On the differentiability of solutions of stochastic evolution equations with respect to their initial values

Nonlinear Analysis 162 (2017) 128–161

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Nonlinear Analysis www.elsevier.com/locate/na

On the differentiability of solutions of stochastic evolution equations with respect to their initial values Adam Anderssona , Arnulf Jentzenb , Ryan Kurniawanb , Timo Weltib, * a b

Syntronic Software Innovations, 417 56 Göteborg, Sweden Seminar for Applied Mathematics, ETH Zürich, 8092 Zürich, Switzerland

article

info

Article history: Received 23 December 2016 Accepted 11 March 2017 Communicated by Enzo Mitidieri

abstract In this article we study the differentiability of solutions of parabolic semilinear stochastic evolution equations (SEEs) with respect to their initial values. We prove that if the nonlinear drift coefficients and the nonlinear diffusion coefficients of the considered SEEs are n-times continuously Fréchet differentiable, then the solutions of the considered SEEs are also n-times continuously Fréchet differentiable with respect to their initial values. Moreover, a key contribution of this work is to establish suitable enhanced regularity properties of the derivative processes of the considered SEE in the sense that the dominating linear operator appearing in the SEE smoothes the higher order derivative processes. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction In this article we study the differentiability of solutions of parabolic semilinear stochastic evolution equations (SEEs) with respect to their initial values. (Semilinear) SEEs have been extensively studied in the last decades by means of several different approaches; see, e.g., the monographs by Rozovski˘ı [22], Pr´evˆot & R¨ ockner [20], and Liu & R¨ ockner [19] for results on SEEs in the context of the so-called “variational approach” for SEEs, see, e.g., Da Prato & Zabczyk [8] for results on semilinear SEEs in the context of the so-called “semigroup approach” for SEEs, and see, e.g., Walsh [25] for results on semilinear SEEs in the context of the so-called “martingale measure approach”. In this paper we employ the semigroup approach to establish differentiability of solutions of parabolic semilinear SEEs with respect to their initial values. More precisely, we prove that the smoothness of the coefficients of the considered SEEs transfers to the smoothness of the solutions of the SEEs with respect to their initial values. We demonstrate that if the nonlinear drift coefficients and the nonlinear diffusion coefficients of the considered SEEs are n-times continuously Fr´echet differentiable, then the solutions of the considered SEEs are also n-times continuously

*

Corresponding author. E-mail addresses: [email protected] (A. Andersson), [email protected] (A. Jentzen), [email protected] (R. Kurniawan), [email protected] (T. Welti). http://dx.doi.org/10.1016/j.na.2017.03.003 0362-546X/© 2017 Elsevier Ltd. All rights reserved.

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

129

Fr´echet differentiable with respect to their initial values. In addition, a key contribution of this work is to establish suitable enhanced regularity properties of the derivative processes of the considered SEE in the sense that the dominating linear operator appearing in the SEE smoothes the higher order derivative processes (see (3)–(6)). In the following theorem we summarize some of the key findings of this article. Theorem 1.1. Let (H, ∥·∥H , ⟨·, ·⟩H ) and (U, ∥·∥U , ⟨·, ·⟩U ) be nontrivial separable R-Hilbert spaces, let n ∈ N = {1, 2, . . .}, T ∈ (0, ∞), η ∈ R, let F : H → H and B : H → HS(U, H) be n-times continuously Fr´echet differentiable functions with globally bounded derivatives, let (Ω , F, P) be a probability space with a normal filtration (Ft )t∈[0,T ] , let (Wt )t∈[0,T ] be an IdU -cylindrical (Ω , F, P, (Ft )t∈[0,T ] )-Wiener process, let A : D(A) ⊆ H → H be a generator of a strongly continuous analytic semigroup with spectrum(A) ⊆ {z ∈ C : Re(z) < η}, let (Hr , ∥·∥Hr , ⟨·, ·⟩Hr ), r ∈ R, be a family of interpolation spaces associated to η − A (cf., e.g., [23, Section 3.7]), let M be the set of all F/ B(H)-measurable functions from Ω to H, and for every X ∈ M let |[X]| be the set given by |[X]| = {Y ∈ M : P(X = Y ) = 1}. Then (i) there exist up-to-modifications unique (Ft )t∈[0,T ] / B(H)-predictable stochastic processes X 0,x : [0, T ] × ∫t Ω → H, x ∈ H, which fulfill for all p ∈ [2, ∞), x ∈ H, t ∈ [0, T ] that 0 ∥e(t−s)A F (Xs0,x )∥H + [ 2 p ] ∥e(t−s)A B(Xs0,x )∥HS(U,H) ds < ∞, sups∈[0,T ] E ∥Xs0,x ∥H < ∞, and ⏐ [ ]⏐ ∫ t ∫ t ⏐ ⏐ e(t−s)A B(Xs0,x ) dWs , |[Xt0,x ]| = ⏐⏐ etA x + e(t−s)A F (Xs0,x ) ds ⏐⏐ +

(1)

0

0

(ii) it holds for all p ∈ [2, ∞), t ∈ [0, T ] that H ∋ x ↦→ |[Xt0,x ]| ∈ Lp (P; H) is n-times continuously Fr´echet differentiable with globally bounded derivatives, (iii) there exist up-to-modifications unique (Ft )t∈[0,T ] / B(H)-predictable stochastic processes X k,u : [0, T ] × Ω → H, u ∈ H k+1 , k ∈ {1, 2, . . . , n}, which fulfill for all p ∈ [2, ∞), k ∈ {1, 2, . . . , n}, x, u1 , u2 , . . . , uk ∈ [ k,(x,u1 ,u2 ,...,uk ) p ] H, t ∈ [0, T ] that sups∈[0,T ] E ∥Xs ∥H < ∞ and (

) dk |[Xt0,x ]| (u1 , u2 , . . . , uk ) dxk

k,(x,u1 ,u2 ,...,uk )

= |[Xt

(2)

]|,

∑k (iv) it holds for all p ∈ (0, ∞), k ∈ {1, 2, . . . , n}, δ1 , δ2 , . . . , δk ∈ [0, 1/2) with i=1 δi < 1/2 that ) ⎡ (∑k ( [ k,u p ])1/p ⎤ δ −1/2 1[2,∞) (k) i=1 i t E ∥Xt ∥H ⎦ < ∞, sup sup ⎣ ∏k u=(u0 ,u1 ,...,uk )∈H×(H\{0})k t∈(0,T ] i=1 ∥ui ∥H−δ

(3)

i

and ∑k (v) it holds for all p ∈ (0, ∞), k ∈ {1, 2, . . . , n}, δ1 , δ2 , . . . , δk ∈ [0, 1/2) with i=1 δi < 1/2, |F |Lipk (H,H) < ∞, and |B|Lipk (H,HS(U,H)) < ∞ that ) ⎡ (∑k ⎤ ( [ k,(x,u) k,(y,u) p ])1/p δ −1/2 i=1 i E ∥Xt − Xt ∥H t ⎦ < ∞. sup sup sup ⎣ ∏k x,y∈H, u=(u ,u ,...,u )∈(H\{0})k t∈(0,T ] ∥x − y∥H i=1 ∥ui ∥H−δ 1 2 k x̸=y

(4)

i

In Theorem 1.1 we denote for nontrivial R-Banach spaces (V, ∥·∥V ) and (W, ∥·∥W ), a natural number k ∈ N, and a k-times continuously Fr´echet differentiable function f : V → W by |f |Lipk (V,W ) the k-Lipschitz semi-norm associated to f (see (9) in Section 1.1 for details). Theorem 1.1 is an immediate consequence of items (i), (ii), (iv), (ix), and (x) of Theorem 2.1 below. In Theorem 2.1 we also specify explicitly for every natural number k ∈ N the SEEs which the kth derivative processes in (2) are solutions of (see item (i) of Theorem 2.1 for details). Moreover, Theorem 2.1 provides explicit bounds for the left hand sides of (3) and (4) (see items (ii) and (iv) of Theorem 2.1) and establishes several further regularity

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A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

properties for the derivative processes in item (iii) of Theorem 1.1. Next we would like to emphasize that Theorem 1.1 and Theorem 2.1, respectively, prove finiteness of (3) and (4) even though the denominators in (3) and (4) contain rather weak norms from negative Sobolev-type spaces for the multilinear arguments of the derivative processes. In particular, item (iv) of Theorem 1.1 and item (ii) of Theorem 2.1, respectively, reveal for every p ∈ [1, ∞), k ∈ {1, 2, . . . , n}, δ1 , δ2 , . . . , δk ∈ [0, 1/2), x ∈ H that the derivative process ( k ) k,(x,u1 ,u2 ,...,uk ) H ∋ (u1 , u2 , . . . , uk ) ↦→ |[Xt ]| ∈ Lp (P; H) ∈ L(k) (H k , Lp (P; H)), t ∈ (0, T ], even takes values in the continuously embedded subspace L(k) (H−δ1 × H−δ2 × · · · × H−δk , Lp (P; H))

(5)

of L(k) (H k , Lp (P; H)) = L(k) (H × H × · · · × H, Lp (P; H)) provided that the hypothesis k ∑

δi < 1/2

(6)

i=1

is satisfied. Here we denote for every k ∈ N and all R-Banach spaces (V1 , ∥·∥V1 ), . . . , (Vk , ∥·∥Vk ), (W, ∥·∥W ) by L(k) (V1 ×V2 ×· · ·×Vk , W ) the R-Banach space of all continuous k-linear functions from V1 ×V2 ×· · ·×Vk to W . Items (iv)–(v) of Theorem 1.1 and items (ii) and (iv) of Theorem 2.1, respectively, are of major importance for establishing essentially sharp probabilistically weak convergence rates for numerical approximation processes since the analytically weak norms for the multilinear arguments of the derivative processes (see the denominators in (3) and (4)) translate in analytically weak norms for the approximation errors in the probabilistically weak error analysis which, in turn, result in essentially sharp probabilistically weak convergence rates for the numerical approximation processes (cf., e.g., Theorem 2.2 in Debussche [10], Theorem 2.1 in Wang & Gan [27], Theorem 1.1 in Andersson & Larsson [2], Theorem 1.1 in Br´ehier [3], Theorem 5.1 in Br´ehier & Kopec [4], Corollary 1 in Wang [26], Corollary 5.2 in Conus et al. [7], Theorem 6.1 in Kopec [18], and Corollary 8.2 in [15]). In the following we briefly relate Theorem 1.1 and Theorem 2.1 with results from the literature. Item (i) of Theorem 1.1 is well-known and can, e.g., be found in Theorem 7.4 in Da Prato & Zabczyk [8] (cf., e.g., Theorem 4.3 in Brze´zniak [5], Theorem 7.3.5 in Da Prato & Zabczyk [9], Theorem 6.2 in Van Neerven et al. [24], and Theorem 6.2.3 in Liu & R¨ockner [19]). Items (ii)– (iii) of Theorem 1.1 and items (i), (vii), and (viii) of Theorem 2.1 are generalizations and enhancements of Theorem 7.3.6 in Da Prato & Zabczyk [9]. In particular, we allow F and B to grow linearly (cf. (8) in Section 1.1), we prove continuous Fr´echet differentiability (cf. item (ii) of Theorem 1.1), and we develop the combinatorics (cf., e.g., Theorem 2 in Clark & Houssineau [6]) to explicitly specify the SEEs which the derivative processes of any order are solutions of (cf. item (i) of Theorem 2.1). Nonetheless, the main contribution of this paper is to establish that the derivative processes even take values in the space (5) provided that the assumption (6) is fulfilled. We would like to emphasize that the assumption (6) can essentially not be improved and is thus essentially sharp. More specifically, it holds for every real number T ∈ (0, ∞), every infinite-dimensional separable R-Hilbert space (H, ∥·∥H , ⟨·, ·⟩H ), every nontrivial separable R-Hilbert space (U, ∥·∥U , ⟨·, ·⟩U ), every probability space (Ω , F, P) with a normal filtration (Ft )t∈[0,T ] , and every IdU -cylindrical (Ω , F, P, (Ft )t∈[0,T ] )-Wiener process (Wt )t∈[0,T ] that there exist a generator A : D(A) ⊆ H → H of a strongly continuous analytic semigroup with spectrum(A) ⊆ {z ∈ C : Re(z) < 0} and infinitely often Fr´echet differentiable functions F : H → H and B : H → HS(U, H) with all derivatives being globally bounded such that it holds for all p ∈ [2, ∞), k ∈ N, q ∈ [0, ∞), δ1 , δ2 , . . . , δk ∈ R, t ∈ (0, T ] ∑k with i=1 δi > 1/2 that ⎡( [ ])1/p ⎤ p E ∥Xtk,u ∥H−q ⎦=∞ ⎣ ∏ (7) sup k u=(u0 ,u1 ,...,uk )∈((∩r∈R Hr )\{0})k+1 i=1 ∥ui ∥H−δ i

(see Corollary 1.2 in Hefter et al. [12] for the precise statement).

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1.1. Notation In this section we introduce some of the notation which we employ throughout this article (cf., e.g., Section 1.1 in [1]). For two measurable spaces (A, A) and (B, B) we denote by M(A, B) the set of all A/B-measurable functions. For a set A we denote by P(A) the power set of A and we denote by #A ∈ N0 ∪ {∞} the number of elements of A. For an R-vector space V we denote by V [k] ⊆ V , k ∈ N0 , the sets which satisfy for all k ∈ N that V [0] = V and V [k] = V \ {0}. For a real number T ∈ (0, ∞), a set Ω , and a family Ft ∈ P(P(Ω )), t ∈ [0, T ], of ({ sigma-algebras on Ω we denote by Pred((F given by } {t )t∈[0,T ] ) the sigma-algebra }) Pred((Ft )t∈[0,T ] ) = σ[0,T ]×Ω (s, t]×A : s ∈ [0, T ), t ∈ (s, T ], A ∈ Fs ∪ {0}×A : A ∈ F0 (the predictable sigma-algebra associated to (Ft )t∈[0,T ] ). For R-Banach spaces (V, ∥·∥V ) and (W, ∥·∥W ) with #V > 1 and a natural number n ∈ N we denote by |·|C n (V,W ) : C n (V, W ) → [0, ∞] and ∥·∥C n (V,W ) : C n (V, W ) → [0, ∞] the b b functions which satisfy for all f ∈ C n (V, W ) that n ∑   |f |C k (V,W ) |f |C n (V,W ) = sup f (n) (x)L(n) (V,W ) , ∥f ∥C n (V,W ) = ∥f (0)∥W + (8) b

b

x∈V

k=1

b

and we denote by Cbn (V, W ) the set given by Cbn (V, W ) = {f ∈ C n (V, W ) : ∥f ∥C n (V,W ) < ∞}. For Rb Banach spaces (V, ∥·∥V ) and (W, ∥·∥W ) with #V > 1 and a nonnegative integer n ∈ N0 we denote by |·|Lipn (V,W ) : C n (V, W ) → [0, ∞] and ∥·∥Lipn (V,W ) : C n (V, W ) → [0, ∞] the functions which satisfy for all f ∈ C n (V, W ) that ( ) ⎧ ∥f (x) − f (y)∥W ⎪ ⎪ sup :n=0 ⎪ ⎨x,y∈V, x̸=y ∥x − y∥V ) ( |f |Lipn (V,W ) = ∥f (n) (x) − f (n) (y)∥L(n) (V,W ) ⎪ ⎪ ⎪ : n ∈ N, sup ⎩ (9) ∥x − y∥V x,y∈V, x̸=y n ∑ |f |Lipk (V,W ) . ∥f ∥Lipn (V,W ) = ∥f (0)∥W + k=0

For an R-Hilbert space (H, ∥·∥H , ⟨·, ·⟩H ), real numbers r ∈ [0, 1], η ∈ R, T ∈ (0, ∞), and a generator of a strongly continuous analytic semigroup A : D(A) ⊆ H → H with spectrum(A) ⊆ {z ∈ C : Re(z) < η} r,T r r tA we denote by χr,T A,η ∈ [0, ∞) the real number given by χA,η = supt∈(0,T ] t ∥(η − A) e ∥L(H) (cf., e.g., [21, 2 Lemma 11.36]). We denote by B : (0, ∞) → (0, ∞) the function which satisfies for all x, y ∈ (0, ∞) that ∫ 1 (x−1) (y−1) B(x, y) = 0 t (1 − t) dt (Beta function). We denote by Eα,β : [0, ∞) → [0, ∞), α, β ∈ ((−∞, 1), the ∑∞ ∏n−1 functions which satisfy for all α, β ∈ (−∞, 1), x ∈ [0, ∞) that Eα,β [x] = 1 + n=1 xn k=0 B 1 − β, k(1 − ) β) + 1 − α (generalized exponential function; cf., e.g., Exercise 3 in Chapter 7 in Henry [13], (1.0.3) in Chapter 1 in Gorenflo et al. [11], and (16) in [1]). For real numbers T ∈ (0, ∞), η ∈ R, p ∈ [1, ∞), a ∈ [0, 1), b ∈ [0, 1/2), λ ∈ (−∞, 1), an R-Hilbert space (H, ∥·∥H , ⟨·, ·⟩H ), and a generator A : D(A) ⊆ H → H of a strongly continuous analytic semigroup with spectrum(A) ⊆ {z ∈ C : Re(z) < η} we denote by a,b,λ ˆ ∈ [0, ∞) that ΘA,η,p,T : [0, ∞)2 → [0, ∞] the function which satisfies for all L, L a,b,λ ˆ ΘA,η,p,T (L, L) ⎧ ⏐ [⏐ a,T √ (1−a) √ ⏐2 ]⏐⏐1/2 √ ⏐ ⎪ ⏐ ⏐ ⏐ χA,η L√ 2 T b,T ˆ ⎪ (1−2b) ⏐ ⎪ 2 E + χ L p (p − 1) T ⏐ ⏐ ⎨ ⏐ 2λ,max{a,2b} ⏐ A,η 1−a [ ] = (1−a) ⎪ Eλ,a χa,T ⎪ A,η L T ⎪ ⎩ ∞

) ( ˆ ∈ −∞, 1 × (0, ∞) : (λ, L) 2 ˆ=0 :L : otherwise

(10) ( ( )) ∗ (see (17) in [1]). We denote {{ by Πk , Πk ∈ }}P P P(N) , k ∈ N0 , the sets which satisfy for all k ∈ N that ∗ ∗ Π0 = Π0 = ∅, Πk = Πk \ {1, 2, . . . , k} , and { [ ] } ⋃ Πk = A ⊆ P(N) : [∅ ̸∈ A] ∧ a = {1, 2, . . . , k} ∧ [∀ a, b ∈ A : (a ̸= b ⇒ a ∩ b = ∅)] (11) a∈A

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

132

{{ }} {{ } { }} (cf., e.g., [6, Theorem 2]). Observe, for example, that Π0 = ∅, Π1 = {1} , Π2 = {1, 2} , {1}, {2} , {{ } { } { } { } { }} and Π3 = {1, 2, 3} , {1, 2}, {3} , {1, 3}, {2} , {1}, {2, 3} , {1}, {2}, {3} and note that for every k ∈ N it holds that Πk is the set of all partitions of {1, 2, . . . , k}. For a natural number k ∈ N and a set ϖ ∈ Πk ( ) ( ) ( ϖ ) ϖ we denote by I1ϖ , I2ϖ , . . . , I# ∈ ϖ the sets which satisfy that min I1ϖ < min I2ϖ < · · · < min I# . ϖ ϖ For a natural number k ∈ N, a set ϖ ∈ Πk , and a natural number i ∈ {1, 2, . . . , #ϖ } we denote by ϖ ϖ ϖ ϖ ϖ ϖ ϖ Ii,1 , Ii,2 , . . . , Ii,# ϖ ∈ Ii the natural numbers which satisfy that Ii,1 < Ii,2 < · · · < Ii,# ϖ . For a measure I

I

i

i

space (Ω , F, µ), a measurable space (S, S), a set R, and a function f : Ω → R we denote by [f ]µ,S the set given by [f ]µ,S = {g ∈ M(F, S) : (∃ A ∈ F : µ(A) = 0 and {ω ∈ Ω : f (ω) ̸= g(ω)} ⊆ A)} .

