Putting it all together: proof of Theorem 6.12

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

We now finally combine all of the results we have just accumulated to give the proof of the main theorem of these sections. Proof of Theorem 6.12. We are going to prove the statement by showing that there exists θ 0 and, for every α 0, a Ψ θ -dominated family of negligible events such that, modulo this family, the assumption inf ϕ∈S α 〈ϕ, M T ϕ〉 ≤ ǫkϕk 2 leads to a contradiction for all ǫ sufficiently small. From now on, fix N as in Assumption C.2. By Lemmas 6.15 and 6.17, we see that there exist constants θ , q, r 0 such that, modulo some Ψ θ -dominated family of negligible events, one has the implication ϕ ∈ S α 〈ϕ, M T ϕ〉 ≤ ǫkϕk 2 « =⇒ ¨ K T −δ,T ϕ ∈ S c α and kΠK T −δ,T ϕk ≥ α 2 kϕk 〈K T −δ,T ϕ, Q N u T −δ K T −δ,T ϕ〉 ≤ ǫ q kϕk 2 , provided that we choose r ≤ r in the definition 77 of δ. By Assumption C.2, this in turn implies modulo the same family of negligible events · · · =⇒ α 2 kϕk ≤ kΠK T −δ,T ϕk ≤ Λ −1 c α u ǫ q 2 kϕk . On the other hand, it follows from Lemma 6.13 and the assumption on the inverse moments of Λ c α that, modulo some Ψ 4 q -dominated family of negligible events, one has the bound Λ −1 c α u ≤ ǫ − q 4 . Possibly making θ smaller, it follows that, modulo some Ψ θ -dominated family of negligible events, one has the implication ϕ ∈ S α 〈ϕ, M T ϕ〉 ≤ ǫkϕk 2 « =⇒ α 2 ≤ ǫ q 4 , which cannot hold for ǫ small enough, thus concluding the proof of Theorem 6.12 713 7 Bounds on Wiener polynomials We will use the terminology of “negligible sets” introduced in Definition 6.9. We will always work on the time interval [0, 1], but all the results are independent modulo change of constants of the time interval, provided that its length is bounded from above and from below by two positive constants independent of ǫ. This is seen easily from the scaling properties of the Wiener process. The results of this section are descendents of similar results obtained in [MP06, BM07] by related techniques. In [BM07] it was proven that if a Wiener polynomial, with continuous, bounded variation coefficients, is identically zero on an interval then so are its coefficients. This is enough to prove the almost sure invertibility of projections of the Malliavin matrix, which in turn implies the existence of a density for the projections of the transition probabilities. To prove smoothness of the densities or the ergodic results of this paper, more quantitative control is needed. In [BM07], a result close to 7.1 is claimed. However an error in Lemma 9.12 of that article leaves the proof incomplete. Arguing along similar, though slightly different lines, we prove the needed result below. We build upon the presentation in [BM07] but simplify it significantly. The presentation in [BM07] was already a significant simplification over that in [MP06]. Theorem 7.1 Let {W k } d k=1 be a family of i.i.d. standard Wiener processes and, for every multi-index α = α 1 , . . . , α ℓ , define W α = W α 1 . . . W α ℓ with the convention that W α = 1 if α = φ. Let furthermore A α be a family of not necessarily adapted stochastic processes with the property that there exists m ≥ 0 such that A α = 0 whenever |α| m and set Z A t = P α A α tW α t. Then, there exists a universal family of negligible events Osc m W depending only on m such that the implication kZ A k L ∞ ≤ ǫ =⇒ ¨ either sup α kA α k L ∞ ≤ ǫ 3 −m or sup α kA α k Lip ≥ ǫ −3 −m+1 92 holds modulo Osc m W . The supremum norms are taken on the interval [0, 1]. Remark 7.2 Informally, we can read the statement of Theorem 7.1 as “if Z A is small, then either all of the coefficients A α are small, or at least one of them oscillates very fast.” The exponents appearing in the statement of Theorem 7.1 are somewhat arbitrary. By going through the proof more carefully, we can see that for any κ 2, it is possible to find a constant C κ 0 such that the exponents in 92 can be replaced by κ −m and −C κ κ −m respectively. Here, the coefficient C κ tends to 0 as κ → 2. While the precise values of the exponents in 92 arising from our proof are unlikely to be sharp, they are not far from it, as can be seen by looking at processes of the form Zt = ǫ 1 − θ 2 W θ t − W t, where W θ is the linear interpolation of the Wiener process W over intervals of size ǫ θ . Remark 7.3 The reason why the family of negligible sets appearing in this statement is called Osc m W is that it relies on the fact that the Wiener processes typically fluctuate sufficiently fast on every small time interval so that their effects can be distinguished from those of the multiplicators A α which fluctuate over much longer timescales. It is important to note that Osc m W depends on the processes A α only through the value of m. Before we start with the proof, we show the following result, which is essentially the particular case of Theorem 7.1 where m = 1 and where the coefficients A α do not depend on time. Here, 〈·, ·〉 denotes the scalar product in R d . 