nk ( α, β ) · max[ c α ˜ M (1) α , ( a ∆ , + b )] (cid:17) /β = T C T , and C T = (cid:16) n k ( α, β ) · max[( c α ˜ M (1) α ) , ( a ∆ , + b )] (cid:17) /β . Since we have all the appropriate notation at hand, we now take a moment off the proof ofTheorem 3.1 to complete the remaining issues in the proof of Lemma 3.1
Proof of Lemma 3.1 (continued):
To prove the final claim of Lemma 3.1, note that forspecific choice ( t − s ) = ∆, together with (16) and (21) we have K ∗ s,t = K s,t (1 − α ) ≤ − α ) · c α n k ( α, β ) M α = 32(1 − α ) · c α n k ( α, β ) X γ ∈ I M γ k B Hγ k β,T − α , and so there exists a constant K ( α, β ), dependent only on α and β , such that K ∗ s,t ( t − s ) β ≤ K ( α, β ) P γ ∈ I M γ k B Hγ k β,T P γ ∈ I M (1) γ k B Hγ k β,T ≤ K ( α, β ) X γ ∈ I M γ M (1) γ . Consequently, a ∆ , can also be bounded above by a constant a , hence we shall replace a ∆ , by a , in the following discussion. ✷ Returning to the proof of Theorem 3.1, note that( C T ) β ≤ n c α k ( α, β ) X γ ∈ I ( ˜ M (1) α,γ + a ,γ + b ,γ ) k B Hγ k β,T , M (1) α,γ , a ,γ , and b ,γ are the coefficients of k B Hγ k β,T in the constants ˜ M (1) α , a and b ,respectively.Now using the bound on S available due to the specific choice of ∆ completes the proof. ✷ The estimates in Theorem 3.1 in turn imply similar bounds on the Hausdorff measure of the m -dimensional manifold M t , evolving under the flow Φ t . More precisely, let { v xi } mi =1 be anorthonormal basis of the tangent space T x M , at the point x ∈ M . Then, writing, as usual, H m ( M t ) for m -th Hausdorff measure of M t , we have the following result. Theorem 3.5
Let M be a C , m -dimensional manifold, evolving under the flow Φ t definedin (3) . Then under the conditions ( A − ( A , and for α = 1 − H + δ , β = H − ε , suchthat (1 − H ) < α < / and δ > ε , there exists a constant c , and a random variable C ,T ,such that sup t ∈ [0 ,T ] H m ( M t ) ≤ c H m ( M ) 2 C ,T T . Here C ,T depends on α , β , n , I , and {k B Hγ k β,T , M γ , M (1) γ , M (2) γ } γ ∈ I , and satsifies E [ C ,T ] β ≤ C · E [ k B H k β,T ] , with the constant C dependent only on α , β , n , | I | and { M γ , M (1) γ , M (2) γ } γ ∈ I . Proof:
Consider the pushforwards { v xi,t } mi =1 of the tangent vectors { v xi } mi =1 under the flowΦ t . Then by using a simple formula for change of variables on a manifold, we have H m ( M t ) = Z M t H m ( dy )= Z M k α x ( t ) kH m ( dx ) , where k α x ( t ) k = q | det( h v xi,t , v xj,t i ) | . By the Cauchy-Schwartz inequality we know that h v xi,t , v xj,t i ≤ k v xi,t k k v xi,t k . Therefore, using Theorem 3.1 and the above expression, we obtainsup t ∈ [0 ,T ] k α x ( t ) k ≤ m !( sup t ∈ [0 ,T ] k v xi,t k ) m ≤ c m ! 2 m T C T , which proves the required result. ✷ Finally, note that for the case corresponding to | I | = 1, the above simplifies tosup t ∈ [0 ,T ] H m ( M t ) ≤ c C T k B H k /ββ,T , for some constants c and C dependent only on the various Lipschitz coefficients of thevector field and its partial derivatives, and this proves Theorem 1.1.18 eferences [1] Adler, R. J. and Taylor, J. E. (2007). Random Fields and their Geometry,
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