(12)

2. Stochastic evolution equations with smooth coefficients 2.1. Setting Consider the notation in Section 1.1, let T ∈ (0, ∞), η ∈ R, let (H, ∥·∥H , ⟨·, ·⟩H ) and (U, ∥·∥U , ⟨·, ·⟩U ) be separable R-Hilbert spaces with #H > 1, let (Ω , F, P) be a probability space with a normal filtration (Ft )t∈[0,T ] , let (Wt )t∈[0,T ] be an IdU -cylindrical (Ω , F, P, (Ft )t∈[0,T ] )-Wiener process, let A : D(A) ⊆ H → H be a generator of a strongly continuous analytic semigroup with spectrum(A) ⊆ {z ∈ C : Re(z) < η}, let (Hr , ∥·∥Hr , ⟨·, ·⟩Hr ), r ∈ R, be a family of interpolation spaces associated to η − A, for every k ∈ N, # ϖ +1

k+1 be the function which satisfies for all u = ϖ ∈ Πk , i ∈ {1, 2, . . . , #ϖ } let [·]ϖ → H Ii i : H [ ] k+1 ϖ ϖ , uI ϖ , . . . , uI ϖ ), let [·] : M(Pred((Ft )t∈[0,T ] ), B(H)) → (u0 , u1 , . . . , uk ) ∈ H that [u]i = (u0 , uIi,1 i,2 i,#I ϖ i ( ) P M(Pred((Ft )t∈[0,T ] ), B(H)) be the function which satisfies for all X ∈ M(Pred((Ft )t∈[0,T ] ), B(H)) that [ ] { } [X] = Y ∈ M(Pred((Ft )t∈[0,T ] ), B(H)) : inf t∈[0,T ] P(Yt = Xt ) = 1 , for every p ∈ (0, ∞) let Lp { } and Lp be the sets given by Lp = X ∈ M(Pred((Ft )t∈[0,T ] ), B(H)) : supt∈[0,T ] ∥Xt ∥Lp (P;H) < ∞ and {[ ] } Lp = [X] : X ∈ Lp and let ∥·∥Lp : Lp → [0, ∞) be the function which satisfies for all X ∈ Lp [ ] that  [X] Lp = supt∈[0,T ] ∥Xt ∥Lp (P;H) , and for every separable R-Banach space (V, ∥·∥V ) and every a ∈ R, ∫b b ∈ (a, ∞), A ∈ B(R), X ∈ M(B(A)⊗F, B(V )) with (a, b) ⊆ A let a Xs ds ∈ {[Y ]P,B(V ) : Y ∈ M(F, B(V ))} [∫ b ] ∫b be the set given by a Xs ds = a 1{∫ b ∥Xu ∥V du<∞} Xs ds P,B(V ) . a

2.2. Differentiability with respect to the initial values Theorem 2.1 (Differentiability with Respect to the Initial Value). Assume the setting in Section 2.1, let n ∈ N, F ∈ Cbn (H, H), B ∈ Cbn (H, HS(U, H)), α ∈ [0, 1), β ∈ [0, 1/2), and for every k ∈ N, ∑ δ = (δ1 , δ2 , . . . , δk ) ∈ Rk , J ∈ P(R) let ιδJ ∈ R be the real number given by ιδJ = i∈J∩{1,2,...,k} δi − 1 1[2,∞) (#J∩{1,2,...,k} ) min{1 − α, /2 − β}. Then (i) there exist up-to-modifications unique (Ft )t∈[0,T ] / B(H)-predictable stochastic processes X k,u : [0, T ] × Ω → H, u ∈ H k+1 , k ∈ {0, 1, . . . , n}, which fulfill for all k ∈ {0, 1, . . . , n}, p ∈ [2, ∞), u = p (u0 , u1 , . . . , uk ) ∈ H k+1 , t ∈ [0, T ] that sups∈[0,T ] E[∥Xsk,u ∥H ] < ∞ and [Xtk,u − etA 1{0,1} (k) uk ]P,B(H)

∫

t (t−s)A

e

=

[

1{0} (k) F (Xs0,u0 )

+

0

∑

F

(#ϖ )

(Xs0,u0 )

(

#I ϖ ,[u]ϖ 1

Xs

1

#I ϖ ,[u]ϖ 2

, Xs

2

#I ϖ

, . . . , Xs

#ϖ

,[u]ϖ #

ϖ

)

] ds

ϖ∈Πk

∫

t (t−s)A

e

+ 0

[

1{0} (k) B(Xs0,u0 )

+

∑

B

(#ϖ )

(Xs0,u0 )

(

#I ϖ ,[u]ϖ 1

Xs

1

#I ϖ ,[u]ϖ 2

, Xs

2

#I ϖ

, . . . , Xs

#ϖ

,[u]ϖ #

ϖ

)

] dWs ,

ϖ∈Πk

(13)

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

(ii) for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), δ = (δ1 , δ2 , . . . , δk ) ∈ [0, 1/2)k with ⎤ ⎡ k,u ιδ N t ∥X ∥Lp (P;H) ⎦ sup ⎣ ∏k t sup k [i] u=(u0 ,u1 ,...,uk )∈(×i=0 H ) t∈(0,T ] i=1 ∥ui ∥H−δ [i

133

∑k

i=1 δi

< 1/2 it holds that

α,β,ιδ

N 1 ,T ≤ ΘA,η,p,T (|F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) ) χδA,η 1{1} (k) b

b

[ ( ) ∑k k + max{T , 1} χα,T B 1 − α, 1 − δ ∥F ∥C k (H,H−α ) i A,η i=1 b √ ] ( ) ∑k p (p−1) β,T B 1 − 2β, 1 − 2 i=1 δi ∥B∥C k (H,HS(U,H−β )) + χA,η 2 b

·

[

∑ ∏

sup

sup

ϖ∈Πk∗ I∈ϖ u=(ui )i∈I∪{0} ∈(×i∈I∪{0} H

[i] ) t∈(0,T ]

ιδ I

#I ,u

t ∥Xt ∏

i∈I

∥Lp#ϖ (P;H)

]] < ∞,

∥ui ∥H−δ

(14)

i

( [ ] ) (iii) for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), x ∈ H it holds that H k ∋ u ↦→ [X k,(x,u) ] ∈ Lp ∈ L(k) (H, Lp ), ∑k (iv) for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), δ = (δ1 , δ2 , . . . , δk ) ∈ [0, 1/2)k with i=1 δi < 1/2, |F |Lipk (H,H−α ) < ∞, and |B|Lipk (H,HS(U,H−β )) < ∞ it holds that ⎡ sup

sup

sup ⎣

x,y∈H, u=(u ,u ,...,u )∈(H\{0})k t∈(0,T ] 1 2 k x̸=y (δ,0) α,β,ιN

≤ max{T k , 1} ΘA,η,p,T (

(

⎤ k,(y,u) − Xt ∥Lp (P;H) ⎦ ∏k ∥x − y∥H i=1 ∥ui ∥H−δ

(δ,0)

tιN

k,(x,u)

∥Xt

i

|F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) b

b

α,β,0 χ0,T A,η ΘA,η,p(k+1),T |F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β ))

(

·

b

·

sup

sup

ϖ∈Πk I∈ϖ u=(ui )i∈I∪{0} ∈(×i∈I∪{0} H

[i] ) t∈(0,T ]

+

ϖ∈Πk∗ I∈ϖ

sup

sup

# ,u

∏

sup

sup

sup

]

i

ι

t

(δ,0) I∪{k+1}

x,y∈H, u=(u ) # t∈(0,T ] i i∈I ∈(H\{0}) I x̸=y

[ ·

δ

tιI ∥Xt I ∥Lp(#ϖ +1) (P;H) ∏ i∈I ∥ui ∥H−δ

[ ∑ ∑

)

b

[ ∑ ∏

)

#I ,(x,u)

# ,(y,u)

− Xt I ∥Lp#ϖ (P;H) ∏ ∥x − y∥H i∈I ∥ui ∥H−δ

∥Xt

]

i

ιδ J

t

# ,u ∥Xt J ∥Lp#ϖ (P;H)

])

∏

i∈J ∥ui ∥H−δi [ ( ) ∑k · χα,T B 1 − α, 1 − δ ∥F ∥Lipk (H,H−α ) i A,η i=1 √ ] ) ( ∑k p (p−1) β,T + χA,η B 1 − 2β, 1 − 2 i=1 δi ∥B∥Lipk (H,HS(U,H−β )) < ∞, 2 J∈ϖ\{I} u=(ui )i∈J∪{0} ∈(×i∈J∪{0} H

[i] ) t∈(0,T ]

(15)

( ( [ ] ) (v) for all k ∈) {1, 2, . . . , n}, p ∈ [2, ∞) it holds that H ∋ x ↦→ H k ∋ u ↦→ [X k,(x,u) ] ∈ Lp ∈ L(k) (H, Lp ) ∈ C(H, L(k) (H, Lp )), (vi) for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), x ∈ H it holds that ⎧ 0,x+uk 1,(x,uk ) ∥Xt −Xt0,x −Xt ∥Lp (P;H) ⎪ ⎪ sup =0 :k=1 ⎨ lim sup ∥uk ∥H H\{0}∋uk →0 t∈[0,T ]

⎪ ⎪ ⎩ lim sup

sup

sup

H\{0}∋uk →0 u=(u1 ,u2 ,...,uk−1 )∈(H\{0})k−1 t∈[0,T ]

k−1,(x+uk ,u) k,(x,u,uk ) k−1,(x,u) −Xt −Xt ∥Lp (P;H) k ∥ui ∥H i=1

∥Xt

∏

( [ ] ) (vii) for all p ∈ [2, ∞) it holds that H ∋ x ↦→ [X 0,x ] ∈ Lp ∈ Cbn (H, Lp ),

(16)

=0

: k > 1,

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

134

(viii) for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), x, u1 , u2 , . . . , uk ∈ H it holds that dk dxk

[ 0,x ]) ( [ ] )(k) [X ] (u1 , u2 , . . . , uk ) = H ∋ y ↦→ [X 0,y ] ∈ Lp (x)(u1 , u2 , . . . , uk ) (17) [ k,(x,u ,u ,...,u ) ] 1 2 k = [X ] , ( ) (ix) for all p ∈ [2, ∞), t ∈ [0, T ] it holds that H ∋ x ↦→ [Xt0,x ]P,B(H) ∈ Lp (P; H) ∈ Cbn (H, Lp (P; H)), and (x) for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), x, u1 , u2 , . . . , uk ∈ H, t ∈ [0, T ] it holds that (

(

) dk [X 0,x ]P,B(H) (u1 , u2 , . . . , uk ) dxk ( t )(k) = H ∋ y ↦→ [Xt0,y ]P,B(H) ∈ Lp (P; H) (x)(u1 , u2 , . . . , uk )

k,(x,u1 ,u2 ,...,uk )

= [Xt

(18) ]P,B(H) .

Proof . Throughout this proof let r0 , r1 ∈ [0, 1) be the real numbers given by r0 = α and r1 = β, let 0k ∈ Rk , k ∈ N, be the vectors which satisfy for all k ∈ N that 0k = (0, 0, . . . , 0), let (Vl,r , ∥·∥Vl,r , ⟨·, ·⟩Vl,r ), l ∈ {0, 1}, r ∈ [0, ∞), be the R-Hilbert spaces which satisfy for all r ∈ [0, ∞) that (V0,r , ∥·∥V0,r , ⟨·, ·⟩V0,r ) = (H−r , ∥·∥H−r , ⟨·, ·⟩H−r )

(19)

(V1,r , ∥·∥V1,r , ⟨·, ·⟩V1,r ) = (HS(U, H−r ), ∥·∥HS(U,H−r ) , ⟨·, ·⟩HS(U,H−r ) ),

(20)

and

let Gl : H → Vl,0 , l ∈ {0, 1}, be the functions given by G0 = F and G1 = B, let ⌊·⌋ : R → R and ⌈·⌉ : R → R be the functions which satisfy for all t ∈ R that ⌊t⌋ = max ((−∞, t] ∩ {0, 1, −1, 2, −2, . . .}) = max ((−∞, t] ∩ Z)

(21)

⌈t⌉ = min ([t, ∞) ∩ {0, 1, −1, 2, −2, . . .}) = min ([t, ∞) ∩ Z) ,

(22)

and

let θlm : H m+1 → H m , m ∈ N, l ∈ {0, 1}, be the functions which satisfy for all l ∈ {0, 1}, m ∈ N, u = (u0 , u1 , . . . , um ) ∈ H m+1 that { u0 + lu1 :m=1 θlm (u) = (23) (u0 + lum , u1 , u2 , . . . , um−1 ) : m > 1, and let Dk ∈ P(Rk ), k ∈ N, be the sets which satisfy for all k ∈ N that { } k ∑ Dk = (δ1 , δ2 , . . . , δk ) ∈ [0, 1/2)k : δi < 1/2 .

(24)

i=1

Next we claim that for every k ∈ {1, 2, . . . , n} there exist up-to-modifications unique (Ft )t∈[0,T ] /B(H)predictable stochastic processes X l,u : [0, T ] × Ω → H, u ∈ H l+1 , l ∈ {0, 1, . . . , k}, which fulfill for all p l ∈ {0, 1, . . . , k}, p ∈ [2, ∞), u = (u0 , u1 , . . . , ul ) ∈ H l+1 , t ∈ [0, T ] that sups∈[0,T ] E[∥Xsl,u ∥H ] < ∞ and [Xtl,u − etA 1{0,1} (l) ul ]P,B(H)

∫ =

t

[

e(t−s)A 1{0} (l) F (Xs0,u0 ) +

0

∑

#I ϖ ,[u]ϖ 1

F (#ϖ ) (Xs0,u0 ) Xs

(

1

#I ϖ ,[u]ϖ 2

, Xs

2

#I ϖ

, . . . , Xs

#ϖ

,[u]ϖ #

ϖ

)

] ds

ϖ∈Πl

∫ + 0

t

[

e(t−s)A 1{0} (l) B(Xs0,u0 ) +

∑

#I ϖ ,[u]ϖ 1

B (#ϖ ) (Xs0,u0 ) Xs

(

1

#I ϖ ,[u]ϖ 2

, Xs

2

#I ϖ

, . . . , Xs

#ϖ

,[u]ϖ #

ϖ

)

] dWs .

ϖ∈Πl

(25)

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

135

We now prove (25) by induction on k ∈ {1, 2, . . . , n}. For the base case k = 1 note that, e.g., item (i) of Corollary 2.10 in [1] (with H = H, U = U , T = T , η = η, α = 0, β = 0, W = W , A = A, F = F , B = B, δ = 0 in the notation of Corollary 2.10 in [1]) ensures the existence of up-to-modifications unique (Ft )t∈[0,T ] /B(H)-predictable stochastic processes X 0,x : [0, T ] × Ω → H, x ∈ H, which fulfill for p all p ∈ [2, ∞), x ∈ H, t ∈ [0, T ] that sups∈[0,T ] E[∥Xs0,x ∥H ] < ∞ and [Xt0,x

∫

tA

t

− e x]P,B(H) =

e

(t−s)A

F (Xs0,x ) ds

t

∫

e(t−s)A B(Xs0,x ) dWs .

+

(26)

0

0

Next we note that for all l ∈ {0, 1}, p ∈ [2, ∞), u ∈ H, Y, Z ∈ Lp (P; H), t ∈ (0, T ] it holds that ∥G′l (Xt0,u )Y − G′l (Xt0,u )Z∥Lp (P;Vl,0 ) ≤ |Gl |C 1 (H,Vl,0 ) ∥Y − Z∥Lp (P;H) ∥G′l (Xt0,u )0∥Lp (P;Vl,0 )

and

(27)

b

= 0.

This allows us to apply item (i) of Theorem 2.9 in [1] (with H = H, U = U , T = T , η = η, p = p, ˆ 0 = 0, L1 = |B| 1 ˆ α = 0, α ˆ = 0, β = 0, βˆ = 0, L0 = |F |C 1 (H,H) , L Cb (H,HS(U,H)) , L1 = 0, W = W , A = A, b ) ( ( 0,u0 ′ F = [0, T)] × Ω × H ∋ (t, ω, x) ↦→ F (Xt (ω)) x ∈ H , B = [0, T ] × Ω × H ∋ (t, ω, x) ↦→ B ′ (Xt0,u0 (ω)) x ∈ HS(U, H) , δ = 0, λ = 0, ξ = (Ω ∋ ω ↦→ u1 ∈ H) for u0 , u1 ∈ H, p ∈ [2, ∞) in the notation of Theorem 2.9 in [1]) to obtain that there exist up-to-modifications unique (Ft )t∈[0,T ] /B(H)-predictable stochastic processes X 1,u : [0, T ] × Ω → H, u ∈ H 2 , which fulfill for all p ∈ [2, ∞), u = (u0 , u1 ) ∈ H 2 , t ∈ [0, T ] that p sups∈[0,T ] E[∥Xs1,u ∥H ] < ∞ and [Xt1,u − etA u1 ]P,B(H) =

t

∫

e(t−s)A F ′ (Xs0,u0 ) Xs1,u ds +

∫

t

e(t−s)A B ′ (Xs0,u0 ) Xs1,u dWs .