714 Lemma 7.4 Let {W k } d k=1 be a collection of i.i.d. standard Wiener processes. Then, for any exponent κ 0, there exists a universal family Osc W of negligible events such that the bound sup t ∈[0,1] 〈A,Wt〉 ≥ ǫ κ |A| , 93 holds modulo Osc W for any choice of coefficients A ∈ R d . Remark 7.5 We would like to stress again the fact that the family of events Osc W is independent of the choice of coefficients A and depends only on the realisation of the W k ’s. Proof. Fix κ 0 and define a family of events B by B ǫ = {sup t ∈[0,1] |W t| ≥ ǫ −κ }. It follows immediately from the fact that the supremum of a Wiener process has Gaussian tails that the family B is negligible. Consider now the unit sphere S d in R d . For every A ∈ S d , the process W A t = 〈A, W t〉 is a standard Wiener process and so P sup t ∈[0,1] |W A t| ≤ 2ǫ κ ≤ C 1 exp −C 2 ǫ −2κ for some constants C 1 and C 2 that are independent of A. Denote this event by H ǫ A . Choose now a collection {A k } of points in S d such that sup A ∈S d inf k |A − A k | ≤ ǫ 2 κ and define H ǫ = S k H ǫ A k . Since this can be achieved with O ǫ −2κd−1 points, the family H is negligible by Lemma 6.10. We now define Osc W = H ∪ B and we note that, modulo Osc W , one has for every ¯ A ∈ R d the bound sup t ∈[0,1] 〈¯A,Wt〉 ≥ |¯A| inf A ∈S d sup t ∈[0,1] 〈A,Wt〉 ≥ | ¯ A | inf k sup t ∈[0,1] 〈A k , W t 〉 − ǫ κ ≥ | ¯ A |ǫ κ , as required. We now turn to the Proof of Theorem 7.1. The proof proceeds by induction on the parameter m. For m = 0, the statement is trivial since in this case one has Z A t = A φ t, so that one can take Osc W = φ. Fix now a value m ≥ 1 and assume that, for some ǫ, both inequalities kZ A k L ∞ ≤ ǫ , 94a sup |α|≤m kA α k Lip ≤ ǫ −3 −m+1 94b hold. Our aim is to find a universal family of negligible sets Osc m W such that, modulo Osc m W , these two bounds imply the bound sup α kA α k L ∞ ≤ ǫ 3 −m . Before we proceed, we localise our argument to Wiener processes that do not behave too “wildly.” Using the fact that the Hölder norm of a Wiener process has Gaussian tails for every Hölder exponent smaller than 1 2, we see that the bounds sup t ∈[0,1] sup |α|≤m |W α t| ≤ ǫ −110 , sup s 6=t sup |α|≤m |W α t − W α s| |t − s| 2 5 ≤ ǫ −130 , 95 both hold modulo some universal family Wien of negligible events. The reason for these particular choices of exponents will become clearer later on, but any two negative exponents would have been admissible. 715 Choose an exponent κ to be determined later and define a sequence of times t ℓ = ℓǫ κ for ℓ = 0, . . . , ǫ −κ , so that the interval [0, 1] gets divided into ǫ −κ subintervals of the form [t ℓ , t ℓ+1 ]. We define A ℓ α = A α t ℓ and similarly for W ℓ α . We also define the Wiener increments ¯ W ℓ i t = W i t−W i t ℓ and their products ¯ W ℓ α = Π j ∈α ¯ W ℓ j . With these notations, one has for t ∈ [t ℓ , t ℓ+1 ] the equality Z A t = Z A t ℓ + X α6=φ A ℓ α W α t − W ℓ α + X α A α t − A ℓ α W α t 96 = Z A t ℓ + X α6=φ X σ⊂α σ6=φ A ℓ α W ℓ α\σ ¯ W ℓ σ t + X α A α t − A ℓ α W α t = Z A t ℓ + X ν X σ6=φ C ν,σ A ℓ ν∪σ W ℓ ν ¯ W ℓ σ t + X α A α t − A ℓ α W α t ≡ Z A t ℓ + X ν d X j=1 C ν, j A ℓ ν∪ j W ℓ ν ¯ W ℓ j t + E ℓ t , for some “error term” E ℓ that will be analysed later. Here, the combinatorial factor C α,σ counts the number of ways in which the multi-index σ can appear in the multi-index α ∪ σ for example C i, j, j is equal to 2 if i 6= j and 3 if i = j. Using the Brownian scaling and the fact that the supremum of a Wiener process has Gaussian tails, we see that for every κ ′ κ, the bound sup ℓ≤ǫ −κ sup t ∈[0,ǫ κ ] sup j ∈{1,...,d} | ¯ W ℓ j t| ≤ ǫ κ ′ 2 , 97 holds modulo some universal family Wien κ ′ ,m of negligible events. Note now that all the terms appearing in E ℓ are up to combinatorial factors either