The iteration step getdoc4164. 719KB Jun 04 2011 12:04:18 AM

Applying Lemma 6.14 with f t = R t T 2 |〈g k , K s,T ϕ〉| ds and α = 1, it follows that there exists a constant C 0 such that, for every k = 1 . . . d, either sup t ∈I δ |〈g k , K t,T ϕ〉| ≤ ǫ 1 4 , or |||〈g k , K ·,T ϕ〉||| 1 ≥ Cǫ −14 . 80 Therefore, to complete the proof, we need only to show that the latter events form a Ψ 4 -dominated negligible family for every k. Since |||〈g k , K ·,T ϕ〉||| 1 ≤ kg k k −β |||K ·,T ϕ||| 1, β , the bound 80 implies that sup ϕ∈H : kϕk=1 |||K t,T ϕ||| 1, β ≥ C ǫ −14 g ∗ , 81 where g ∗ = max k kg k k −β which is finite since we have by assumption that −β ≤ γ + 1 γ ⋆ + 1 and since g k ∈ H γ ⋆ +1 for every k . This event depends only on the initial condition u and on the model under consideration. In particular, it is independent of ϕ. The claim now follows from the a priori bound 72b and Lemma 6.13 with q = 1 4 and b = ¯ p β .

6.9 The iteration step

Recall that we consider evolution equations of the type du t = F u t d t + d X k=1 g k dW k t , 82 where F is a “polynomial” of degree n. The aim of this section is to implement the following recursion: if, for any given polynomial Q, the expression 〈Qu t , K t,T ϕ〉 is “small” in the supremum norm, then both the expression 〈[Q, F]u t , K t,T ϕ〉 and 〈[Q, g k ]u t , K t,T ϕ〉 must be small in the supremum norm as well. The main technical tool used in this section will be the estimates on “Wiener polynomials” from Section 7. Using the notation W α t def = W α 1 t W α 2 t · · · W α ℓ t , for a multi-index α = α 1 , . . . , α ℓ , this estimate states that if an expression of the type P |α|≤m A α tW α t is small, then, provided that the processes A α are sufficiently regular in time, each of the A α must be small. In other words, two distinct monomials in a Wiener polynomial cannot cancel each other out. Here, the processes A α do not have to be adapted to the filtration generated by the W k , so this gives us some kind of anticipative replacement of Norris’ lemma. The main trick that we use in order to take advantage of such a result is to switch back and forth between considering the process u t solution to 82 and the process v t defined by v t def = u t − d X k=1 g k W k t , 708 which has more time-regularity than u t . Recall furthermore that given a polynomial Q and a multi- index α, we denote by Q α the corresponding term 12 appearing in the finite Taylor expansion of Q. Recall the definition Poly m γ, β = Poly m H γ , H −β−1 ∩ Poly m H γ+1 , H −β . We first show that if Q ∈ Poly m γ, β and 〈Qu t , K t,T ϕ〉 is small, then the expression 〈Q α v t , K t,T ϕ〉 note the appearance of v t rather than u t must be small as well for every multi-index α: Lemma 6.19 Let Q ∈ Poly m γ, β for some m ≥ 0 and for γ and β as chosen in 73. Let furthermore q 0 an set ¯ q = q3 −m . Then, the implication sup t ∈I δ |〈Qu t , K t,T ϕ〉| ≤ ǫ q kϕk =⇒ sup α sup t ∈I δ |〈Q α v t , K t,T ϕ〉| ≤ ǫ ¯ q kϕk , holds modulo some Ψ 6m+1 ¯ q -dominated negligible family of events, provided that r ¯ q 6¯ p β . Proof. Note first that both inner products appearing in the implication are well-defined by Propo- sition 6.2 and the assumptions on Q. By homogeneity, we can assume that kϕk = 1. Since Q is a polynomial, 11 implies that 〈Qu t , K t,T ϕ〉 = X α 〈Q α v t , K t,T ϕ〉W α t . Applying Theorem 7.1, we see that, modulo some negligible family of events Osc m W , sup t ∈I δ |〈Qu t , K t,T ϕ〉| ≤ ǫ q implies that either sup α sup t ∈I δ |〈Q α v t , K t,T ϕ〉| ≤ ǫ ¯ q , 83 or there exists some α such that |||〈Q α v t , K ·,T ϕ〉||| 1 ≥ ǫ −¯q3 . 