(28)

0

0

This and (26) prove (25) in the base case k = 1. For the induction step {1, 2, . . . , n − 1} ∋ k → k + 1 ∈ {2, 3, . . . , n} we introduce more notation. Assume that there exists a natural number k ∈ {1, 2, . . . , n − 1} such that (25) holds for k = k, let X l,u : [0, T ] × Ω → H, u ∈ H l+1 , l ∈ {2, 3, . . . , k} = N ∩ [2, ∞) ∩ [1, k], be the family of up-to-modifications unique (Ft )t∈[0,T ] /B(H)-predictable stochastic processes which fulfill for p all l ∈ {2, 3, . . . , k}, p ∈ [2, ∞), u = (u0 , u1 , . . . , ul ) ∈ H l+1 , t ∈ [0, T ] that sups∈[0,T ] E[∥Xsl,u ∥H ] < ∞ and [Xtl,u ]P,B(H) =

ϖ

t

∫

e(t−s)A

0 t

∫

ϖ∈Πl

e(t−s)A

+

#I ϖ ,[u]# ) #I ϖ ,[u]ϖ ( #I ϖ ,[u]ϖ ϖ 2 1 ds , Xs 2 , . . . , Xs #ϖ F (#ϖ ) (Xs0,u0 ) Xs 1

∑

0

ϖ

∑

#I ϖ ,[u]# ) #I ϖ ,[u]ϖ ( #I ϖ ,[u]ϖ ϖ 1 2 dWs , B (#ϖ ) (Xs0,u0 ) Xs 1 , Xs 2 , . . . , Xs #ϖ

(29)

ϖ∈Πl

let Glu : [0, T ] × Ω × H → Vl,0 , u ∈ H k+2 , l ∈ {0, 1}, be the functions which satisfy for all l ∈ {0, 1}, u = (u0 , u1 , . . . , uk+1 ) ∈ H k+2 , t ∈ [0, T ], x ∈ H that ϖ

Glu (t, x) = G′l (Xt0,u0 ) x +

∑

(#ϖ )

Gl

#I ϖ ,[u]# ) #I ϖ ,[u]ϖ ( #I ϖ ,[u]ϖ ϖ 1 2 (Xt0,u0 ) Xt 1 , Xt 2 , . . . , X t #ϖ ,

(30)

∗ ϖ∈Πk+1

¯ u,p ∈ [0, ∞), u ∈ H k+2 , p ∈ [2, ∞), l ∈ {0, 1}, be the real numbers which satisfy for all l ∈ {0, 1}, and let L l p ∈ [2, ∞), u ∈ H k+2 that ¯ u,p = L l

∑ ∗ ϖ∈Πk+1

|Gl |C #ϖ (H,V b

l,0 )

# ϖ ∏

[ #I ϖ ,[u]ϖ ] i  [X i ] Lp#ϖ .

(31)

i=1

Next we note that H¨ older’s inequality implies for all l ∈ {0, 1}, p ∈ [2, ∞), u = (u0 , u1 , . . . , uk+1 ) ∈ H k+2 , p Y, Z ∈ L (P; H), t ∈ (0, T ] that ∥Glu (t, Y ) − Glu (t, Z)∥Lp (P;Vl,0 ) ≤ |Gl |C 1 (H,Vl,0 ) ∥Y − Z∥Lp (P;H) b

(32)

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

136

and  u  Gl (t, 0) p L (P;Vl,0 ) #I ϖ ,[u]ϖ ∑  (# ) 0,u ( #I ϖ ,[u]ϖ #I ϖ ,[u]ϖ #ϖ ) 1 2 ϖ 0 1 2   p Gl (Xt ) Xt ≤ , Xt , . . . , X t #ϖ L (P;V ∗ ϖ∈Πk+1

∑

≤

∗ ϖ∈Πk+1

# ϖ ∏

|Gl |C #ϖ (H,V

l,0 )

b

#I ϖ ,[u]ϖ i i

 Xt

 

Lp#ϖ (P;H)

l,0 )

(33)

¯ u,p . ≤L l

i=1

We can hence apply item (i) of Theorem 2.9 in [1] (with H = H, U = U , T = T , η = η, p = p, α = 0, α ˆ = 0, ˆ1 = L ¯ u,p , W = W , A = A, F = G u , ˆ0 = L ¯ u,p , L1 = |B| 1 ,L β = 0, βˆ = 0, L0 = |F |C 1 (H,H) , L 0 1 0 C (H,HS(U,H)) b b B = G1u , δ = 0, λ = 0, ξ = (Ω ∋ ω ↦→ 0 ∈ H) for u ∈ H k+2 , p ∈ [2, ∞) in the notation of Theorem 2.9 in [1]) to obtain that there exist up-to-modifications unique (Ft )t∈[0,T ] /B(H)-predictable stochastic processes X k+1,u : [0, T ] × Ω → H, u ∈ H k+2 , which fulfill for all p ∈ [2, ∞), u = (u0 , u1 , . . . , uk+1 ) ∈ H k+2 , t ∈ [0, T ] p that sups∈[0,T ] E[∥Xsk+1,u ∥H ] < ∞ and ∫

[Xtk+1,u ]P,B(H)

e

=

(t−s)A

0

t

∫ 0

G0u (s, Xsk+1,u ) ds

t

∫

e(t−s)A G1u (s, Xsk+1,u ) dWs

+ 0

∑

e(t−s)A

=

t

ϖ

#I ϖ ,[u]# ) #I ϖ ,[u]ϖ ( #I ϖ ,[u]ϖ ϖ 2 1 ds , Xs 2 , . . . , X s #ϖ F (#ϖ ) (Xs0,u0 ) Xs 1

ϖ∈Πk+1

∫ +

t

e(t−s)A

0

∑

(34)

ϖ

#I ϖ ,[u]# ) #I ϖ ,[u]ϖ ( #I ϖ ,[u]ϖ ϖ 1 2 dWs . B (#ϖ ) (Xs0,u0 ) Xs 1 , Xs 2 , . . . , Xs #ϖ

ϖ∈Πk+1

This proves (25) in the case k + 1. Induction hence establishes (25). The proof of item (i) is thus completed. For our proof of items (ii)–(x) we introduce further notation. Let X k,u : [0, T ] × Ω → H, u ∈ H k+1 , k ∈ {0, 1, . . . , n}, be (Ft )t∈[0,T ] /B(H)-predictable stochastic processes which fulfill for all k ∈ {0, 1, . . . , n}, p p ∈ [2, ∞), u = (u0 , u1 , . . . , uk ) ∈ H k+1 , t ∈ [0, T ] that sups∈[0,T ] E[∥Xsk,u ∥H ] < ∞ and [Xtk,u − etA 1{0,1} (k) uk ]P,B(H)

∫ =

t

[

e(t−s)A 1{0} (k) F (Xs0,u0 ) +

0

∑

#I ϖ ,[u]ϖ 1

F (#ϖ ) (Xs0,u0 ) Xs

(

1

#I ϖ ,[u]ϖ 2

, Xs

2

#I ϖ ,[u]ϖ #ϖ

, . . . , Xs

#ϖ

)

] ds

ϖ∈Πk

∫

t

e

+

(t−s)A

[

1{0} (k) B(Xs0,u0 )

+

0

∑

B

(#ϖ )

(Xs0,u0 )

(

#I ϖ ,[u]ϖ #I ϖ ,[u]ϖ #I ϖ ,[u]ϖ #ϖ 1 2 Xs 1 , Xs 2 , . . . , Xs #ϖ

)

] dWs ,

ϖ∈Πk

(35)

( ) let Lδϖ,p ∈ [0, ∞], ϖ ∈ P P({1, 2, . . . , k}) \ {∅} , δ ∈ Dk , p ∈ (0, ∞), k ∈ {1, 2, . . . , n}, be the extended ( real numbers which ) satisfy for all k ∈ {1, 2, . . . , n}, p ∈ (0, ∞), δ = (δ1 , δ2 , . . . , δk ) ∈ Dk , ϖ ∈ P P({1, 2, . . . , k}) \ {∅} \ {∅} that Lδ∅,p = 1 and [ Lδϖ,p

=

∏

sup

sup

I∈ϖ u=(ui )i∈I∪{0} ∈(×i∈I∪{0} H

[i] ) t∈(0,T ]

δ

] ∥Lp (P;H) , i∈I ∥ui ∥H−δ #I ,u

tιI ∥Xt ∏

(36)

i

˜ p ∈ [0, ∞], p ∈ (0, ∞), be the extended real numbers which satisfy for all p ∈ (0, ∞) that let L [ ˜ p = sup L

sup

sup

u0 ∈H u1 ∈H\{0} t∈(0,T ]

] ∥Xt0,u0 +u1 − Xt0,u0 ∥Lp (P;H) , ∥u1 ∥H

(37)

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

137

ˆ δ,u,p ∈ [0, ∞], u ∈ H k+1 , δ ∈ Dk , p ∈ (0, ∞), l ∈ {0, 1}, k ∈ {1, 2, . . . , n}, be the extended real numbers let L k,l which satisfy for all k ∈ {1, 2, . . . , n}, l ∈ {0, 1}, p ∈ (0, ∞), δ ∈ Dk , u ∈ H k+1 that ˆ δ,u,p = |T ∨ 1|⌊k/2⌋ min{1−α,1/2−β} L k,l # ϖ ] [ ιδϖ ∑ ∏ #I ϖ ,[u]ϖ i |Gl |C #ϖ (H,V ) · sup t Ii ∥Xt i ∥Lp#ϖ (P;H) , l,rl

b

ϖ∈Πk∗

(38)

i=1 t∈(0,T ]

for every k ∈ {1, 2, . . . , n}, l ∈ {0, 1}, u = (u0 , u1 , . . . , uk ) ∈ H k+1 let Gu k,l : [0, T ] × Ω × H → Vl,0 and ¯ u : [0, T ] × Ω × H → Vl,0 be the functions which satisfy for all t ∈ [0, T ], x ∈ H that G k,l ϖ

0,u0 ′ Gu )x + k,l (t, x) = Gl (Xt

∑

(#ϖ )

Gl

#I ϖ ,[u]# ) #I ϖ ,[u]ϖ ( #I ϖ ,[u]ϖ ϖ 1 2 (Xt0,u0 ) Xt 1 , Xt 2 , . . . , X t #ϖ

(39)

ϖ∈Πk∗

and ¯ u (t, x) G k,l ⎧∫ 1 ) ( ⎪ ⎪ G′l Xt0,u0 + ρ[Xt0,u0 +u1 − Xt0,u0 ] x dρ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪G′l (Xt0,u0 ) x ⎪ ∫ 1 ⎪ ⎪ k ) ( 0,u0 ⎪ 0,u0 +uk 0,u +u 0,u0 )( k−1,θ1 (u) ⎪ ′′ ⎪ + G X + ρ[X − X ] X , Xt 0 k − Xt0,u0 dρ ⎪ t t t t l ⎪ ⎪ 0 [∫ ⎪ ⎪ 1 ⎪ ∑ )( #I ϖ ,[θ1k (u)]ϖ ⎪ 1 0,u +u (# +1) ( 0,u0 ⎨ + Gl ϖ Xt + ρ[Xt 0 k − Xt0,u0 ] Xt 1 , = 0 ∗ ϖ∈Π ⎪ k−1 ⎪ ⎪ ⎪ #I ϖ ,[θ1k (u)]ϖ ⎪ #I ϖ ,[θ1k (u)]ϖ ) #ϖ 2 ⎪ 0,u +u #ϖ 2 ⎪ ⎪ X , Xt 0 k − Xt0,u0 dρ , . . . , X t t ⎪ ⎪ ⎪ #I ϖ ,[θ1k (u)]ϖ ⎪ #I ϖ ,[θ1k (u)]ϖ #I ϖ ,[θ1k (u)]ϖ #ϖ ) 1 2 ⎪ (#ϖ ) 0,u0 ( #ϖ 1 2 ⎪ ⎪ + G (X ) X , X , . . . , X t t t t ⎪ l ] ⎪ ⎪ #I ϖ ,[θ0k (u)]ϖ ⎪ #I ϖ ,[θ0k (u)]ϖ #I ϖ ,[θ0k (u)]ϖ ( #ϖ ) ⎪ 1 2 (# ) 0,u # ⎪ − G ϖ (X 0 ) X 1 ⎪ , Xt 2 , . . . , Xt ϖ ⎩ t t l

:k=1

(40) : k > 1,

and for every k ∈ {1, 2, . . . , n}, p ∈ (0, ∞) let dk,p : H 2 → [0, ∞] and d˜k,p : H × (H \ {0}) → [0, ∞] be the functions which satisfy for all x, y ∈ H, v ∈ H \ {0} that [ ] k,(x,u) k,(y,u) ∥Xt − Xt ∥Lp (P;H) (41) dk,p (x, y) = sup sup ∏k u=(u1 ,u2 ,...,uk )∈(H\{0})k t∈(0,T ] i=1 ∥ui ∥H and d˜k,p (x, v) ] [ ⎧ 1,(x,v) ∥Xt0,x+v − Xt0,x − Xt ∥Lp (P;H) ⎪ ⎪ ⎪ sup ⎪ ⎨t∈(0,T ∥v∥H ] [ ] = k−1,(x+v,u) k−1,(x,u) k,(x,u,v) ⎪ p (P;H) ∥X − X − X ∥ ⎪ L t t t ⎪ ⎪ sup sup ∏k−1 ⎩ ∥v∥H i=1 ∥ui ∥H u=(u1 ,u2 ,...,uk−1 )∈(H\{0})k−1 t∈(0,T ]

:k=1 (42) : k > 1.

In the next step we prove item (ii) and the fact that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), x ∈ H, t ∈ [0, T ] it holds that k,(x,u)

H k ∋ u ↦→ [Xt is a k-linear function.

]P,B(H) ∈ Lp (P; H)

(43)

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

138

We prove item (ii) and (43) by induction on k ∈ {1, 2, . . . , n}. Note that for all l ∈ {0, 1}, p ∈ [2, ∞), u ∈ H 2 , Y, Z ∈ Lp (P; H), t ∈ (0, T ] it holds that u ∥Gu 1,l (t, Y ) − G1,l (t, Z)∥Lp (P;Vl,r

l

∥Gu 1,l (t, 0)∥Lp (P;Vl,r

l

)

≤ |Gl |C 1 (H,Vl,r ) ∥Y − Z∥Lp (P;H)

)

= 0.

b

and

l

(44)

Moreover, observe that (35) and (39) ensure that for all u = (u0 , u1 ) ∈ H 2 , t ∈ [0, T ] it holds that [Xt1,u ]P,B(H) = [etA u1 ]P,B(H) +

t

∫

1,u e(t−s)A Gu 1,0 (s, Xs ) ds +

0

t

∫

1,u e(t−s)A Gu 1,1 (s, Xs ) dWs .

(45)

0

Combining (44)–(45) with items (i)–(ii) of Theorem 2.9 in [1] (with H = H, U = U , T = T , η = η, p = p, ˆ 0 = 0, L1 = |B| 1 ˆ α = α, α ˆ = 0, β = β, βˆ = 0, L0 = |F |C 1 (H,H−α ) , L Cb (H,HS(U,H−β )) , L1 = 0, W = W , A = A, b ( ) ( F = [0, T ] × Ω × H ∋ (t, ω, x) ↦→ G)u 1,0 (t, ω, x) ∈ H−α , B = [0, T ] × Ω × H ∋ (t, ω, x) ↦→ (U ∋ u ↦→ 2 Gu 1,1 (t, ω, x) u ∈ H−β ) ∈ HS(U, H−β ) , δ = δ, λ = δ, ξ = (Ω ∋ ω ↦→ u1 ∈ H−δ ) for u = (u0 , u1 ) ∈ H , 1 1 δ ∈ [0, /2), p ∈ [2, ∞) in the notation of Theorem 2.9 in [1]) implies that for all p ∈ [2, ∞), δ ∈ [0, /2) it holds that ] [ 1,(u ,u ) tδ ∥Xt 0 1 ∥Lp (P;H) sup sup sup ∥u1 ∥H−δ u0 ∈H u1 ∈H\{0} t∈(0,T ] [ ] supt∈(0,T ] (tδ ∥etA u1 ∥H ) α,β,δ ≤ ΘA,η,p,T (|F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) ) sup sup (46) b b ∥u1 ∥H−δ u0 ∈H u1 ∈H\{0} [ ] α,β,δ ≤ sup tδ ∥(η − A)δ etA ∥L(H) ΘA,η,p,T (|F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) ) =

b t∈(0,T ] δ,T α,β,δ χA,η ΘA,η,p,T (|F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) ) b b

b

< ∞.

This proves item (ii) in the base case k = 1. Next we observe that (35) shows that for all p ∈ [2, ∞), [ 1,(x,u) p 1,(x,˜ u) p ] ∥H < ∞ and ∥H + ∥Xs x, u, u ˜ ∈ H, λ ∈ R, t ∈ [0, T ] it holds that sups∈[0,T ] E ∥Xs 1,(x,˜ u)

1,(x,u)

+ λXt

[Xt

∫ +

t

]P,B(H) = [etA (u + λ˜ u)]P,B(H)

e(t−s)A F ′ (Xs0,x )(Xs1,(x,u) + λXs1,(x,˜u) ) ds +

t

∫

0

(47) e(t−s)A B ′ (Xs0,x )(Xs1,(x,u) + λXs1,(x,˜u) ) dWs .

0

Item (i) therefore ensures for all x, u, u ˜ ∈ H, λ ∈ R, t ∈ [0, T ] that 1,(x,u+λ˜ u)

[Xt

1,(x,u)

]P,B(H) = [Xt

1,(x,˜ u)

+ λXt

]P,B(H) .

