Linearization and its adjoint

3.3 Linearization and its adjoint

In this section, we study how the solutions to 8 depend on their initial conditions. Since the map from 14 used to construct the solutions to 8 is Fréchet differentiable it is actually infinitely differentiable and since it is a contraction for sufficiently small values of t, we can apply the implicit functions theorem see for example [RR04] for a Banach space version to deduce that for every realisation of the driving noise, the map u s 7→ u t is Fréchet differentiable, provided that t s is sufficiently close to s. Iterating this argument, one sees that, for any s ≤ t τ, the map u s 7→ u t given by the solutions to 8 is Fréchet differentiable in H . Inspecting the expression for the derivative given by the implicit functions theorem, we conclude that the derivative J s,t ϕ in the direction ϕ ∈ H satisfies the following random linear equation in its mild formulation: ∂ t J s,t ϕ = −LJ s,t ϕ + DN u t J s,t ϕ , J s,s ϕ = ϕ . 17 Note that, by the properties of monomials, it follows from Assumption A.1.2 that kDNuvk γ ≤ C1 + kuk γ+a n −1 kvk γ+a , for every γ ∈ [−a, γ ⋆ . A fixed point argument similar to the one in Proposition 3.4 shows that the solution to 17 is unique, but note that it does not allow us to obtain bounds on its moments. We only have that for any T smaller than the explosion time to the solutions of 8, there exists a random constant C such that sup ≤stT sup kϕk≤1 kJ s,t ϕk ≤ C . 18 The constant C depends exponentially on the size of the solution u in the interval [0, T ]. However, if we obtain better control on J s,t by some means, we can then use the following bootstrapping argument: Proposition 3.7 For every γ ∈ 0, γ ⋆ + 1, there exists an exponent ˜ q γ ≥ 0, and constants C 0 and γ γ such that we have the bound kJ t,t+s ϕk γ ≤ Cs −γ sup r ∈[ s 2 ,s] 1 + ku t+r k γ ˜ q γ kJ t,t+r ϕk , 19 for every ϕ ∈ H and every t, s 0. If γ 1 − a, then one can choose γ = 0 and ˜ q γ = n − 1. Since an almost identical argument will be used in the proof of Proposition 3.9 below, we refer the reader there for details. We chose to present that proof instead of this one because the presence of an adjoint causes slight additional complications. For s ≤ t, let us define operators K s,t via the solution to the random PDE ∂ s K s,t ϕ = LK s,t ϕ − DN ∗ u s K s,t ϕ , K t,t ϕ = ϕ , ϕ ∈ H . 20 Note that this equation runs backwards in time and is random through the solution u t of 8. Here, DN ∗ u denotes the adjoint in H of the operator DNu defined earlier. Fixing the terminal time t and setting ϕ s = K t −s,t ϕ, we obtain a more usual representation for ϕ s : ∂ s ϕ s = −Lϕ s + DN ∗ u t −s ϕ s . 21 677 The remainder of this subsection will be devoted to obtaining regularity bounds on the solutions to 20 and to the proof that K s,t is actually the adjoint of J s,t . We start by showing that, for γ sufficiently close to but less than γ ⋆ + 1, 20 has a unique solution for every path u ∈ C R, H γ and ϕ ∈ H . Proposition 3.8 There exists γ γ ⋆ + 1 such that, for every ϕ ∈ H , equation 20 has a unique continuous H -valued solution for every s t and every u ∈ C R, H γ . Furthermore, K s,t depends only on u r for r ∈ [s, t] and the map ϕ 7→ K s,t ϕ is linear and bounded. Proof. As in Proposition 3.4, we define a map Φ T,u : H × C [0, T ], H → C [0, T ], H by Φ T,u ϕ , ϕ t = e −Lt ϕ + Z t e −Lt−s DN ∗ u s ϕ s ds . It follows from Assumption A.1.3 with β = −a that there exists γ γ ⋆ + 1 such that DN ∗ u: H → H −a is a bounded linear operator for every u ∈ H γ . Proceeding as in the proof of Proposition 3.4, we see that Φ is a contraction for sufficiently small T . Similarly to before, we can use a bootstrapping argument to show that K s,t ϕ actually has more regularity than stated in Proposition 3.8. Proposition 3.9 For every β ∈ 0, β ⋆ + 1, there exists γ γ ⋆ + 1, an exponent ¯ q β 0, and a constant C such that kK t −s,t ϕk β ≤ Cs −β sup r ∈[ s 2 ,s] 1 + ku t −r k γ ¯ q β kK t −r,t ϕk , 22 for every ϕ ∈ H , every t, s 0, and every u ∈ C R, H γ . Proof. Fix β β ⋆ + a and δ ∈ 0, 1− a and assume that the bound 22 holds for kK s,t ϕk β . Since we run s “backwards in time” from s = t, we consider again t as fixed and use the notation ϕ s = K t −s,t ϕ. We then have, for arbitrary α ∈ 0, 1, kϕ s k β+δ ≤ Cs −δ kϕ αs k β + C Z s αs s − r −δ+a kDN ∗ u t −r ϕ r k β−a d r , provided that γ is sufficiently close to γ ⋆ +1 such that DN ∗ : H γ → L H β , H β−a by Assumption A.1.3. Furthermore, the operator norm of DN ∗ v is bounded by C1 + kvk γ n −1 , yielding kϕ s k β+δ ≤ Cs −δ kϕ αs k β + Cs −δ+a−1 sup r ∈[αs,s] 1 + ku r k γ n −1 kϕ r k β ≤ Cs −δ sup r ∈[αs,s] 1 + ku r k γ n −1 kϕ r k β . Iterating these bounds as in Proposition 3.6 concludes the proof. The following lemma appears also in [MP06, BM07]. It plays a central role in establishing the representation of the Malliavin matrix given in 37 on which this article as well as [MP06, BM07] rely heavily. 678 Proposition 3.10 For every 0 ≤ s t, K s,t is the adjoint of J s,t in H , that is K s,t = J ∗ s,t . Proof. Fixing 0 ≤ s t and ϕ, ψ ∈ H ∞ , we claim that the expression 〈J s,r ϕ, K r,t ψ〉 , 23 is independent of r ∈ [s, t]. Evaluating 23 at both r = s and r = t then concludes the proof. We now prove that 23 is independent of r as claimed. It follows from 20 and Proposition 3.6 that, with probability one, the map r 7→ K r,t ϕ is continuous with values in H β+1 and differentiable with values in H β , provided that β β ⋆ . Similarly, the map r 7→ J s,r ψ is continuous with values in H γ+1 and differentiable with values in H γ , provided that γ γ ⋆ . Since γ ⋆ + β ⋆ −1 by assumption, it thus follows that 23 is differentiable in r for r ∈ s, t with ∂ r 〈J s,r ϕ, K r,t ψ〉 = 〈 L + DNu r J s,r ϕ, K r,t ψ〉 − 〈J s,r ϕ, L + DN ∗ u r K r,t ψ〉 = 0 . Since furthermore both r 7→ K r,t ϕ and r 7→ J s,r ψ are continuous in r on the closed interval, the proof is complete. See for example [DL92, p. 477] for more details.

3.4 Higher order variations