84 We begin by arguing that the second event is negligible. Since Q is of degree m, there exists a constant C such that |||〈Q α v t ,K ·,T ϕ〉||| 1 ≤ sup t ∈I δ kK t,T ϕk β+1 |||Q α v · ||| 1, −β−1 + sup t ∈I δ kQ α v t k −β |||K ·,T ϕ||| 1, β ≤ C sup t ∈I δ kK t,T ϕk β+1 sup t ∈I δ kv t k m −1 γ |||v||| 1, γ + C sup t ∈I δ kv t k m γ+1 |||K ·,T ϕ||| 1, β . Here, we used the fact that Q α ∈ Poly m H γ , H −1−β to bound the first term and the fact that Q α ∈ Poly m H γ+1 , H −β to bound the second term. The fact that Q α belongs to these spaces is a consequence of g k ∈ H γ ⋆ +1 and of the definition 11 of Q α . Therefore, 84 implies that either X δ def = sup ϕ∈H : kϕk=1 sup t ∈I δ kK t,T ϕk β+1 sup t ∈I δ kv t k m −1 γ |||v||| 1, γ ≥ 1 2C ǫ −¯q3 85 or Y δ def = sup ϕ∈H : kϕk=1 sup t ∈I δ kv t k m γ+1 |||K ·,T ϕ||| 1, β ≥ 1 2C ǫ −¯q3 . 86 Combining the Cauchy-Schwarz inequality with 72b of Proposition 6.2, we see that X δ and Y δ satisfy the assumptions of Lemma 6.13 with Φ = Ψ m+1 and b = ¯ p β , thus showing that the families of events 85 and 86 are both Ψ 6m+1 ¯ q -dominated negligible, provided that r ¯ q 6¯p β . 709 In the sequel, we will need the follow simple result which is, in some way, a converse to Theorem 7.1. Lemma 6.20 Given any integer N 0 and any two exponents 0 ¯ q q, there exists a universal family of negligible events Sup N W such that the implication sup α sup t ∈I δ |A α t| ǫ q =⇒ sup t ∈I δ X α:|α|≤N A α tW α t ǫ ¯ q holds modulo Sup N W for any collection of processes {A α t : |α| ≤ N}. Proof. Observe that sup t ∈I δ X α:|α|≤N A α tW α t ≤ ‚ sup α sup t ∈I δ |A α t| Œ‚ X α:|α|≤N sup t ∈I δ |W α | Œ Since for any p 0, X α:|α|≤N sup t ∈I δ |W α | ǫ −p is a negligible family of events, the claim follows at once. As a corollary to Lemmas 6.19 and 6.20, we now obtain the key estimate for Lemma 6.17 in the particular case where the commutator is taken with one of the constant vector fields: Lemma 6.21 Let Q ∈ Poly m γ, β be a polynomial of degree m and let q 0. Then, for ¯ q = q3 −m+1 , the implication sup t ∈I δ |〈Qu t , K t,T ϕ〉| ≤ ǫ q kϕk =⇒ sup α sup t ∈I δ |〈Q α u t , K t,T ϕ〉| ≤ ǫ ¯ q kϕk , holds for all ϕ ∈ H modulo some Ψ 2m+1 ¯ q -dominated negligible family of events, provided that r ¯ q 2¯ p β . Proof. Since it follows from 11 that Q α β = Q α∪β , we have the identity Q α u t = X β Q α β v t W β = X β Q α∪β v t W β . Combining Lemma 6.19 and Lemma 6.20 with N = m proves the claim. In the next step, we show a similar result for the commutators between Q and F . We are going to use the fact that if a function f is differentiable with Hölder continuous derivative, then f being small implies that ∂ t f is small as well, as made precise by Lemma 6.14. As previously, we start by showing a result that involves the process v t instead of u t : 710 Lemma 6.22 Let Q be as in Lemma 6.19 and such that [Q α , F σ ] ∈ Polyγ, β for any two multi-indices α, σ. Let furthermore q 0 and set ¯ q = q3 −2m 8. Then the implication sup t ∈I δ |〈Qu t , K t,T ϕ〉| ≤ ǫ q kϕk =⇒ sup α,σ sup t ∈I δ |〈[Q α , F σ ]v t , K t,T ϕ〉| ≤ ǫ ¯ q kϕk , holds modulo some Ψ 6m+1 ¯ q -dominated negligible family of events, provided that r ¯ q 6¯p β . As before the empty multi-indices are included in the supremum. Proof. By homogeneity, we can assume that kϕk = 1. Combining Lemma 6.19 with Lemma 6.14 and defining ˆ q = q3 −m , we obtain that sup t ∈I δ |〈Qu t , K t,T ϕ〉| ≤ ǫ q implies for f α,ϕ t def = ∂ t 〈Q α v t , K t,T ϕ〉 the bound sup t ∈I δ | f α,ϕ t| ≤ C max ǫ ˆ q , ǫ ˆ q 4 ||| f α,ϕ ||| 3 4 1 3 , 87 modulo some Ψ 6m+1 ˆ q -dominated negligible family of events, provided that r ≤ ˆq6¯p β . Note that this family is in particular independent of both α and ϕ. Here and in the sequel, we use the letter C to denote a generic constant depending on the details of the problem that may change from one expression to the next. One can see that 〈Q α v t , K t,T ϕ〉 is differentiable in t by combining Proposition 