(48)

This proves (43) in the base case k = 1. For the induction step {1, 2, . . . , n − 1} ∋ k → k + 1 ∈ {2, 3, . . . , n} of item (ii) and (43) assume that there exists a natural number k ∈ {1, 2, . . . , n − 1} such that item (ii) and (43) hold for k = 1, k = 2, . . . , k = k. This ensures that for all l ∈ {0, 1}, p ∈ [2, ∞), δ ∈ Dk+1 , u ∈ H k+2 it holds that ˆ δ,u,p < ∞. L k+1,l

(49)

This and H¨ older’s inequality imply that for all l ∈ {0, 1}, p ∈ [2, ∞), δ = (δ1 , δ2 , . . . , δk+1 ) ∈ Dk+1 , u = (u0 , u1 , . . . , uk+1 ) ∈ H k+2 , Y, Z ∈ Lp (P; H), t ∈ (0, T ] it holds that  u  Gk+1,l (t, Y ) − Gu  k+1,l (t, Z) Lp (P;V

l,rl )

≤ |Gl |C 1 (H,Vl,r ) ∥Y − Z∥Lp (P;H) b

l

(50)

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

139

and   u Gk+1,l (t, 0) p L (P;Vl,r ) l #I ϖ ,[u]ϖ ∑  (# ) 0,u ( #I ϖ ,[u]ϖ #I ϖ ,[u]ϖ #ϖ ) 1 2 ϖ 0 1 2   p Gl ≤ (Xt ) Xt , Xt , . . . , Xt #ϖ L (P;V ∗ ϖ∈Πk+1

∑

≤

|Gl |C #ϖ (H,V

t

∗ ϖ∈Πk+1

(δ1 +δ2 +···+δk+1 )

# ϖ ∏

t

⎜ ⎝

#I ϖ ,[u]ϖ i

ιδ Iϖ

⎞

i

∥Xt

i

∥Lp#ϖ (P;H) ⎟ ⎠ )

ιδ ϖ −(δI ϖ +δI ϖ +...+δI ϖ I

i=1

i,1

i

i,2

i,# ϖ

I i (# t ) ϖ ϖ δ ∏ # ,[u] ϖ ι ) I 1[2,∞) (#I ϖ ) min{1−α,1/2−β} i l,rl Iϖ b i i t i ∥Xt ∥Lp#ϖ (P;H) t +δ +···+δ )

(51)

|Gl |C #ϖ (H,V

∑

t(δ1

∗ ϖ∈Πk+1

∑

≤

⎛

l,rl )

b

 #I ϖ ,[u]ϖ i  Xt i  p# L ϖ (P;H)

i=1

|Gl |C #ϖ (H,V

∑

=

l,rl )

b

∗ ϖ∈Πk+1

=

# ϖ ∏

l,rl )

2

k+1

|Gl |C #ϖ (H,V

t(δ1 +δ2 +···+δk+1 )

i=1

l,rl )

b

∗ ϖ∈Πk+1

i=1 # ϖ ∏

⎡ ⎢ ∑ ≤⎣

∗ ϖ∈Πk+1

t

# ϖ ∏

|Gl |C #ϖ (H,V

l,rl )

b

[

#I ϖ ,[u]ϖ i 

 ιδ Iϖ i

Xt

i



Lp#ϖ (P;H)

|T ∨ 1|

]

i

⎤ sup

[

ιδ Iϖ i

s

i=1 s∈(0,T ]

] #I ϖ ,[u]ϖ i ⎥ ∥Xs i ∥Lp#ϖ (P;H) ⎦

⌊(k+1)/2⌋ min{1−α,1/2−β} −(δ1 +δ2 +···+δk+1 )

· |T ∨ 1|

1[2,∞) (#I ϖ ) min{1−α,1/2−β}

t

ˆ δ,u,p t−(δ1 +δ2 +···+δk+1 ) < ∞. =L k+1,l

In addition, note that (35) and (39) demonstrate that for all u ∈ H k+2 , t ∈ [0, T ] it holds that [Xtk+1,u ]P,B(H) =

t

∫

∫

k+1,u e(t−s)A Gu ) ds + k+1,0 (s, Xs

0

t k+1,u e(t−s)A Gu ) dWs . k+1,1 (s, Xs

(52)

0

Combining (49)–(52) of Theorem 2.9 in [1] (with H = H, U = U , T = T , η = η, p = p, ∑k+1 with itemsˆ (i)–(ii) ∑k+1 ˆ0 = L ˆ δ,u,p , L1 = |B| 1 α = α, α ˆ = δ , β = β, β = , L 1 (H,H i=1 i i=1 δi , L0 = |F |Cb Cb (H,HS(U,H−β )) , k+1,0 −α ) ( ) δ,u,p u ˆ ˆ L1 =( Lk+1,1 , W = W , A = A, F = [0, T ] × Ω × H ∋ (t, ω, x) ↦→ Gk+1,0 (t, ω, x) ∈ H−α , ) δ 1 B = [0, T ] × Ω × H ∋ (t, ω, x) ↦→ (U ∋ u ↦→ Gu k+1,1 (t, ω, x) u ∈ H−β ) ∈ HS(U, H−β ) , δ = − /2, λ = ιN , ξ = (Ω ∋ ω ↦→ 0 ∈ H) for u ∈ H k+2 , δ = (δ1 , δ2 , . . . , δk+1 ) ∈ Dk+1 , p ∈ [2, ∞) in the notation of item (ii) of Theorem 2.9 in [1]) ensures that for all p ∈ [2, ∞), δ = (δ1 , δ2 , . . . , δk+1 ) ∈ Dk+1 , u ∈ H k+2 it holds that [ δ ] α,β,ιδ N sup tιN ∥Xtk+1,u ∥Lp (P;H) ≤ ΘA,η,p,T (|F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) ) b b t∈(0,T ] [ ( ) ∑ ˆ δ,u,p (1−α−min{1−α,1/2−β}) B 1 − α, 1 − k+1 δi · χα,T A,η Lk+1,0 T i=1 √ ] ( ∑k+1 ) p (p−1) β,T ˆ δ,u,p (1/2−β−min{1−α,1/2−β}) +χ L T B 1 − 2β, 1 − 2 δi A,η

k+1,1

(k+1)

≤ |T ∨ 1| ∑ ·

∗ ϖ∈Πk+1

+ χβ,T A,η ·

# ϖ ∏

i=1

2

α,β,ιδ N

ΘA,η,p,T (|F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) ) b b [ ∑k+1 ) α,T ( χA,η B 1 − α, 1 − i=1 δi |F |C #ϖ (H,H )

√

−α

b

p (p−1) 2

sup

i=1 t∈(0,T ]

( ∑k+1 ) B 1 − 2β, 1 − 2 i=1 δi |B|C #ϖ (H,HS(U,H b

#I ϖ ,[u]ϖ i i

[ ιδ ϖ t Ii ∥Xt

] ∥Lp#ϖ (P;H) .

(53)

] −β ))

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

140

This implies that for all p ∈ [2, ∞), δ = (δ1 , δ2 , . . . , δk+1 ) ∈ Dk+1 it holds that [ sup

sup

u=(u0 ,u1 ,...,uk+1 )∈(×k+1 H [i] ) t∈(0,T ] i=0 (k+1)

δ

tιN ∥Xtk+1,u ∥Lp (P;H) ∏k+1 i=1 ∥ui ∥H−δ i

α,β,ιδ N , 1} ΘA,η,p,T (|F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) ) b b

≤ max{T ∑ [ α,T ( ∑k+1 ) · χA,η B 1 − α, 1 − i=1 δi |F |C #ϖ (H,H b

∗ ϖ∈Πk+1

+ χβ,T A,η ·

√

]

p (p−1) 2

∏

−α )

(54) ]

( ∑k+1 ) B 1 − 2β, 1 − 2 i=1 δi |B|C #ϖ (H,HS(U,H )) −β b ] [ ιδI # ,u t ∥Xt I ∥Lp#ϖ (P;H) ∏ . sup sup i∈I ∥ui ∥H−δ ∈(× H [i] ) t∈(0,T ]

I∈ϖ u=(ui )i∈I∪{0}

i

i∈I∪{0}

This and the induction hypothesis imply item (ii) in the case k + 1 and thus complete the induction ∗ step for item (ii). In the next step we note that for all λ ∈ R, i ∈ {1, 2, . . . , k + 1}, ϖ ∈ Πk+1 and all (m) (3) (1) (2) (m) k+2 u = (u0 , u1 , . . . , ui−1 , ui , ui+1 , ui+2 , . . . , uk+1 ) ∈ H , m ∈ {1, 2, 3}, with ui = ui + λui it holds that there exists a unique natural number j ∈ {1, 2, . . . , #ϖ } such that there exists a natural number q ∈ {1, 2, . . . , #Ijϖ } such that for all l ∈ {1, 2, . . . , #ϖ } \ {j} it holds that ϖ Ij,q = i,

(2) ϖ [u(1) ]ϖ ]l = [u(3) ]ϖ l = [u l ,

(55)

and ( (1) (2) ϖ , uI ϖ , . . . , uI ϖ ϖ ϖ ϖ [u(3) ]ϖ , ui + λui , uIj,q+1 , uIj,q+2 , . . . , uIj,# j = u0 , uIj,1 j,2 j,q−1

Iϖ j

)

.

(56)

∗ In addition, observe that for all ϖ ∈ Πk+1 , j ∈ {1, 2, . . . , #ϖ } it holds that

#Ijϖ ∈ {1, 2, . . . , k}.

(57)

Moreover, observe that the induction hypothesis establishes that for all m ∈ {1, 2, . . . , k}, p ∈ [2, ∞), x ∈ H, t ∈ [0, T ] it holds that m,(x,u)

H m ∋ u ↦→ [Xt

]P,B(H) ∈ Lp (P; H)

(58)

is an m-linear function. Combining (55) and (56) with (57) hence assures that for all λ ∈ R, i ∈ (m) ∗ {1, 2, . . . , k + 1}, ϖ ∈ Πk+1 , t ∈ [0, T ] and all u(m) = (u0 , u1 , . . . , ui−1 , ui , ui+1 , ui+2 , . . . , uk+1 ) ∈ H k+2 , (3) (1) (2) m ∈ {1, 2, 3}, with ui = ui + λui it holds that there exists a unique natural number j ∈ {1, 2, . . . , #ϖ } such that for all l ∈ {1, 2, . . . , #ϖ } \ {j} it holds that i ∈ Ijϖ ,

#I ϖ ,[u(1) ]ϖ l

Xt

l

#I ϖ ,[u(2) ]ϖ l

= Xt

l

#I ϖ ,[u(3) ]ϖ l

= Xt

l

,

(59)

and #I ϖ ,[u(1) ]ϖ j

[Xt

j

#I ϖ ,[u(2) ]ϖ j

+ λXt

j

#I ϖ ,[u(3) ]ϖ j

]P,B(H) = [Xt

j

]P,B(H) .

(60)

This shows that for all λ ∈ R, l ∈ {0, 1}, i ∈ {1, 2, . . . , k + 1}, t ∈ [0, T ] and all u(m) = (m) (3) (1) (2) (u0 , u1 , . . . , ui−1 , ui , ui+1 , ui+2 , . . . , uk+1 ) ∈ H k+2 , m ∈ {1, 2, 3}, with ui = ui + λui it holds that

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

141

∗ there exist jϖ ∈ {1, 2, . . . , #ϖ }, ϖ ∈ Πk+1 , such that

[ u(3) (1) (2) ] Gk+1,l (t, Xtk+1,u + λXtk+1,u ) P,B(V

l,0 )

[ (1) (2) = G′l (Xt0,u0 )(Xtk+1,u + λXtk+1,u ) +

(#ϖ )

∑

Gl

( #I ϖ ,[u(1) ]ϖ 1 (Xt0,u0 ) Xt 1 ,

∗ ϖ∈Πk+1

#I ϖ ,[u(1) ]ϖ 2

Xt

2

#I ϖ

Xt

jϖ +2

#I ϖ

, . . . , Xt

,[u(1) ]ϖ jϖ +2

jϖ −1

,[u(1) ]ϖ jϖ −1

#I ϖ

#ϖ

, . . . , Xt

#I ϖ ,[u(3) ]ϖ j

,[u(1) ]ϖ #

ϖ

)

ϖ

jϖ

, Xt

#I ϖ

, Xt

jϖ +1

,[u(1) ]ϖ jϖ +1

,

] P,B(Vl,0 )

[ (1) (2) = G′l (Xt0,u0 )(Xtk+1,u + λXtk+1,u ) +

(61)

( #I ϖ ,[u(1) ]ϖ 1 (# ) Gl ϖ (Xt0,u0 ) Xt 1 ,

∑ ∗ ϖ∈Πk+1

#I ϖ ,[u(1) ]ϖ 2

Xt

2

#I ϖ

, . . . , Xt

#I ϖ ,[u(1) ]ϖ jϖ +2 jϖ +2

Xt

jϖ −1

,[u(1) ]ϖ jϖ −1

#I ϖ

, . . . , Xt

#ϖ

#I ϖ ,[u(1) ]ϖ j

,[u(1) ]ϖ #ϖ

)

ϖ

jϖ

, Xt

#I ϖ ,[u(2) ]ϖ j

+ λXt

ϖ

jϖ

#I ϖ

, Xt

jϖ +1

,[u(1) ]ϖ jϖ +1

,

] P,B(Vl,0 )

[ (1) (2) k+1,u(1) k+1,u(2) ] = Gu ) + λ Gu ) P,B(V k+1,l (t, Xt k+1,l (t, Xt

l,0 )

.

This, (52), Itˆ o’s isometry, and, e.g., Lemma 3.1 in Jentzen & Puˇsnik [16] (with (Ω , F, µ) = (Ω , F, P),  ( (2) (3) k+1,u(1) (ω) + λXsk+1,u (ω)) − T = t, Y = Ω × [0, t] ∋ (ω, s) ↦→ e(t−s)A Gu k+1,l (s, ω, Xs (l+1) ) (1) (1) (2) (2) e(t−s)A Gu (s, ω, Xsk+1,u (ω)) − λe(t−s)A Gu (s, ω, Xsk+1,u (ω)) ∈ R , Z = (Ω × [0, t] ∋ k+1,l

k+1,l

Vl,0 (1)

(2)

(ω, s) ↦→ 0 ∈ R) for t ∈ (0, T ], l ∈ {0, 1}, u(3) = (u0 , u1 , . . . , ui−1 , ui + λui , ui+1 , ui+2 , . . . , uk+1 ), (2) (1) u(2) = (u0 , u1 , . . . , ui−1 , ui , ui+1 , ui+2 , . . . , uk+1 ), u(1) = (u0 , u1 , . . . , ui−1 , ui , ui+1 , ui+2 , . . . , uk+1 ) ∈ k+2 H , i ∈ {1, 2, . . . , k + 1}, λ ∈ R in the notation of Lemma 3.1 in Jentzen & Puˇsnik [16]) prove that for all (m) λ ∈ R, i ∈ {1, 2, . . . , k + 1}, t ∈ [0, T ] and all u(m) = (u0 , u1 , . . . , ui−1 , ui , ui+1 , ui+2 , . . . , uk+1 ) ∈ H k+2 , (3) (1) (2) m ∈ {1, 2, 3}, with ui = ui + λui it holds that ∫ t (3) (2) k+1,u(1) k+1,u(2) k+1,u(1) [Xt + λXt ]P,B(H) = e(t−s)A Gu + λXsk+1,u ) ds k+1,0 (s, Xs 0 ∫ t (62) (3) k+1,u(1) k+1,u(2) + e(t−s)A Gu (s, X + λX ) dW . s k+1,1 s s 0

This, (39), and item (i) imply for all λ ∈ R, i ∈ {1, 2, . . . , k + 1}, t ∈ [0, T ] and all u(m) = (m) (3) (1) (2) (u0 , u1 , . . . , ui−1 , ui , ui+1 , ui+2 , . . . , uk+1 ) ∈ H k+2 , m ∈ {1, 2, 3}, with ui = ui + λui that [Xtk+1,u

(3)

(1)

]P,B(H) = [Xtk+1,u

(2)

+ λXtk+1,u

]P,B(H) .

(63)

This proves (43) in the case k + 1 and hence completes the induction step for (43). Induction thus completes the proof of item (ii) and (43). Combining (43) with item (ii) establishes item (iii). Next we prove item ( (iv). We first note that ) item (ii) implies that for all k ∈ {1, 2, . . . , n}, l ∈ {0, 1}, p ∈ (0, ∞), δ ∈ Dk , ϖ ∈ P P({1, 2, . . . , k}) \ {∅} , u ∈ H k+1 it holds that ˆ δ,u,p < ∞. Lδϖ,p + L k,l

(64)

This, the Burkholder–Davis–Gundy type inequality in Lemma 7.7 in Da Prato & Zabczyk [8], (44), (45), (50), (51), (52), and Proposition 2.7 in [1] (with H = H, U = U , T = T , η = η, p = p, α = α,

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

142

ˆ0 = L ˆ 0k ,y,p , L1 = |B| 1 ˆ ˆ 0k ,y,p , α ˆ = 0, β = β, βˆ = 0, L0 = |F |C 1 (H,H−α ) , L Cb (H,HS(U,H−β )) , L1 = Lk,1 k,0 b ( ) ( W = W , A = A, F = [0, T ] × Ω × H ∋ (t, ω, z) ↦→ Gyk,0)(t, ω, z) ∈ H−α , B = [0, T ] × Ω × H ∋ (t, ω, z) ↦→ (U ∋ u ↦→ Gyk,1 (t, ω, z) u ∈ H−β ) ∈ HS(U, H−β ) , δ = 0, Y 1 = X k,x , Y 2 = X k,y , λ = λ for x = (x, u1 , u2 , . . . , uk ), y = (y, u1 , u2 , . . . , uk ) ∈ H k+1 , λ ∈ (−∞, 1/2), p ∈ [2, ∞), k ∈ {1, 2, . . . , n} in the notation of Proposition 2.7 in [1]) show that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), λ ∈ (−∞, 1/2), x = (x, u1 , u2 , . . . , uk ), y = (y, u1 , u2 , . . . , uk ) ∈ H k+1 it holds that   ) α,β,λ ( |F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) sup tλ Xtk,x − Xtk,y Lp (P;H) ≤ ΘA,η,p,T b b t∈(0,T ] [ ∫ t ( )  e(t−s)A Gxk,0 (s, Xsk,x ) − Gyk,0 (s, Xsk,x ) ds · sup tλ   t∈(0,T ] 0 ]  ∫ t ( )  y (t−s)A x k,x k,x  + e Gk,1 (s, Xs ) − Gk,1 (s, Xs ) dWs  0 Lp (P;H) ) ( (65) α,β,λ ≤ ΘA,η,p,T |F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) b { b [ } ∫ ·

tλ χα,T A,η

sup t∈(0,T ]

0

{ + sup t∈(0,T ]

t ∥Gx (s,X k,x )−Gy (s,X k,x )∥ s s Lp (P;H−α ) k,0 k,0 (t−s)α

[ λ

t

χβ,T A,η

p (p−1) 2

∫

k,x

t ∥Gx k,1 (s,Xs

y

k,x

)−Gk,1 (s,Xs

ds

(t−s)2β

0

]1/2 }]

2

)∥ p L (P;HS(U,H−β ))

ds

.