of the form A ℓ α∪σ W ℓ α ¯ W σ t with |σ| ≥ 2, or of the form A α t − A ℓ α W α t. Together with 97 and the first bound in 95, this shows that there exists a constant C depending only on m such that 94b implies sup ℓ≤ǫ −κ sup t ∈[t ℓ ,t ℓ+1 ] |E ℓ t| ≤ C ǫ κ ′ −127−110 + ǫ κ−19−110 , 98 modulo Wien κ ′ ,m . Here we used the fact that 94b implies in particular that the bound kA α k L ∞ ≤ ǫ −127 holds for every α with |α| ≥ 2 note that these terms are non-zero only if m ≥ 2 and that kA α k Lip ≤ ǫ −19 , since we assumed m ≥ 1. At this point, we fix κ = 5 4 and κ ′ = 6 5 , so that in particular both exponents appearing in 98 are greater than 1. We then define Wien ′ = Wien ∪ Wien κ ′ ,m so that, modulo Wien ′ , 94a and 96 imply sup t ∈[t ℓ ,t ℓ+1 ] X α d X j=1 C α, j A ℓ α∪ j W ℓ α ¯ W j t ≤ 2ǫ + sup ℓ≤ǫ −κ sup t ∈[t ℓ ,t ℓ+1 ] |E ℓ t| ≤ Cǫ . 99 The left hand side of this expression motivates the introduction of operators M j acting on the set of families of stochastic processes by M j A α = C α, j A α∪ j . Note that M j lowers the “degree” of A by one in the sense that if A α = 0 for every |α| ≥ m, then M j A α = 0 for every |α| ≥ m − 1. 716 With this notation, we can rewrite 99 as sup t ∈[t ℓ ,t ℓ+1 ] d X j=1 Z M j A t ℓ ¯ W j t ≤ Cǫ . 100 Using the Brownian scaling and applying Lemma 7.4, combined with Lemma 6.10, shows the existence of a family Osc W of negligible events such that 100 implies |Z M j A t ℓ | ≤ ǫ 7 20 , ∀ℓ ≤ ǫ −54 . Here, we used the fact that our choice of κ implies that 1 − κ2 720. This shows that the statements 94 imply kZ M j A k L ∞ ≤ ǫ 7 20 + C m sup α ǫ κ kA α k Lip kW α k L ∞ + ǫ 2 κ5 kA α k L ∞ kW α k C 2 5 ≤ ǫ 7 20 + C m ǫ κ−19−10 + ǫ 1 2−19−130 ≤ C m ǫ 7 20 , 101 modulo Wien ′ ∪ Osc W . Here, the constant C m 1 depends only on m. We now finally arrived at the stage where we are able to apply our induction hypothesis to each of the processes Z M j A . Note that since 7 20 13, 94b implies that sup α, j kM j A α k Lip ≤ C m ǫ 7 20 −3 −m , for all sufficiently small ǫ. Therefore, outside of the event Osc m −1 W C m ǫ 7 20 , one has the implication n sup j kZ M j A k L ∞ ≤ C m ǫ 7 20 o n sup α kA α k Lip ≤ ǫ −3 −m+1 o =⇒ n sup α, j kM j A α k L ∞ ≤ C ′ m ǫ 7 20 3 −m−1 o , 102 for some different constant C ′ m depending also only on m. Since 7 20 13 and since kM j A α k L ∞ ≥ kA α∪ j k L ∞ , this implies in particular that kA α k L ∞ ≤ ǫ 3 −m for every α 6= φ. In order to conclude the proof of the theorem, it therefore only remains to obtain a similar bound on kA φ k L ∞ . We define a family of negligible events Wien ′′ m so that Wien ′ ⊂ Wien ′′ m and such that the bound sup t ∈[0,1] sup |α|≤m |W α t| ≤ ǫ − 1 70 3 −m−1 , 103 holds modulo Wien ′′ m . We claim that if we define recursively Osc m W ǫ = Osc m −1 W C ǫ 7 20 ∪ Wien ′′ m ǫ , the family Osc m W has the requested properties. It follows indeed from 94a, 103 and the definition of Z A that, modulo Osc m W , 94 imply the bound kA φ k L ∞ ≤ ǫ + X α6=φ kA α k L ∞ kW α k L ∞ ≤ ǫ + C ′ m ǫ 720−1703 −m−1 . 104 Since we choose the bound 103 in such a way that 7 20 − 170 13, we obtain kA φ k L ∞ ≤ ǫ 1 3 for sufficiently small ǫ. Together with the remark following 102, this concludes the proof of Theorem 7.1. 717 8 Examples In this section, we apply the abstract framework developed in this article to two concrete examples: the stochastic Navier-Stokes equations on a sphere and a class of stochastic reaction-diffusion equations. The examples are chosen in order to highlight the techniques that can be used to verify the assumptions of our results and to get some idea of their scope of applicability. In particular, the Navier-Stokes equations provide an example where bounds on the Jacobian are not very uniform, so that an initial condition dependent control is required in Assumption C.1. The stochastic reaction- diffusion system on the other hand satisfies very strong a priori bounds, but Assumption A.1 is not verified with the usual choice H = L 2 , so that one has to work a bit more to fit the equations into the framework presented here. Our strategy is as follows: in a first section, we provide a simplified version of our results. We tried to find a formulation that strikes a balance between powerful results and easily verifiable assumptions. This general formulation will then be used by both of the examples mentioned above.

8.1 A general formulation