6.2 with the fact that Q α ∈ PolyH γ , H −1−β ∩ PolyH γ+1 , H −β as in the proof of Lemma 6.19. See [DL92] for a more detailed proof of a similar statement. Computing the derivative explicitly, we obtain f α,ϕ t = 〈DQ α v t F u t − DFu t Q α v t , K t,T ϕ〉 def = 〈B α t, K t,T ϕ〉 . The function B α can be further expanded to B α t = X σ DQ α v t F σ v t − DF σ v t Q α v t W σ t = X σ [Q α , F σ ]v t W σ t . Notice that, by the assumption that [Q α , F σ ] ∈ Polyγ, β, one has |||[Q α , F σ ]v · W σ ·||| 1 3 , −1−β ≤ C1 + sup t ∈I δ kv t k γ n+m −2−|α|−|σ| k∂ t v t k γ sup t ∈I δ |W σ t| + C|||W σ ||| 1 3 1 + sup t ∈I δ kv t k γ n+m −1−|α|−|σ| , k[Q α , F σ ]v t W σ tk −β ≤ C1 + kv t k γ+1 n+m −1−|α|−|σ| |W σ t| . Here it is understood that if one of the exponents of the norm of v t is negative, the term in question actually vanishes. It therefore follows from Proposition 6.2 that E |||B α ||| p 1 3 , −1−β ≤ C p Ψ n+m−1p u , E sup t ∈I δ kB α tk p −β ≤ C p Ψ n+m−1p u , for every p ≥ 1 and some constants C p . Since the Hölder norm of f α,ϕ is bounded by |||〈B α ·, K ·,T ϕ〉||| 1 3 ≤ |||B α ||| 1 3 , −1−β sup t ∈I δ kK t,T k β+1 + |||K ·,T ||| 1 3 , β sup t ∈I δ kB α tk −β , 711 we can use the bounds on B α just obtained, the Cauchy-Schwarz inequality, Proposition 6.2, and Lemma 6.13, to obtain sup α sup kϕk≤1 ||| f α,ϕ ||| 3 4 1 3 ≤ ǫ − ˆ q 8 , 88 modulo some Ψ 12n+m ˆ q -dominated negligible family of events, provided that r ≤ min {ˆq12, ˆq6¯p β }. As a consequence, modulo this family, we obtain from 87 the bound sup α sup t ∈I δ | f α,ϕ t| ≤ Cǫ ˆ q 8 which can be rewritten as sup α sup t ∈I δ X σ 〈[Q α , F σ ]v t , K t,T ϕ〉W σ t ≤ Cǫ ˆ q 8 . 89 Since [Q α , F σ ] ∈ Polyγ, β the same reasoning as in Lemma 6.19 combined with Theorem 7.1 on Wiener polynomials implies that modulo some negligible family of events Osc m W , the estimate 89 implies that either sup α,σ sup t ∈I δ |〈[Q α , F σ ]v t , K t,T ϕ〉| ≤ ǫ ¯ q , 90 or there exists some α and σ such that |||〈[Q α , F σ ]v t , K ·,T ϕ〉||| 1 ≥ ǫ −¯q3 . 91 Again following the same logic as Lemma 6.19, we see that the family of events in 91 is Φ 6m+1 ¯ q - dominated negligible provided that r ¯ q 6¯p β . In order to turn this result into a result involving the process u t , we need the following expansion: Lemma 6.23 Given any two multi-indices α and σ including the empty indices, there exist an integer N and a collection of multi-indices {α i , σ i , ζ i : i = 1 . . . N } and constants {c i : i = 1 . . . N } so that [Q α , F σ ]u t = N X i=1 c i [Q α i , F σ i ]v t W ζ i t Proof. First observe that [Q α , F σ ]u t = X ζ [Q α , F σ ] ζ v t W ζ t . The Jacobi identity for Lie bracket states that D g k [Q α , F σ ] = [g k , [Q α , F σ ]] = [[g k , Q α ], F σ ] + [Q α , [g k , F σ ]] = |α| + 1[Q α∪k , F σ ] + |σ| + 1[Q α , F σ∪k ] . By iterating this calculation, we see that for any multi-index ζ, [Q α , F σ ] ζ is equal to some linear combination of a finite number of terms of the form [Q α i , F σ i ] for some multi-indices α i and σ i . In very much the same way as before, it then follows that: 712 Corollary 6.24 Let Q be as in Lemma 6.19 and such that [Q α , F σ ] ∈ Polyγ, β for any two multi- indices α, σ. Let furthermore q 0 and set ¯ q = q3 −2m+1 8. Then the implication sup t ∈I δ |〈Qu t , K t,T ϕ〉| ≤ ǫ q kϕk =⇒ sup α,σ sup t ∈I δ |〈[Q α , F σ ]u t , K t,T ϕ〉| ≤ ǫ ¯ q kϕk , holds modulo some Ψ 2m+1 3¯ q -dominated negligible family of events, provided that r 3¯ q 2¯p β . Proof. It follows from Lemma 6.23 that 〈[Q α , F σ ]u t , K T,t ϕ〉 = N X i=1 c i 〈[Q α i , F σ i ]v t , K T,t ϕ〉W γ i t . Combining the control of the 〈[Q α i , F σ i ]v t , K T,t ϕ〉 obtained in Lemma 6.22 with Lemma 6.20 gives the quoted result.

6.10 Putting it all together: proof of Theorem 6.12