Moreover, observe that (39) ensures that for all k ∈ {1, 2, . . . , n}, l ∈ {0, 1}, x = (x, u1 , u2 , . . . , uk ), y = (y, u1 , u2 , . . . , uk ) ∈ H k+1 , t ∈ [0, T ] it holds that ) ( Gxk,l (t, Xtk,x ) − Gyk,l (t, Xtk,x ) = G′l (Xt0,x ) − G′l (Xt0,y ) Xtk,x #I ϖ ,[y]ϖ ∑ [( (# ) 0,x #I ϖ ,[y]ϖ #I ϖ ,[y]ϖ #ϖ ) 1 2 (#ϖ ) 0,y )( ϖ 1 2 + Gl (Xt ) − Gl (Xt ) Xt , Xt , . . . , X t #ϖ +

ϖ∈Πk∗ #ϖ ∑

(66)

#I ϖ ,[x]ϖ #I ϖ ,[y]ϖ #I ϖ ,[x]ϖ #I ϖ ,[x]ϖ ( #I ϖ ,[x]ϖ i−1 i i 1 2 , (Xt0,x ) Xt 1 − Xt i , Xt 2 , Xt i , . . . , Xt i−1 i=1 ] #I ϖ ,[y]ϖ #I ϖ ,[y]ϖ #I ϖ ,[y]ϖ #ϖ ) i+1 i+2 #ϖ i+1 i+2 Xt , Xt , . . . , Xt . (#ϖ )

Gl

Next note that H¨ older’s inequality demonstrates that for all k ∈ {1, 2, . . . , n}, l ∈ {0, 1}, p ∈ [2, ∞), δ = (δ1 , δ2 , . . . , δk ) ∈ Dk , ϖ ∈ Πk , x ∈ H, y = (y, u1 , u2 , . . . , uk ) ∈ ×ki=0 H [i] , t ∈ (0, T ] it holds that ϖ

#I ϖ ,[y]# ) #I ϖ ,[y]ϖ ( (#ϖ ) 0,x )( #I ϖ ,[y]ϖ ϖ 1 2 (# )  G  (Xt ) − G ϖ (Xt0,y ) Xt 1 , Xt 2 , . . . , Xt #ϖ l

Lp (P;Vl,r )

l

l

∏k

i=1 ∥ui ∥H−δ

i

 (# )  (# ) ≤ Gl ϖ (Xt0,x ) − Gl ϖ (Xt0,y )Lp(#ϖ +1) (P;L(#ϖ ) (H,V

l,rl ))

# ϖ ∏ i=1

#I ϖ ,[y]ϖ i

∥Xt i [∏#I ϖ i

∥Lp(#ϖ +1) (P;H) ] ϖ ∥H ∥u I m=1 −δ ϖ i,m I

[  (# )  (# ) = Gl ϖ (Xt0,x ) − Gl ϖ (Xt0,y )Lp(#ϖ +1) (P;L(#ϖ ) (H,V

l,rl ))

ιδ Iϖ i

] ϖ ∏ ∏ 1 # t

I∈ϖ

t

ιδ I

i=1

i,m

#I ϖ ,[y]ϖ i

∥Xt i [∏#I ϖ i

∥Lp(#ϖ +1) (P;H) ] ϖ m=1 ∥uIi,m ∥H−δ ϖ I

i,m

⌊k/2⌋ min{1−α,1/2−β}

|T ∨ 1| ≤ t(δ1 +δ2 +···+δk )

 (# )  (# ) Lδϖ, p(#ϖ +1) Gl ϖ (Xt0,x ) − Gl ϖ (Xt0,y )Lp(#ϖ +1) (P;L(#ϖ ) (H,V

l,rl ))

. (67)

In addition, H¨ older’s inequality establishes that for all k ∈ {1, 2, . . . , n}, l ∈ {0, 1}, p ∈ [2, ∞), γ ∈ [0, ∞), δ = (δ1 , δ2 , . . . , δk ) ∈ Dk , ϖ ∈ Πk∗ , j ∈ {1, 2, . . . , #ϖ }, x = (x, u1 , u2 , . . . , uk ), y = (y, u1 , u2 , . . . , uk ) ∈

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

143

×ki=0 H [i] , t ∈ (0, T ] it holds that

∏k i=1

#I ϖ ,[x]ϖ #I ϖ ,[x]ϖ #I ϖ ,[y]ϖ #I ϖ ,[x]ϖ  (#ϖ ) 0,x ( #I ϖ ,[x]ϖ j−1 j j 1 2 j−1 j j 2 1 G (X ) X , X , . . . , X , X − X , t t t t t t l

1 ∥ui ∥H

−δi

#I ϖ

Xt

j+1

,[y]ϖ j+1

#I ϖ

j+2

, Xt

,[y]ϖ j+2

#I ϖ

#ϖ

, . . . , Xt

,[y]ϖ #

ϖ

) 

Lp (P;Vl,r ) l

#I ϖ ,[x]ϖ #I ϖ ,[y]ϖ [j−1 ][ # ] i i ϖ ∏ ∥Xt i ∏ ∥Lp#ϖ (P;H) ∥Xt i ∥Lp#ϖ (P;H) ≤ |Gl |C #ϖ (H,V ) ∏#Iiϖ ∏#Iiϖ l,rl b i=1 i=j+1 ϖ ϖ m=1 ∥uIi,m ∥H−δ ϖ m=1 ∥uIi,m ∥H−δ ϖ I

#I ϖ ,[x]ϖ j

·

j

∥Xt

#I ϖ ,[y]ϖ j j

− Xt

∏#I ϖ j

m=1

ϖ ∥H ∥uIj,m −δ

l,rl

b

Iϖ j,m

1

⎢ )⎣

·

t

i=j+1

∥Xt ∏#Iiϖ

i

m=1

i

m=1

j

(δ,0)

]

γ+ι ϖ I ∪{k+1}

∥Lp#ϖ (P;H) t

j

ϖ ∥H ∥uIi,m −δ

(δ,0)

·

j

∏

#I ϖ ,(y,v)

− Xs

∥vi ∥H−δ

i∈Ijϖ

ϖ ∥H ∥uIj,m −δ

b

#I ϖ s∈(0,T ] v=(vi )i∈I ϖ ∈(H\{0}) j j

j

j

∥Lp#ϖ (P;H)

Iϖ j,m

|Gl |C #ϖ (H,V

sup

sup

#I ϖ ,(x,v)

∥Xs

#I ϖ ,[y]ϖ j

− Xt

{

⌈k/2⌉ min{1−α,1/2−β}

γ+ι ϖ I ∪{k+1}

−δI ϖ i,m

i,m

#I ϖ ,[x]ϖ j

∥Xt j ∏#Ijϖ

m=1

Iϖ i,m

|T ∨ 1| ≤ t(γ+δ1 +δ2 +···+δk )

s

(68)

∏

(δ,0) γ+ι ϖ I ∪{k+1} j

#I ϖ ,[y]ϖ i

ιδ Iϖ

# ϖ ∏

#I ϖ ,[x]ϖ [j−1 ιδI ϖ ] i i ∥Lp#ϖ (P;H) 1 ⎥ ∏ t i ∥Xt ⎦ ∏#Iiϖ ιδ I i=1 I∈ϖ\{I ϖ } t ∥uI ϖ ∥H

⎤

t [

i,m

∥Lp#ϖ (P;H)

⎡ = |Gl |C #ϖ (H,V

I

i,m

j

∥Lp#ϖ (P;H)

l,rl )

Lδϖ\{I ϖ }, p#ϖ j

} .

i

Combining (66)–(68) yields that for all k ∈ {1, 2, . . . , n}, l ∈ {0, 1}, p ∈ [2, ∞), γ ∈ [0, ∞), δ = (δ1 , δ2 , . . . , δk ) ∈ Dk , x = (x, u1 , u2 , . . . , uk ), y = (y, u1 , u2 , . . . , uk ) ∈ ×ki=0 H [i] , t ∈ (0, T ] it holds that ⌈k/2⌉ min{1−α,1/2−β} ∥Gxk,l (t, Xtk,x ) − Gyk,l (t, Xtk,x )∥Lp (P;Vl,r ) |T ∨ 1| l ≤ ∏k t(δ1 +δ2 +···+δk ) i=1 ∥ui ∥H−δi ( ∑ [  (#ϖ ) 0,x  (# ) G Lδ · (Xt ) − G ϖ (Xt0,y ) p(# +1) (# ϖ, p(#ϖ +1)

l

l

ϖ

L

ϖ∈Πk

(69)

1 ∑ ∑ + γ t ∗ v=(v ϖ∈Πk I∈ϖ

(δ,0) γ+ι I∪{k+1}

·

s

(P;L

]

ϖ ) (H,V l,rl ))

{ sup i )i∈I

#I ,(x,v)

∥Xs

∏

i∈I

sup

∈(H\{0})#I s∈(0,T ] #I ,(y,v)

− Xs

∥vi ∥H−δ

|Gl |C #ϖ (H,V

∥Lp#ϖ (P;H)

l,rl )

b

Lδϖ\{I}, p#ϖ

}) .

i

This and Minkowski’s inequality imply ∑k that for all k ∈ {1, 2, . . . , n}, l ∈ {0, 1}, p ∈ [2, ∞), δ = (δ1 , δ2 , . . . , δk ) ∈ Dk , γ ∈ [0, 1/2 − i=1 δi ), x = (x, u1 , u2 , . . . , uk ), y = (y, u1 , u2 , . . . , uk ) ∈ ×ki=0 H [i] ,

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

144

t ∈ (0, T ] it holds that

[∫ ( t

∥Gxk,l (s, Xsk,x ) − Gyk,l (s, Xsk,x )∥Lp (P;Vl,r

t

·

w

·

s(δ1 +δ2 +···+δk )

ϖ∈Πk

ϖ∈Π ∗ k γ+ι

≤ |T ∨ 1|⌈k/2⌉ min{1−α, /2−β} 1

ds

∥ui ∥H−δi i=1

s(γ+δ1 +δ2 +···+δk ) (t − s)rl

I∈ϖ

(δ,0) I∪{k+1}

# ,(x,v)

∥Xw I

∏ i∈I

(t −

l

))

s)rl

{

1

∑ ∑

+

]1/(l+1)

)(l+1)

ϖ) (Xs0,y )∥Lp(#ϖ +1) (P;L(#ϖ ) (H,Vl,r ∑ Lδϖ, p(#ϖ +1) ∥Gl(#ϖ ) (Xs0,x ) − G(# l

[∫ ( 0

)

∏k

(t − s)rl

0

l

# ,(y,v)

− Xw I

sup

sup

v=(vi )i∈I ∈(H\{0})#I w∈(0,T ]

l,rl )

b

Lδϖ\{I}, p#ϖ

]1/(l+1)

})(l+1)

∥Lp#ϖ (P;H)

|Gl |C #ϖ (H,V

ds

∥vi ∥H−δi

≤ |T ∨ 1|⌈k/2⌉ min{1−α, /2−β} 1

(

[∫ ( t

∑

·

Lδϖ, p(#ϖ +1)

(#ϖ )

∥Gl

∑ ∑ (1/(l+1)−r −γ−∑k l

t

(Xs0,y )∥Lp(#ϖ +1) (P;L(#ϖ ) (H,Vl,r

l

))

i=1

δi

)[ (

(

B 1 − (l + 1)rl , 1 − (l + 1) γ +

∑k i=1

δi

]1/(l+1)

)(l+1) ds

s(δ1 +δ2 +···+δk ) (t − s)rl

0

ϖ∈Πk

+

(#ϖ )

(Xs0,x ) − Gl

))]1/(l+1)

ϖ∈Π ∗ I∈ϖ k

{ ·

|Gl |C #ϖ (H,V ) Lδϖ\{I}, p#ϖ l,rl b

sup

sup

v=(vi )i∈I ∈(H\{0})#I w∈(0,T ]

w

γ+ι

(δ,0) I∪{k+1}

# ,(x,v)

∥Xw I

∏ i∈I

# ,(y,v)

− Xw I

∥Lp#ϖ (P;H)

}) .

∥vi ∥H−δi

(70)

(δ,0)

Hence, we obtain that for all k ∈ {1, 2, . . . , n}, l ∈ {0, 1}, p ∈ [2, ∞), δ = (δ1 , δ2 , . . . , δk ) ∈ Dk , λ ∈ [ιN , 1/2), ∑k (δ,0) γ ∈ [0, λ − ιN ] ∩ [0, 1/2 − i=1 δi ), x = (x, u1 , u2 , . . . , uk ), y = (y, u1 , u2 , . . . , uk ) ∈ ×ki=0 H [i] it holds that

{ t

sup t∈(0,T ]

λ

rl ,T χA,η

[

p (p−1) 2

[ ( ) ]1/(l+1) } ]l/2 ∫ t ∥Gxk,l (s, Xsk,x ) − Gy (s, Xsk,x )∥Lp (P;V ) (l+1) k,l l,rl ds ∏k r (t − s)

0

l

∥ui ∥H−δi

i=1

( ∑

≤ |T ∨ 1|⌈k/2⌉ min{1−α, /2−β} 1

r ,T

l Lδϖ, p(#ϖ +1) χA,η

ϖ∈Πk

{ · sup

t

λ

[

t∈(0,T ]

+

p (p−1) 2

[ ]l/2 ∫

t

(#ϖ )

∥Gl

(#ϖ )

(Xs0,x ) − Gl

rl ,T χA,η

( T

l,rl ))

s(l+1)(δ1 +δ2 +···+δk ) (t − s)(l+1)rl

0

∑ ∑

(l+1)

(Xs0,y )∥Lp(#ϖ +1) (P;L(#ϖ ) (H,V

∑k

λ+1/(l+1)−rl −γ−

i=1

δi

)[

p (p−1) 2

]l/2 [ (

]1/(l+1) } ds

(

B 1 − (l + 1)rl , 1 − (l + 1) γ +

∑k i=1

δi

))]1/(l+1)

ϖ∈Π ∗ I∈ϖ k

{ ·

sup

sup

v=(vi )i∈I ∈(H\{0})#I w∈(0,T ]

|Gl |C #ϖ (H,V ) Lδϖ\{I}, p#ϖ l,rl b

w

γ+ι

(δ,0) I∪{k+1}

# ,(x,v)

∥Xw I

∏ i∈I

# ,(y,v)

− Xw I

∥vi ∥H−δi

∥Lp#ϖ (P;H)

}) . (71)

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

145

(δ,0) 1 , /2),

This shows that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), δ = (δ1 , δ2 , . . . , δk ) ∈ Dk , λ ∈ [ιN ∑k (δ,0) γ ∈ [0, λ − ιN ] ∩ [0, 1/2 − i=1 δi ), x, y ∈ H it holds that {

1 ∑

r ,T

l tλ χA,η

sup

sup

k l=0 u=(u1 ,u2 ,...,uk )∈(H\{0}) t∈(0,T ]

[∫ ( t

·

k,(x,u)

(x,u)

∥Gk,l (s, Xs

0

[

p (p−1) 2

]l/2

k,(x,u)

(y,u)

) − Gk,l (s, Xs ∏k (t − s)rl i=1 ∥ui ∥H−δ

)∥Lp (P;Vl,r

l

)

)(l+1)

]1/(l+1) } ds

i

⌈k/2⌉ min{1−α,1/2−β}

≤ |T ∨ 1| ( [ { ∫ ∑ α,T δ Lϖ,p(#ϖ +1) χA,η sup tλ · t∈(0,T ]

ϖ∈Πk

+ χβ,T A,η +

−α ))

(t−s)α s(δ1 +δ2 +···+δk )

0

t ∥B (#ϖ ) (Xs0,x )−B (#ϖ ) (Xs0,y )∥2 p(# +1) ϖ L (P;L(#ϖ ) (H,HS(U,H

{ [ ∫ sup tλ p (p−1) 2

t∈(0,T ]

t ∥F (#ϖ ) (Xs0,x )−F (#ϖ ) (Xs0,y )∥ p(# +1) ϖ L (P;L(#ϖ ) (H,H

(t−s)2β s2(δ1 +δ2 +···+δk )

0

{

∑ ∑

sup

sup

# t∈(0,T ] ϖ∈Πk∗ I∈ϖ u=(ui )i∈I ∈(H\{0}) I

[ ( ) ∑k λ+1−α−γ− δ i=1 i |F | #ϖ · χα,T T A,η (H,H C b

+ χβ,T A,η

T

(

λ+1/2−β−γ−

∑k i=1

−β )))

δi

)

γ+ι

Lδϖ\{I},p #ϖ

−α

t

} ds

]1/2 } ] ds

(72)

(δ,0) I∪{k+1} ∥X #I ,(x,u) −X #I ,(y,u) ∥ t t Lp#ϖ (P;H)

∏ i∈I

∥ui ∥H

−δi

( ) ∑k B 1 − α, 1 − γ − δ i i=1 ) √

|B|C #ϖ (H,HS(U,H b

−β ))

p (p−1) 2

) ( )]} ∑k B 1 − 2β, 1 − 2γ − 2 i=1 δi .

Combining (65) with (72) yields that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), δ = (δ1 , δ2 , . . . , δk ) ∈ Dk , ∑k (δ,0) (δ,0) λ ∈ [ιN , 1/2), γ ∈ [0, λ − ιN ] ∩ [0, 1/2 − i=1 δi ), x, y ∈ H it holds that [ sup

sup

k,(x,u)

tλ ∥Xt

∏k

u=(u1 ,u2 ,...,uk )∈(H\{0})k t∈(0,T ] ⌈k/2⌉ min{1−α,1/2−β}

k,(y,u)

− Xt

i=1

α,β,λ ΘA,η,p,T

∥Lp (P;H)

]

∥ui ∥H−δ

i

(

) ≤ |T ∨ 1| |F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) b b } ( [ { ∫ t ∥F (#ϖ ) (Xs0,x )−F (#ϖ ) (Xs0,y )∥ p(# +1) ∑ ϖ L (P;L(#ϖ ) (H,H−α )) α,T δ λ · Lϖ,p(#ϖ +1) χA,η sup t ds α (δ1 +δ2 +···+δk ) t∈(0,T ]

ϖ∈Πk

+ χβ,T A,η

+

{ [ ∫ λ p (p−1) sup t 2

t∈(0,T ]

t ∥B (#ϖ ) (Xs0,x )−B (#ϖ ) (Xs0,y )∥2 p(# +1) ϖ L (P;L(#ϖ ) (H,HS(U,H (t−s)2β s2(δ1 +δ2 +···+δk )

0

{

∑ ∑

sup

sup

# t∈(0,T ] ϖ∈Πk∗ I∈ϖ u=(ui )i∈I ∈(H\{0}) I

[

· χα,T A,η T

(

+ χβ,T A,η

(

T

(t−s) s

0

∑k

λ+1−α−γ−

δ i=1 i

)

|F |C #ϖ (H,H b

λ+1/2−β−γ−

∑k i=1

δi

)

Lδϖ\{I},p #ϖ

−α

]1/2 } ] ds

(73)

(δ,0) γ+ι I∪{k+1} ∥X #I ,(x,u) −X #I ,(y,u) ∥ t t t Lp#ϖ (P;H)

∏ i∈I

∥ui ∥H

−δi

( ) ∑k B 1 − α, 1 − γ − δ i i=1 ) √

|B|C #ϖ (H,HS(U,H b

−β )))

−β ))

p (p−1) 2

) ( )]} ∑k B 1 − 2β, 1 − 2γ − 2 i=1 δi .

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

146

In particular, this shows that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), δ = (δ1 , δ2 , . . . , δk ) ∈ Dk , x, y ∈ H it holds that ⎤ ⎡ (δ,0) k,(x,u) k,(y,u) tιN ∥Xt − Xt ∥Lp (P;H) ⎦ sup ⎣ sup ∏k ∥u ∥ u=(u1 ,u2 ,...,uk )∈(H\{0})k t∈(0,T ] i H i=1 −δ i

⌈k/2⌉ min{1−α,1/2−β}

≤ |T ∨ 1| ( ·

∑

χα,T A,η

Lδϖ,p(#ϖ +1) {

+ χβ,T A,η

t

sup

(δ,0)

ιN

(δ,0)

(

[

(δ,0)

ιN

sup

ι

Lδϖ\{I},p #ϖ

[ ( (δ,0) ) ∑k ιN +1−α− δ i=1 i |F | #ϖ · χα,T T A,η (H,H C b

T

ιN

+1/2−β−

∑k i=1

δi

)

−β )))

(t−s)2β s2(δ1 +δ2 +···+δk )

0

sup

( (δ,0)

(t−s)α s(δ1 +δ2 +···+δk )

t ∥B (#ϖ ) (Xs0,x )−B (#ϖ ) (Xs0,y )∥2 p(# +1) ϖ L (P;L(#ϖ ) (H,HS(U,H

∫

# t∈(0,T ] ϖ∈Πk∗ I∈ϖ u=(ui )i∈I ∈(H\{0}) I

+ χβ,T A,η

−α ))

0

{

∑ ∑

t ∥F (#ϖ ) (Xs0,x )−F (#ϖ ) (Xs0,y )∥ p(# +1) ϖ L (P;L(#ϖ ) (H,H

t

sup

)

b

∫

t∈(0,T ]

p (p−1) 2

t∈(0,T ]

|F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) b

{

[ ϖ∈Πk

+

α,β,ι

N ΘA,η,p,T

−α

} ds

]1/2 } ] ds

(δ,0)

# ,(x,u) # ,(y,u) t I∪{k+1} ∥Xt I −Xt I ∥ p#ϖ L (P;H)

∏ i∈I

∥ui ∥H

−δi

( ) ∑k B 1 − α, 1 − δ i i=1 ) √

|B|C #ϖ (H,HS(U,H b

−β ))

p (p−1) 2

) ( )]} ∑k B 1 − 2β, 1 − 2 i=1 δi . (74)

Furthermore, we note that Corollary 2.8 in [1] (with H = H, U = U , T = T , η = η, p = p, α = α, α ˆ = 0, ˆ 0 = ∥F (0)∥H , L1 = |B| 1 ˆ 1 = ∥B(0)∥HS(U,H ) , β = β, βˆ = 0, L0 = |F |C 1 (H,H−α ) , L , L −α −β −β )) b ( ) Cb (H,HS(U,H ( W = W , A = A, F = [0, T ] × Ω ×) H ∋ (t, ω, z) ↦→ F (z) ∈ H−α , B = [0, T ] × Ω × H ∋ (t, ω, z) ↦→ (U ∋ u ↦→ B(z) u ∈ H−β ) ∈ HS(U, H−β ) , δ = 0, X 1 = X 0,x , X 2 = X 0,y , λ = 0 for x, y ∈ H, p ∈ [2, ∞) in the notation of Corollary 2.8 in [1]) and (35) show that for all p ∈ [2, ∞), x, y ∈ H it holds that ) α,β,0 ( sup ∥Xt0,x − Xt0,y ∥Lp (P;H) ≤ χ0,T A,η ∥x − y∥H ΘA,η,p,T |F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) < ∞. b

t∈(0,T ]

(75)

b

This implies that for all k ∈ {1, 2, . . . , n}, m ∈ {1, 2, . . . , k}, p ∈ [2, ∞), l ∈ {0, 1}, δ = (δ1 , δ2 , . . . , δk ) ∈ Dk , x, y ∈ H, t ∈ (0, T ] with x = ̸ y it holds that

t

(δ,0) ιN

⎡ ∫ ⎣

(m)

t ∥Gl

(δ,0)

(m)

)−Gl

0,y

(Xs

[∫ 0

l

t 1 (t−s)(l+1)rl s(l+1)(δ1 +δ2 +···+δk ) (m)

(Xs0,x ) − Gl

(m)

(Xs0,x ) − Gl

⎤1/(l+1)

(l+1)

)∥ p L (P;L(m) (H,Vl,r ))

(t−s)(l+1)rl s(l+1)(δ1 +δ2 +···+δk )

0

≤ tιN

0,x

(Xs

ds⎦

]1/(l+1) ds

(m)

(Xs0,y )∥Lp (P;L(m) (H,V )) l,rl ⏐ ( )⏐1 ∑k ⏐ ⏐ /(l+1) ≤ T (1/(l+1)−rl −min{1−α,1/2−β}) ⏐B 1 − (l + 1)rl , 1 − (l + 1) i=1 δi ⏐ · sup ∥Gl s∈(0,T ]

(m)

(Xs0,y )∥Lp (P;L(m) (H,V )) l,rl ⏐ ( )⏐1 ∑k ⏐ ⏐ /(l+1) ≤ T (1/(l+1)−rl −min{1−α,1/2−β}) ⏐B 1 − (l + 1)rl , 1 − (l + 1) i=1 δi ⏐ · sup ∥Gl s∈(0,T ]

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

· |Gl |Lipm (H,Vl,r

l

)

147

sup ∥Xs0,x − Xs0,y ∥Lp (P;H) s∈(0,T ]

) α,β,0 ( ≤ T (1/(l+1)−rl −min{1−α,1/2−β}) ΘA,η,p,T |F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) b ⏐ ( )⏐1 b ∑k ⏐ /(l+1) 0,T ⏐ χA,η |Gl |Lipm (H,Vl,r ) ∥x − y∥H . · ⏐B 1 − (l + 1)rl , 1 − (l + 1) i=1 δi ⏐

(76)

l

Combining this with (74) establishes that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), δ = (δ1 , δ2 , . . . , δk ) ∈ Dk with |F |Lipk (H,H−α ) + |B|Lipk (H,HS(U,H−β )) < ∞ it holds that ⎡ sup

sup ⎣

sup

⎤ k,(y,u) − Xt ∥Lp (P;H) ⎦ ∏k ∥x − y∥H i=1 ∥ui ∥H−δ

(δ,0)

tιN

x,y∈H, u=(u ,u ,...,u )∈(H\{0})k t∈(0,T ] 1 2 k x̸=y

k,(x,u)

∥Xt

i

(δ,0)

α,β,ιN ⌈k/2⌉ min{1−α,1/2−β} 1| ΘA,η,p,T

) ≤ |T ∨ |F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) b b ( ∑ ) ( 0,T α,β,0 δ · Lϖ,p(#ϖ +1) χA,η ΘA,η,p(#ϖ +1),T |F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) (

b

ϖ∈Πk

b

[ ( ) ∑k (1−α−min{1−α,1/2−β}) |F |Lip#ϖ (H,H−α ) B 1 − α, 1 − i=1 δi · χα,T A,η T √ ( + χβ,T A,η +

T

(1/2−β−min{1−α,1/2−β})

∑ ∑ ϖ∈Πk∗ I∈ϖ

|B|Lip#ϖ (H,HS(U,H−β )) {

sup

p (p−1) 2

sup

sup

x,y∈H, u=(u ) # t∈(0,T ] i i∈I ∈(H\{0}) I x̸=y

ι

Lδϖ\{I},p #ϖ

B 1 − 2β, 1 − 2

∑k

i=1 δi

)]

(77)

(δ,0)

# ,(y,u) # ,(x,u) t I∪{k+1} ∥Xt I −X I ∥ p#ϖ L (P;H) ∏ t ∥x−y∥H ∥ui ∥H i∈I

−δi

[ (1−α−min{1−α,1/2−β}) · χα,T |F |C #ϖ (H,H A,η T

( ) ∑k B 1 − α, 1 − δ i i=1 −α ) √ (

b

+ χβ,T A,η

T

(1/2−β−min{1−α,1/2−β})

|B|C #ϖ (H,HS(U,H b

p (p−1) 2

−β ))

B 1 − 2β, 1 − 2

) )]} . i=1 δi

∑k

Induction and (64) hence imply that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), δ = (δ1 , δ2 , . . . , δk ) ∈ Dk with |F |Lipk (H,H−α ) + |B|Lipk (H,HS(U,H−β )) < ∞ it holds that ⎡ sup

sup

sup ⎣

x,y∈H, u=(u ,u ,...,u )∈(H\{0})k t∈(0,T ] 1 2 k x̸=y (δ,0) α,β,ιN k 1 (H,H A,η,p,T Cb −α )

(δ,0)

t

ιN

k,(x,u) ∥Xt

∥x − y∥H

(

− ∏k

⎤

k,(y,u) Xt ∥Lp (P;H) ⎦

i=1

∥ui ∥H−δ

i

)

≤ |T ∨ 1| Θ |F | , |B|C 1 (H,HS(U,H−β )) b ( ∑ ( ) α,β,0 · Lδϖ,p(#ϖ +1) χ0,T A,η ΘA,η,p(#ϖ +1),T |F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) b

ϖ∈Π

k [ ( ) ∑k α,T · χA,η |F |Lip#ϖ (H,H−α ) B 1 − α, 1 − i=1 δi √ (

+ χβ,T A,η +

|B|Lip#ϖ (H,HS(U,H−β ))

∑ ∑ ϖ∈Πk∗ I∈ϖ

Lδϖ\{I},p #ϖ

p (p−1) 2

b

B 1 − 2β, 1 − 2

∑k

i=1 δi

[ sup

sup

sup

x,y∈H, u=(u ) # t∈(0,T ] i i∈I ∈(H\{0}) I x̸=y

ι

)]

(δ,0)

# ,(x,u) # ,(y,u) t I∪{k+1} ∥Xt I −Xt I ∥ p#ϖ L (P;H)

∥x−y∥H

∏ i∈I

∥ui ∥H

−δi

]

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

148

[ · χα,T A,η |F |C #ϖ (H,H b

+ χβ,T A,η

( ) ∑k B 1 − α, 1 − δ i=1 i −α ) √ (

|B|C #ϖ (H,HS(U,H b

−β ))

p (p−1) 2

B 1 − 2β, 1 − 2

∑k

i=1 δi

) )]

(78) < ∞.

This implies (15) and thus completes the proof of item (iv). To prove item (v) we first observe that (75) ensures that for all x ∈ H, t ∈ [0, T ] it holds that [ ] lim supH∋y→x E min{1, ∥Xt0,x − Xt0,y ∥H } = 0.

(79)

This implies for all x ∈ H, ρ ∈ [0, 1], t ∈ [0, T ] that [ ] lim supH∋y→x E min{1, ∥(Xt0,x + ρ[Xt0,y − Xt0,x ]) − Xt0,x ∥H } = 0.

(80)

(k)

The fact that ∀ k ∈ {1, 2, . . . , n}, l ∈ {0, 1} : Gl ∈ C(H, L(k) (H, Vl,0 )) and, e.g., item (ii) of Theorem 6.12 in Klenke [17] hence ensure that for all k ∈ {1, 2, . . . , n}, l ∈ {0, 1}, ρ ∈ [0, 1], t ∈ [0, T ], (xm )m∈N0 ⊆ H with lim supm→∞ ∥xm − x0 ∥H = 0 it holds that [ (k) (k) lim supm→∞ E min{1, ∥Gl (Xt0,x0 + ρ[Xt0,xm − Xt0,x0 ]) − Gl (Xt0,x0 )∥L(k) (H,V

} l,0 )

]

= 0.

(81)

Combining this and, e.g., Lemma 4.2 in Hutzenthaler, Jentzen & Salimova [14] (with I = {∅}, c = 1, (k) (k) X m (∅, ω) = ∥Gl (Xt0,x (ω) + ρ[Xt0,xm (ω) − Xt0,x (ω)]) − Gl (Xt0,x (ω))∥L(k) (H,V ) for ω ∈ Ω , m ∈ N, l,0 (xj )j∈N ∈ {y ∈ M(N, H) : lim supj→∞ ∥yj −x∥H = 0}, t ∈ [0, T ], ρ ∈ [0, 1], x ∈ H, l ∈ {0, 1}, k ∈ {1, 2, . . . , n} in the notation of Lemma 4.2 in Hutzenthaler, Jentzen & Salimova [14]) establishes that for all ε ∈ (0, ∞), k ∈ {1, 2, . . . , n}, l ∈ {0, 1}, x ∈ H, ρ ∈ [0, 1], t ∈ [0, T ], (xm )m∈N ⊆ H with lim supm→∞ ∥xm − x∥H = 0 it holds that ({ (k) lim supm→∞ P ω ∈ Ω : ∥Gl (Xt0,x (ω) + ρ[Xt0,xm (ω) − Xt0,x (ω)]) (k)

− Gl (Xt0,x (ω))∥L(k) (H,V

l,0 )

(k)

This, the fact that ∀ k ∈ {1, 2, . . . , n}, l ∈ {0, 1} : supx∈H ∥Gl (x)∥L(k) (H,V

l,0 )

≥ε

})

= 0.

(82)

< ∞, and, e.g., Proposition 4.5 (k)

in Hutzenthaler, Jentzen & Salimova [14] (with I = {∅}, p = p, V = R, X m (∅, ω) = ∥Gl (Xt0,x0 (ω) + (k) ρ[Xt0,xm (ω) − Xt0,x0 (ω)]) − Gl (Xt0,x0 (ω))∥L(k) (H,V ) for ω ∈ Ω , m ∈ N0 , (xj )j∈N0 ∈ {y ∈ M(N0 , H) : l,0 lim supj→∞ ∥yj − y0 ∥H = 0}, t ∈ [0, T ], ρ ∈ [0, 1], p ∈ (0, ∞), l ∈ {0, 1}, k ∈ {1, 2, . . . , n} in the notation of Proposition 4.5 in Hutzenthaler, Jentzen & Salimova [14]) ensure that for all k ∈ {1, 2, . . . , n}, l ∈ {0, 1}, p ∈ (0, ∞), ρ ∈ [0, 1], t ∈ [0, T ], (xm )m∈N0 ⊆ H with lim supm→∞ ∥xm − x0 ∥H = 0 it holds that [ p (k) (k) lim supm→∞ E Gl (Xt0,x0 + ρ[Xt0,xm − Xt0,x0 ]) − Gl (Xt0,x0 )L(k) (H,V

l,0 )

]

= 0.

(83)

Combining H¨ older’s inequality and Lebesgue’s theorem of dominated convergence with (83) (with ρ = 1 1 in the notation of (83)) yields that for all k ∈ {1, 2, . . . , n}, l ∈ {0, 1}, p ∈ [2, ∞), q ∈ (1, max{α,2β, 1/2} ),

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

149

λ ∈ [− min{1/q − α, 1/(2q) − β}, ∞), x ∈ H it holds that

{ [∫ lim sup sup tλ

t

(k)

(l+1)

(k)

∥Gl (Xs0,x ) − Gl (Xs0,y )∥Lp (P;L(k) (H,V

l,rl ))

]1/(l+1) } ds

(t − s)(l+1)rl 0 ]1/[q(l+1)] { [∫ t 1 λ ds ≤ lim sup sup t q(l+1)rl H∋y→x t∈(0,T ] 0 (t − s) ](q−1)/[q(l+1)] } [∫ t q(l+1)/(q−1) (k) (k) 0,x 0,y ∥Gl (Xs ) − Gl (Xs )∥Lp (P;L(k) (H,V )) ds · l,rl 0 { (λ+1/[q(l+1)]−rl ) t = lim sup sup 1/[q(l+1)] H∋y→x t∈(0,T ] [1 − q(l + 1)rl ] [∫ t ](q−1)/[q(l+1)] } q(l+1)/(q−1) (k) (k) ∥Gl (Xs0,x ) − Gl (Xs0,y )∥Lp (P;L(k) (H,V )) ds ·

H∋y→x t∈(0,T ]

(84)

l,rl

0

T (λ+1/[q(l+1)]−rl ) = [1 − q(l + 1)rl ]1/[q(l+1)] [ ∫ T q(l+1)/(q−1) (k) (k) · lim sup ∥Gl (Xs0,x ) − Gl (Xs0,y )∥Lp (P;L(k) (H,V H∋y→x

l,rl

0

](q−1)/[q(l+1)] ds = 0. ))

1 1 1 Moreover, observe that the fact that ∀ q ∈ (1, max{α,2β, 1/2} ) : 0 < min{ /q − α, /(2q) − β} < min{1 − α, 1/2 − β} ≤ 1/2 and (73) (with k = k, p = p, δ = 0k , λ = − min{1/q − α, 1/(2q) − β}, γ = 1 min{1 − α, 1/2 − β} − min{1/q − α, 1/(2q) − β}, x = x, y = y for x, y ∈ H, q ∈ (1, max{α,2β, 1/2} ), p ∈ [2, ∞), 1 k ∈ {1, 2, . . . , n} in the notation of (73)) imply that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), q ∈ (1, max{α,2β, 1/2} ), x, y ∈ H it holds that

[ sup

sup

u=(u1 ,u2 ,...,uk )∈(H\{0})k t∈(0,T ] ⌈k/2⌉ min{1−α,1/2−β}

k,(x,u)

∥Xt

k,(y,u)

− Xt

tmin{1/q−α,1/(2q)−β}

1 tmin{1/q−α,1/(2q)−β}

t∈(0,T ] ϖ∈Πk t ∥F (#ϖ ) (Xs0,x )−F (#ϖ ) (Xs0,y )∥ p(# +1) ϖ

·

L (t−s)α

0

+ χβ,T A,η sup

{

∏k

i=1 ∥ui ∥H ) α,β,− min{1/q −α,1/(2q)−β} ( ΘA,η,p,T |F |C 1 (H,H−α ) , |B|C 1 (H,HS(U,H−β )) b b

≤ |T ∨ 1| ( [ { ∑ 0 α,T k · Lϖ,p(#ϖ +1) χA,η sup ∫

]

∥Lp (P;H)

(P;L(#ϖ ) (H,H−α ))

} ds

1 tmin{1/q−α,1/(2q)−β}

t∈(0,T ] t ∥B (#ϖ ) (Xs0,x )−B (#ϖ ) (Xs0,y )∥2 p(# +1) ϖ L (P;L(#ϖ ) (H,HS(U,H

[ ∫ p (p−1) · 2

(t−s)2β

0

+

∑ ∑

]1/2 } ] ds

−β )))

{ sup

sup

# t∈(0,T ] ϖ∈Π ∗ I∈ϖ u=(ui )i∈I ∈(H\{0}) I

#I ,(x,u)

0

k Lϖ\{I},p #ϖ

∥Xt

#I ,(y,u)

− Xt

tmin{1/q−α,1/(2q)−β}

(85)

∥Lp#ϖ (P;H)

∏

i∈I

∥ui ∥H

k [ α,T (1−α−min{1−α,1/2−β}) · χA,η T |F |C #ϖ (H,H ) −α b ( ) · B 1 − α, 1 − min{1 − α, 1/2 − β} + min{1/q − α, 1/(2q) − β}

(1/2−β−min{1−α,1/2−β}) + χβ,T |B|C #ϖ (H,HS(U,H A,η T b

−β ))

)]1/2 ( · p (p−1) B 1 − 2β, 1 − 2 min{1 − α, 1/2 − β} + 2 min{1/q − α, 1/(2q) − β} 2 [

]}) .

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

150

1 Induction and (84)–(85) hence ensure that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), q ∈ (1, max{α,2β, 1/2} ), x ∈ H it holds that ] [ k,(x,u) k,(y,u) ∥Xt − Xt ∥Lp (P;H) = 0. (86) lim sup sup sup ∏k H∋y→x u=(u1 ,u2 ,...,uk )∈(H\{0})k t∈(0,T ] tmin{1/q −α,1/(2q)−β} i=1 ∥ui ∥H 1 This and (35) show that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), q ∈ (1, max{α,2β, 1/2} ), x ∈ H it holds that

] k,(y,u) − Xt ∥Lp (P;H) lim sup sup sup ∏k H∋y→x u=(u1 ,u2 ,...,uk )∈(H\{0})k t∈[0,T ] i=1 ∥ui ∥H [ ] (87) k,(x,u) k,(y,u) ∥Xt − Xt ∥Lp (P;H) min{1/q −α,1/(2q)−β} sup ≤T lim sup sup = 0. ∏k H∋y→x u=(u1 ,u2 ,...,uk )∈(H\{0})k t∈(0,T ] tmin{1/q −α,1/(2q)−β} i=1 ∥ui ∥H [

k,(x,u)

∥Xt

Combining (87) with item (iii) proves item (v). We now prove item (vi) by induction on k ∈ {1, 2, . . . , n}. Note that (75) ensures that for all p ∈ (0, ∞) it holds that ˜ p < ∞. L

(88)

Furthermore, observe that for all l ∈ {0, 1}, u = (u0 , u1 ) ∈ H 2 , t ∈ [0, T ] it holds that ¯ u (t, X 0,u0 +u1 − X 0,u0 ). Gl (Xt0,u0 +u1 ) − Gl (Xt0,u0 ) = G t t 1,l

(89)

This and (35) imply that for all u = (u0 , u1 ) ∈ H 2 , t ∈ [0, T ] it holds that [Xt0,u0 +u1 − Xt0,u0 ]P,B(H) = [etA u1 ]P,B(H) ∫ t ∫ t (t−s)A ¯ u 0,u0 +u1 0,u0 ¯ u (s, X 0,u0 +u1 − X 0,u0 ) dWs . + e G1,0 (s, Xs − Xs ) ds + e(t−s)A G 1,1 s s 0

(90)

0

Combining this with the Burkholder–Davis–Gundy type inequality in Lemma 7.7 in Da Prato & Zabczyk [8], (27), (35), and Proposition 2.7 in [1] (with H = H, U = U , T = T , η = η, p = p, α = 0, α ˆ = 0, β = 0, ˆ 0 = 0, L1 = |B| 1 ˆ 1 = 0, W = W , A = A, F = Gu , B = Gu , βˆ = 0, L0 = |F |C 1 (H,H) , L , L 1,0 1,1 C (H,HS(U,H)) b 1

b

1

δ = 0, Y 1 = X 0,θ1 (u) − X 0,θ0 (u) , Y 2 = X 1,u , λ = 0 for u ∈ H 2 , p ∈ [2, ∞) in the notation of Proposition 2.7 in [1]) ensures that for all p ∈ [2, ∞), u ∈ H 2 it holds that  0,θ1 (u)  ) 0,θ 1 (u) 0,0,0 ( sup Xt 1 − Xt 0 − Xt1,u Lp (P;H) ≤ ΘA,η,p,T |F |C 1 (H,H) , |B|C 1 (H,HS(U,H)) b b t∈[0,T ] [ ∫ 1 1 1 1  t (t−s)A ( u ) ¯ (s, Xs0,θ1 (u) − Xs0,θ0 (u) ) − Gu (s, Xs0,θ1 (u) − Xs0,θ0 (u) ) ds · sup  e G 1,0 1,0  t∈(0,T ] 0 ]  ∫ t  ( 0,θ11 (u) 0,θ01 (u) 0,θ11 (u) 0,θ01 (u) ) (t−s)A ¯ u u + e G1,1 (s, Xs − Xs ) − G1,1 (s, Xs − Xs ) dWs   ≤

Lp (P;H)

0 ) 0,0,0 ( χ0,T 1 (H,H) , |B|C 1 (H,HS(U,H)) A,η ΘA,η,p,T |F |Cb b

[∫ ·

T

0,θ11 (u)

¯ u (s, Xs ∥G 1,0

0,θ01 (u)

− Xs

0,θ11 (u)

) − Gu 1,0 (s, Xs

0,θ01 (u)

− Xs

(91)

)∥Lp (P;H) ds

0

[ +

p (p−1) 2

∫ 0

T

1 ¯ u (s, Xs0,θ1 (u) ∥G 1,1

−

0,θ 1 (u) Xs 0 )

−

0,θ11 (u) Gu 1,1 (s, Xs

−

0,θ 1 (u) 2 Xs 0 )∥Lp (P;HS(U,H))

]1/2 ] ds

.

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

151

In addition, H¨ older’s inequality yields that for all p ∈ [2, ∞), l ∈ {0, 1}, u = (u0 , u1 ) ∈ H × (H \ {0}), t ∈ (0, T ] it holds that ¯ u (t, X 0,u0 +u1 − X 0,u0 ) − Gu (t, X 0,u0 +u1 − X 0,u0 )∥Lp (P;V ) ∥G t t t t 1,l 1,l l,0 ∥u1 ∥H ∫ 1   1  0,u ′  [Gl (Xt 0 + ρ[Xt0,u0 +u1 − Xt0,u0 ]) − G′l (Xt0,u0 )](Xt0,u0 +u1 − Xt0,u0 ) dρ =   p ∥u1 ∥H 0 L (P;Vl,0 ) ∫ 1 0,u 0,u +u 0,u 0,u ˜ 2p ≤L ∥G′l (Xt 0 + ρ[Xt 0 1 − Xt 0 ]) − G′l (Xt 0 )∥L2p (P;L(H,Vl,0 )) dρ.

(92)

0

In the next step we combine (91) with (92) and Jensen’s inequality to obtain that for all p ∈ [2, ∞), u = (u0 , u1 ) ∈ H × (H \ {0}) it holds that 0,θ11 (u)

0,θ01 (u)

∥Xt

sup

t∈[0,T ][

− Xt1,u ∥Lp (P;H)

∥u1 ∥H ∫

T

∫

0

[

) ( ˜ 2p χ0,T Θ 0,0,0 |F | 1 ≤L A,η A,η,p,T C (H,H) , |B|C 1 (H,HS(U,H)) b

b

1

· +

− Xt

∥F ′ (Xs0,u0 + ρ[Xs0,u0 +u1 − Xs0,u0 ]) − F ′ (Xs0,u0 )∥L2p (P;L(H,H)) dρ ds

(93)

0

p (p−1) 2

T

∫

]

1

∫

∥B 0

′

(Xs0,u0

+

ρ[Xs0,u0 +u1

Xs0,u0 ])

−

−B

0

′

2 (Xs0,u0 )∥L2p (P;L(H,HS(U,H)))

]1/2 . dρ ds

Furthermore, Lebesgue’s theorem of dominated convergence and (83) yield that for all m ∈ {1, 2, . . . , n}, l ∈ {0, 1}, p ∈ [2, ∞), u0 ∈ H it holds that T

∫

∫

1

lim sup H∋u1 →0

0

0

(m)

∥Gl

(m)

(Xs0,u0 + ρ[Xs0,u0 +u1 − Xs0,u0 ]) − Gl

(l+1)

(Xs0,u0 )∥Lp (P;L(m) (H,V

l,0 ))

dρ ds = 0.

(94)

Combining (93) with (88) and (94) establishes item (vi) in the base case k = 1. For the induction step {1, 2, . . . , n − 1} ∋ k → k + 1 ∈ {2, 3, . . . , n} assume that there exists a natural number k ∈ {1, 2, . . . , n − 1} such that item (vi) holds for k = 1, k = 2, . . . , k = k. Note that item (ii) ensures that for all m ∈ {1, 2, . . . , n}, p ∈ (0, ∞), x, y ∈ H, v ∈ H \ {0} it holds that dm,p (x, y) + d˜m,p (x, v) < ∞. We also note that (87) and the induction hypothesis assure that for all m ∈ {1, 2, . . . , k}, p ∈ (0, ∞), x ∈ H it holds that lim sup dm,p (x, y) = 0 H∋y→x

and

lim sup d˜m,p (x, v) = 0.

(95)

H\{0}∋v→0

Next observe that (39) shows that for all l ∈ {0, 1}, u = (u0 , u1 , . . . , uk+1 ) ∈ H k+2 , t ∈ [0, T ] it holds that θ k+1 (u)

k,θ k+1 (u)

0,u +u

k,θ k+1 (u)

(t, Xt 1 ) = G′l (Xt 0 k+1 ) Xt 1 #I ϖ ,[θ1k+1 (u)]ϖ ∑ (# ) 0,u0 +u #I ϖ ,[θ1k+1 (u)]ϖ ( #I ϖ ,[θ1k+1 (u)]ϖ #ϖ ) 1 2 k+1 + Gl ϖ (Xt ) Xt 1 , Xt 2 , . . . , X t #ϖ

1 Gk,l

(96)

ϖ∈Πk∗

and θ k+1 (u)

k,θ k+1 (u)

k,θ k+1 (u)

(t, Xt 0 ) = G′l (Xt0,u0 ) Xt 0 #I ϖ ,[θ0k+1 (u)]ϖ ∑ (# ) 0,u ( #I ϖ ,[θ0k+1 (u)]ϖ #I ϖ ,[θ0k+1 (u)]ϖ #ϖ ) 1 2 + Gl ϖ (Xt 0 ) Xt 1 , Xt 2 , . . . , X t #ϖ .

0 Gk,l

ϖ∈Πk∗

(97)

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

152

This implies that for all l ∈ {0, 1}, u = (u0 , u1 , . . . , uk+1 ) ∈ H k+2 , t ∈ [0, T ] it holds that θ k+1 (u)

1 Gk,l

=

k,θ1k+1 (u)

(t, Xt

θ k+1 (u)

0 ) − Gk,l

k,θ k+1 (u) 0,u +u G′l (Xt 0 k+1 ) Xt 1

−

k,θ0k+1 (u)

(t, Xt

)

k,θ k+1 (u) G′l (Xt0,u0 ) Xt 0 k+1

∑ [

+

(#ϖ )

Gl

0,u0 +uk+1

(Xt

#I ϖ ,[θ1 #I ϖ ,[θ1k+1 (u)]ϖ ( #I ϖ ,[θ1k+1 (u)]ϖ 1 2 ) Xt 1 , Xt 2 , . . . , X t #ϖ

(u)]ϖ #

ϖ

)

ϖ∈Πk∗

] #I ϖ ,[θ0k+1 (u)]ϖ #I ϖ ,[θ0k+1 (u)]ϖ #I ϖ ,[θ0k+1 (u)]ϖ #ϖ ) 1 2 (#ϖ ) 0,u0 ( 1 2 − Gl (Xt ) Xt , Xt , . . . , X t #ϖ k,θ k+1 (u)

k,θ k+1 (u)

(98)

k,θ k+1 (u)

0,u +u

= G′l (Xt0,u0 )(Xt 1 − Xt 0 ) + [G′l (Xt 0 k+1 ) − G′l (Xt0,u0 )] Xt 1 ∑ [ (# ) 0,u0 +u #I ϖ ,[θ1k+1 (u)]ϖ ( #I ϖ ,[θ1k+1 (u)]ϖ 1 2 (# ) k+1 + [Gl ϖ (Xt ) − Gl ϖ (Xt0,u0 )] Xt 1 , Xt 2 ,..., ϖ∈Πk∗ #I ϖ

#ϖ

Xt

,[θ1k+1 (u)]ϖ #

(#ϖ )

− Gl

ϖ

#I ϖ ,[θ1k+1 (u)]ϖ #I ϖ ,[θ1k+1 (u)]ϖ #I ϖ ,[θ1k+1 (u)]ϖ #ϖ ) 1 2 (#ϖ ) 0,u0 ( 1 2 + Gl (Xt ) Xt , Xt , . . . , X t #ϖ ] #I ϖ ,[θ0k+1 (u)]ϖ # ϖ ,[θ k+1 (u)]ϖ # ϖ ,[θ k+1 (u)]ϖ # )

)

( (Xt0,u0 ) Xt

I 1

1

0

, Xt

I

2

2

0

, . . . , Xt

ϖ

#ϖ

.

The fundamental theorem of calculus and (40) hence yield that for all l ∈ {0, 1}, u ∈ H k+2 , t ∈ [0, T ] it holds that θ k+1 (u)

1 Gk,l

k,θ1k+1 (u)

(t, Xt

θ k+1 (u)

0 ) − Gk,l

k,θ0k+1 (u)

(t, Xt

k+1

¯ u (t, X k,θ1 )=G t k+1,l

(u)

k,θ0k+1 (u)

− Xt

).

(99)

This, (35), and (39) show that for all u ∈ H k+2 , t ∈ [0, T ] it holds that [ k,θ1k+1 (u) k,θ k+1 (u) ] Xt − Xt 0 = P,B(H)

∫

t

[ θ1k+1 (u) θ0k+1 (u) k,θ k+1 (u) ] k,θ k+1 (u) ) ds ) − Gk,0 (s, Xs 0 e(t−s)A Gk,0 (s, Xs 1 0∫ t [ θ1k+1 (u) θ0k+1 (u) k,θ k+1 (u) ] k,θ k+1 (u) + ) − Gk,1 e(t−s)A Gk,1 (s, Xs 1 (s, Xs 0 ) dWs (100) ∫ t0 k,θ0k+1 (u) k,θ1k+1 (u) (t−s)A ¯ u ) ds − Xs = e Gk+1,0 (s, Xs 0∫ t k+1 k+1 k,θ (u) k,θ1 (u) ¯u ) dWs . − Xs 0 + e(t−s)A G k+1,1 (s, Xs 0

Combining this with (32), (33), (35), and Proposition 2.7 in [1] (with H = H, U = U , T = T , η = η, ϖ ] [ ∏#ϖ  ˆ0 = ∑  [X #Iiϖ ,[u]i ]  p# , p = p, α = 0, α ˆ = 0, β = 0, βˆ = 0, L0 = |F |C 1 (H,H) , L |F |C #ϖ (H,H) i=1 ∗ ϖ∈Π L ϖ b k+1 b ϖ ]  [ ∑ ∏ # ,[u] ϖ # I i ϖ ˆ1 = L1 = |B|C 1 (H,HS(U,H)) , L [X i ] Lp#ϖ , W = W , A = A, ϖ∈Π ∗ |B|C #ϖ (H,HS(U,H)) i=1 b

k+1

b

k+1

k+1

u 1 F = Gu = X k,θ1 (u) − X k,θ0 (u) , Y 2 = X k+1,u , λ = 0 for u ∈ H k+2 , k+1,0 , B = Gk+1,1 , δ = 0, Y p ∈ [2, ∞) in the notation of Proposition 2.7 in [1]) implies that for all p ∈ [2, ∞), u ∈ H k+2 it holds that

 k,θk+1 (u)  ) k,θ k+1 (u) 0,0,0 ( sup Xt 1 − Xt 0 − Xtk+1,u Lp (P;H) ≤ ΘA,η,p,T |F |C 1 (H,H) , |B|C 1 (H,HS(U,H)) b b t∈[0,T ] [ ∫  t (t−s)A ( u k,θ1k+1 (u) k,θ k+1 (u) k,θ1k+1 (u) k,θ k+1 (u) ) ¯ · sup  e G − Xs 0 ) − Gu − Xs 0 ) ds k+1,0 (s, Xs k+1,0 (s, Xs  t∈(0,T ] 0 ∫ t ( u k,θ1k+1 (u) ¯ + e(t−s)A G k+1,1 (s, Xs 0

−

k,θ k+1 (u) Xs 0 )

−

k,θ1k+1 (u) Gu k+1,1 (s, Xs



−

 k,θ k+1 (u) ) Xs 0 ) dWs   p L (P;H)

] .

(101)

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

153

The Burkholder–Davis–Gundy type inequality in Lemma 7.7 in Da Prato & Zabczyk [8] hence shows that for all p ∈ [2, ∞), u ∈ H k+2 it holds that k+1

k,θ sup Xt 1



(u)

k,θ0k+1 (u)

− Xt

T

k,θ1k+1 (u)

¯u ∥G k+1,0 (t, Xt

·



(

b

t∈[0,T ]

[∫

0,0,0 − Xtk+1,u Lp (P;H) ≤ χ0,T A,η ΘA,η,p,T |F |C 1 (H,H) , |B|C 1 (H,HS(U,H)) k,θ0k+1 (u)

− Xt

k,θ1k+1 (u)

) − Gu k+1,0 (t, Xt

)

b

k,θ0k+1 (u)

− Xt

)∥Lp (P;H) dt +

[

p (p−1) 2

]1/2

0

[∫

T

·

k,θ1k+1 (u) ¯u ∥G k+1,1 (t, Xt

−

k,θ k+1 (u) Xt 0 )

−

k,θ1k+1 (u) Gu k+1,1 (t, Xt

−

k,θ k+1 (u) 2 Xt 0 )∥Lp (P;HS(U,H))

]1/2 ] dt

.

0

(102)

Next observe that for all m ∈ N it holds that { } { } Πm+1 = ϖ ∪ {m + 1} : ϖ ∈ Πm } ⋃{{ } ϖ ϖ ϖ ϖ I1ϖ , I2ϖ , . . . , Ii−1 , Iiϖ ∪ {m + 1}, Ii+1 , Ii+2 , . . . , I# : i ∈ {1, 2, . . . , #ϖ }, ϖ ∈ Πm . ϖ

(103)

This implies that for all m ∈ N it holds that

{ } { } ∗ Πm+1 = ϖ ∪ {m + 1} : ϖ ∈ Πm } ⋃{{ } ϖ ϖ ϖ ϖ ∗ I1ϖ , I2ϖ , . . . , Ii−1 , Iiϖ ∪ {m + 1}, Ii+1 , Ii+2 , . . . , I# : i ∈ {1, 2, . . . , # }, ϖ ∈ Π ϖ m ϖ [ {{ ⋃ ({ }} ⋃ { }} {1, 2, . . . , m}, {m + 1} = ϖ ∪ {m + 1} ∗ ϖ∈Πm

⋃{{

ϖ I1ϖ , I2ϖ , . . . , Ii−1 , Iiϖ

∪ {m +

ϖ ϖ ϖ 1}, Ii+1 , Ii+2 , . . . , I# ϖ

}

(104)

] }) : i ∈ {1, 2, . . . , #ϖ } .

This and (39) prove that for all l ∈ {0, 1}, u = (u0 , u1 , . . . , uk+1 ) ∈ H k+2 , t ∈ [0, T ], x ∈ H it holds that

Gu k+1,l (t, x)

=

G′l (Xt0,u0 ) x

+

#I ϖ ,[u]ϖ #I ϖ ,[u]ϖ #I ϖ ,[u]ϖ #ϖ ) 1 2 (#ϖ ) 0,u0 ( 1 2 Gl (Xt ) Xt , Xt , . . . , X t #ϖ

∑ ∗ ϖ∈Πk+1

( k,θk+1 (u) 1,(u0 ,uk+1 ) ) , Xt = G′l (Xt0,u0 ) x + G′′l (Xt0,u0 ) Xt 0 [ #I ϖ ,[θ0k+1 (u)]ϖ ∑ #I ϖ ,[θ0k+1 (u)]ϖ ( #I ϖ ,[θ0k+1 (u)]ϖ #ϖ 1 2 1,(u ,u )) (# +1) , Xt 2 , . . . , Xt #ϖ , Xt 0 k+1 + Gl ϖ (Xt0,u0 ) Xt 1 ϖ∈Πk∗

+

#ϖ ∑

k+1

(#ϖ )

Gl

#I ϖ ,[θ0 #I ϖ ,[θ0k+1 (u)]ϖ ( #I ϖ ,[θ0k+1 (u)]ϖ 1 2 (Xt0,u0 ) Xt 1 , Xt 2 , . . . , Xt i−1

(u)]ϖ i−1

,

i=1 #I ϖ +1,([θ0k+1 (u)]ϖ i ,uk+1 )

Xt

i

#I ϖ ,[θ0k+1 (u)]ϖ i+1

, Xt

i+1

#I ϖ ,[θ0k+1 (u)]ϖ #I ϖ ,[θ0k+1 (u)]ϖ #ϖ ) i+2 , Xt i+2 , . . . , Xt #ϖ

] . (105)

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

154

Moreover, observe that (40) shows that for all l ∈ {0, 1}, u = (u0 , u1 , . . . , uk+1 ) ∈ H k+2 , t ∈ [0, T ], x ∈ H it holds that ¯ u (t, x) = G′ (X 0,u0 ) x G t k+1,l l ∫ 1 ( )( k,θk+1 (u) 0,u0 +uk+1 ) 0,u +u + G′′l Xt0,u0 + ρ[Xt 0 k+1 − Xt0,u0 ] Xt 1 , Xt − Xt0,u0 dρ 0 [ ∑ ∫ 1 (# +1) ( 0,u #I ϖ ,[θ1k+1 (u)]ϖ )( #I ϖ ,[θ1k+1 (u)]ϖ 1 2 0,u +u Gl ϖ Xt 0 + ρ[Xt 0 k+1 − Xt0,u0 ] Xt 1 + , Xt 2 ,..., 0

ϖ∈Πk∗

,[θ1k+1 (u)]ϖ #

#I ϖ

+

Xt #ϖ ∑

(106)

ϖ

#ϖ

(#ϖ )

Gl

0,u +u , Xt 0 k+1

) Xt0,u0 dρ

−

#I ϖ ,[θ0k+1 (u)]ϖ 1 1

( (Xt0,u0 ) Xt

#I ϖ ,[θ0k+1 (u)]ϖ 2 2

, Xt

#I ϖ ,[θ0k+1 (u)]ϖ i−1 i−1

, . . . , Xt

#I ϖ ,[θ1k+1 (u)]ϖ i

, Xt

i

i=1 #I ϖ ,[θ0k+1 (u)]ϖ i i

− Xt

#I ϖ ,[θ1k+1 (u)]ϖ #I ϖ ,[θ1k+1 (u)]ϖ #I ϖ ,[θ1k+1 (u)]ϖ #ϖ ) i+1 i+2 i+1 i+2 , Xt , . . . , X t #ϖ , Xt

] .

This implies that for all l ∈ {0, 1}, u = (u0 , u1 , . . . , uk+1 ) ∈ H k+2 , t ∈ [0, T ] it holds that k+1

k+1

k+1

k+1

¯ u (t, X k,θ1 (u) − X k,θ0 (u) ) − Gu (t, X k,θ1 (u) − X k,θ0 (u) ) G t t t t k+1,l k+1,l #ϖ [ k+1 k+1 ϖ ϖ #I ϖ ,[θ0k+1 (u)]ϖ ∑ ∑ # ,[θ (u)] # ,[θ (u)] i−1 Iϖ Iϖ 1 2 0 0 (#ϖ ) 0,u0 ( 1 2 = Gl (Xt ) Xt , Xt , , . . . , Xt i−1 ϖ∈Πk∗ i=1 #I ϖ ,[θ1k+1 (u)]ϖ i

Xt

i

#I ϖ ,[θ0k+1 (u)]ϖ i

− Xt

#I ϖ ,[θ1k+1 (u)]ϖ i+2

Xt

i+2

i

#I ϖ

, . . . , Xt

#ϖ

#I ϖ +1,([θ0k+1 (u)]ϖ i ,uk+1 ) i

− Xt

,[θ1k+1 (u)]ϖ #

ϖ

#I ϖ ,[θ1k+1 (u)]ϖ i+1

, Xt

i+1

) k+1

(#ϖ )

+ Gl

#I ϖ ,[θ0 #I ϖ ,[θ0k+1 (u)]ϖ ( #I ϖ ,[θ0k+1 (u)]ϖ 1 2 (Xt0,u0 ) Xt 1 , Xt 2 , . . . , Xt i−1

#I ϖ +1,([θ0k+1 (u)]ϖ i ,uk+1 )

Xt

i

#I ϖ ,[θ1k+1 (u)]ϖ i+1 i+1

, Xt

#I ϖ ,[θ1k+1 (u)]ϖ i+2 i+2

, Xt

Xt

i

,

#I ϖ

#I ϖ ,[θ0 #I ϖ ,[θ0k+1 (u)]ϖ ( #I ϖ ,[θ0k+1 (u)]ϖ 1 2 , Xt 2 , . . . , Xt i−1 (Xt0,u0 ) Xt 1

#I ϖ +1,([θ0k+1 (u)]ϖ i ,uk+1 )

(u)]ϖ i−1

, . . . , Xt k+1

(#ϖ )

− Gl

,

#ϖ

(u)]ϖ i−1

,[θ1k+1 (u)]ϖ #

ϖ

)

,

] #I ϖ ,[θ0k+1 (u)]ϖ #I ϖ ,[θ0k+1 (u)]ϖ #I ϖ ,[θ0k+1 (u)]ϖ #ϖ ) i+1 i+2 i+1 i+2 , Xt , Xt , . . . , X t #ϖ

(107)

∑ [ ∫ 1 [ (# +1) ( 0,u #I ϖ ,[θ1k+1 (u)]ϖ ) 1 0,u +u (# +1) ( 0,u0 )]( Gl ϖ Xt 0 + ρ[Xt 0 k+1 − Xt0,u0 ] − Gl ϖ Xt Xt 1 ,

+

ϖ∈Πk

0

#I ϖ ,[θ1k+1 (u)]ϖ 2

Xt

2

#I ϖ

#ϖ

, . . . , Xt

,[θ1k+1 (u)]ϖ #

ϖ

0,u0 +uk+1

, Xt

) − Xt0,u0 dρ

#I ϖ ,[θ0k+1 (u)]ϖ #I ϖ ,[θ0k+1 (u)]ϖ #I ϖ ,[θ0k+1 (u)]ϖ #ϖ 1 2 (#ϖ +1) 0,u0 ( 1 2 , + Gl (Xt ) Xt , Xt , . . . , Xt #ϖ 0,u0 +uk+1

Xt

1,(u0 ,uk+1 ) )

− Xt0,u0 − Xt

k+1

(#ϖ +1)

#I ϖ ,[θ1 #I ϖ ,[θ1k+1 (u)]ϖ ( #I ϖ ,[θ1k+1 (u)]ϖ 1 2 (Xt0,u0 ) Xt 1 , Xt 2 , . . . , X t #ϖ

(#ϖ +1)

#I ϖ ,[θ0 #I ϖ ,[θ0k+1 (u)]ϖ ( #I ϖ ,[θ0k+1 (u)]ϖ 1 2 (Xt0,u0 ) Xt 1 , Xt 2 , . . . , X t #ϖ

+ Gl

k+1

− Gl

(u)]ϖ #

ϖ

(u)]ϖ #

ϖ

0,u0 +uk+1

− Xt0,u0

)

0,u0 +uk+1

− Xt0,u0

)

, Xt , Xt

] .

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

155

This assures that for all l ∈ {0, 1}, u = (u0 , u1 , . . . , uk+1 ) ∈ H k+2 , t ∈ [0, T ] it holds that k+1

k+1

k+1

k+1

¯ u (t, X k,θ1 (u) − X k,θ0 (u) ) − Gu (t, X k,θ1 (u) − X k,θ0 (u) ) G t [ t t t k+1,l k+1,l #ϖ #I ϖ ,[θ0k+1 (u)]ϖ ∑ ∑ #I ϖ ,[θ0k+1 (u)]ϖ #I ϖ ,[θ0k+1 (u)]ϖ ( i−1 1 2 (# ) Gl ϖ (Xt0,u0 ) Xt 1 = , Xt 2 , . . . , Xt i−1 , ϖ∈Πk∗ i=1 #I ϖ ,[θ1k+1 (u)]ϖ i

Xt

#I ϖ ,[θ0k+1 (u)]ϖ i

i

i

− Xt

#I ϖ +1,([θ0k+1 (u)]ϖ i ,uk+1 )

− Xt

i

,[θ1k+1 (u)]ϖ #ϖ

#I ϖ ,[θ1k+1 (u)]ϖ i+1

, Xt

i+1

,

#I ϖ #I ϖ ,[θ1k+1 (u)]ϖ ) i+2 Xt i+2 , . . . , X t #ϖ #ϖ #I ϖ ,[θ0k+1 (u)]ϖ ∑ #I ϖ ,[θ0k+1 (u)]ϖ ( #I ϖ ,[θ0k+1 (u)]ϖ i−1 1 2 (# ) + Gl ϖ (Xt0,u0 ) Xt 1 , Xt 2 , . . . , Xt i−1 , j=i+1 #I ϖ ,[θ0k+1 (u)]ϖ #I ϖ ,[θ0k+1 (u)]ϖ #I ϖ ,[θ0k+1 (u)]ϖ #I ϖ +1,([θ k+1 (u)]ϖ j−1 i+1 i+2 i ,uk+1 )

Xt

0

i

Xt

#I ϖ

Xt

#ϖ

#I ϖ ,[θ0k+1 (u)]ϖ j j

− X]t ϖ)

,[θ1k+1 (u)]ϖ #

[∫

∑

+

i+1

, Xt

#I ϖ ,[θ1k+1 (u)]ϖ j j

1[

0

ϖ∈Πk

i+2

, Xt

#I ϖ ,[θ1k+1 (u)]ϖ j+1 j+1

, Xt

, . . . , Xt

j−1

#I ϖ ,[θ1k+1 (u)]ϖ j+2 j+2

, Xt

,

,..., (108)

(#ϖ +1) (

Gl

#I ϖ ,[θ1k+1 (u)]ϖ 2

0,u0 +uk+1

Xt0,u0 + ρ[Xt

#I ϖ

#I ϖ ,[θ1k+1 (u)]ϖ ) 1 (# +1) ( 0,u0 )]( − Xt0,u0 ] − Gl ϖ Xt Xt 1 ,

,[θ1k+1 (u)]ϖ #

) ϖ 0,u +u , Xt 0 k+1 − Xt0,u0 dρ , . . . , Xt #I ϖ ,[θ0k+1 (u)]ϖ #I ϖ ,[θ0k+1 (u)]ϖ ( #I ϖ ,[θ0k+1 (u)]ϖ #ϖ 1 2 (# +1) , , Xt 2 , . . . , X t #ϖ + Gl ϖ (Xt0,u0 ) Xt 1 0,u0 +uk+1 1,(u0 ,uk+1 ) ) 0,u0 Xt − Xt − Xt #ϖ #I ϖ ,[θ0k+1 (u)]ϖ ∑ #I ϖ ,[θ0k+1 (u)]ϖ ( #I ϖ ,[θ0k+1 (u)]ϖ i−1 1 2 (# +1) + Gl ϖ (Xt0,u0 ) Xt 1 , Xt 2 , , . . . , Xt i−1 Xt

#ϖ

2

i=1 #I ϖ ,[θ1k+1 (u)]ϖ i

Xt

i

0,u0 +uk+1

Xt

#I ϖ ,[θ0k+1 (u)]ϖ i

− Xt] i ) − Xt0,u0 .

#I ϖ ,[θ1k+1 (u)]ϖ i+1

, Xt

i+1

#I ϖ ,[θ1k+1 (u)]ϖ i+2

, Xt

i+2

#I ϖ

#ϖ

, . . . , Xt

,[θ1k+1 (u)]ϖ #

ϖ

,

Furthermore, H¨ older’s inequality shows that for all l ∈ {0, 1}, p ∈ [2, ∞), ϖ ∈ Πk∗ , j ∈ {1, 2, . . . , #ϖ }, [i] u = (u0 , u1 , . . . , uk+1 ) ∈ ×k+1 i=0 H , t ∈ (0, T ] it holds that k+1

#I ϖ ,[θ0 #I ϖ ,[θ0k+1 (u)]ϖ  (#ϖ ) 0,u ( #I ϖ ,[θ0k+1 (u)]ϖ 1 1 2 0 1 2 G , . . . , Xt j−1 (X ) X , X ∏k+1 t t t l i=1 ∥ui ∥H #I ϖ ,[θ1k+1 (u)]ϖ j j

Xt

#I ϖ

j+2

Xt

#I ϖ ,[θ0k+1 (u)]ϖ j

− Xt

,[θ1k+1 (u)]ϖ j+2

j

#I ϖ

, . . . , Xt

#ϖ

≤ |Gl |C #ϖ (H,V

l,0 )

b

ϖ

#I ϖ ,[θ1k+1 (u)]ϖ j

·

∥Xt

j

∏#Iiϖ q=1

j+1

,[θ1k+1 (u)]ϖ j+1

,

) 

∥Lp#ϖ (P;H)

ϖ ∥H ∥uIi,q

#I ϖ ,[θ0k+1 (u)]ϖ j

− Xt

#I ϖ

, Xt

,

Lp (P;Vl,0 )

i

i=1

j

,[θ1k+1 (u)]ϖ #

#I ϖ ,[θ0k+1 (u)]ϖ i

[j−1 ∏ ∥Xt

#I ϖ +1,([θ0k+1 (u)]ϖ j ,uk+1 )

− Xt

(u)]ϖ j−1

j

∥uk+1 ∥H

][

# ϖ ∏ i=j+1

#I ϖ ,[θ1k+1 (u)]ϖ i

∥Xt

i

∏#Iiϖ q=1

#I ϖ +1,([θ0k+1 (u)]ϖ j ,uk+1 )

− Xt

∏#I ϖ j

q=1

j

ϖ ∥H ∥uIj,q

∥Lp#ϖ (P;H)

ϖ ∥H ∥uIi,q

∥Lp#ϖ (P;H)

]

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

156

≤ |Gl |C #ϖ (H,V

l,0 )

b

≤ |T ∨ 1|

0k ˜ Lϖ\{I ϖ },p # d#I ϖ +1,p #ϖ (u0 , uk+1 ) ϖ j

∏

0k

t−ιI

j

⌊k/2⌋ min{1−α,1/2−β}

I∈ϖ\{Ijϖ } 0k |Gl |C #ϖ (H,V ) Lϖ\{I ϖ },p #ϖ d˜#I ϖ +1,p #ϖ (u0 , uk+1 ). l,0 j j b

(109)

Moreover, H¨ older’s inequality proves that for all l ∈ {0, 1}, p ∈ [2, ∞), ϖ ∈ Πk∗ , j ∈ {1, 2, . . . , #ϖ }, [i] m ∈ {j + 1, j + 2, . . . , #ϖ }, u = (u0 , u1 , . . . , uk+1 ) ∈ ×k+1 i=0 H , t ∈ (0, T ] it holds that k+1

#I ϖ ,[θ0 #I ϖ ,[θ0k+1 (u)]ϖ  (#ϖ ) 0,u ( #I ϖ ,[θ0k+1 (u)]ϖ 1 1 2 j−1 0 1 2 G (X ) X , X , . . . , X ∏k+1 t t t t l ∥u ∥ i H i=1 #I ϖ +1,([θ0k+1 (u)]ϖ j ,uk+1 ) j

Xt

#I ϖ ,[θ1k+1 (u)]ϖ m m

Xt

#I ϖ

#ϖ

Xt

#I ϖ

j+1

, Xt

#I ϖ ,[θ0k+1 (u)]ϖ m m

− Xt ) 

,[θ1k+1 (u)]ϖ #ϖ

m+1

, Xt

j+2

,[θ0k+1 (u)]ϖ j+2

,[θ1k+1 (u)]ϖ m+1

#I ϖ ,[θ0k+1 (u)]ϖ i

i∈{1,2,...,m−1}\{j} # ϖ ∏

·

∏#Iiϖ

l,0 )

b

[

i

∥Xt

∏

≤ |Gl |C #ϖ (H,V

#I ϖ ,[θ1k+1 (u)]ϖ i i

∥Xt

∏#Iiϖ

i=m+1

q=1

∥Xt

≤ |Gl |C #ϖ (H,V

l,0 )

b

0k+1 −ι ϖ I

j

∪{k+1}

q=1

#I ϖ

, Xt

#I ϖ

, . . . , Xt

m+2

m−1

,[θ1k+1 (u)]ϖ m+2

∥Lp#ϖ (P;H)

,[θ0k+1 (u)]ϖ m−1

,

,...,

#I ϖ ,[θ0k+1 (u)]ϖ m m

]

ϖ ∥H ∥uIi,q

∥Xt

j

∥uk+1 ∥H

− Xt ϖ ∏#Im ϖ q=1 ∥uIm,q ∥H

0

][

∥Lp#ϖ (P;H)

#I ϖ +1,([θ0k+1 (u)]ϖ j ,uk+1 )

ϖ ∥H ∥uIi,q

#I ϖ ,[θ1k+1 (u)]ϖ m m

·t

#I ϖ

#I ϖ

, Xt

,

Lp (P;Vl,0 )

[

·

,[θ0k+1 (u)]ϖ j+1

(u)]ϖ j−1

∏#Ijϖ q=1

∥Lp#ϖ (P;H)

]

ϖ ∥H ∥uIj,q

∥Lp#ϖ (P;H)

0

k L{Ik+1 ϖ ∪{k+1}},p # Lϖ\{I ϖ , I ϖ },p # d#I ϖ ,p #ϖ (u0 , u0 + uk+1 ) ϖ ϖ j

j

∏

t

m

m

0 −ιI k

ϖ} I∈ϖ\{Ijϖ , Im

≤ |T ∨ 1|

⌊k/2⌋ min{1−α,1/2−β}

|Gl |C #ϖ (H,V b

l,0 )

0

0

k L{Ik+1 ϖ ∪{k+1}},p # Lϖ\{I ϖ , I ϖ },p # d#I ϖ ,p #ϖ (u0 , u0 + uk+1 ). ϖ ϖ j

j

m

m

(110) In addition, H¨ older’s inequality also shows that for all l ∈ {0, 1}, p ∈ [2, ∞), ϖ ∈ Πk , u = (u0 , u1 , . . . , uk+1 ) ∈ [i] ×k+1 i=0 H , t ∈ (0, T ] it holds that ∫ 1 #I ϖ ,[θ1k+1 (u)]ϖ  [ (#ϖ +1) ( 0,u0 1 0,u0 +uk+1 (#ϖ +1) ( 0,u0 )]( 0,u0 ) 1  ] − G X X , G X + ρ[X − X ∏k+1 t t t t t l l  0 i=1 ∥ui ∥H  #I ϖ ,[θ1k+1 (u)]ϖ #I ϖ ,[θ1k+1 (u)]ϖ )  #ϖ 2 0,u +u Xt 2 , . . . , X t #ϖ , Xt 0 k+1 − Xt0,u0 dρ  p L (P;Vl,0 ) ∫ 1  (#ϖ +1) ( 0,u ) ( ) 0,u +u (# +1) G ≤ Xt 0 + ρ[Xt 0 k+1 − Xt0,u0 ] − Gl ϖ Xt0,u0 Lp(#ϖ +2) (P;L(#ϖ +1) (H,V )) dρ l 1

0

#I ϖ ,[θ1k+1 (u)]ϖ [# ] i 0,u +u ϖ ∏ ∥Xt i ∥Lp(#ϖ +2) (P;H) ∥Xt 0 k+1 − Xt0,u0 ∥Lp(#ϖ +2) (P;H) · ∏#Iiϖ ∥uk+1 ∥H i=1 ϖ q=1 ∥uIi,q ∥H

l,0

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

157

1

∫

 (#ϖ +1) ( 0,u ) 0,u +u (# +1) ( 0,u0 ) G  p(# +2) Xt 0 + ρ[Xt 0 k+1 − Xt0,u0 ] − Gl ϖ Xt dρ ϖ l L (P;L(#ϖ +1) (H,Vl,0 )) 0 ∏ 0k 0k ˜ · Lϖ,p(# L t−ιI ϖ +2) p(#ϖ +2) (111) I∈ϖ ∫ 1  (#ϖ +1) ( 0,u ) ( ) 0,u +u (# +1) 0,u 0,u 0 k+1 ϖ G Xt 0 + ρ[Xt ≤ − Xt 0 ] − Gl Xt 0 Lp(#ϖ +2) (P;L(#ϖ +1) (H,V )) dρ l

≤

l,0

0

⌊k/2⌋ min{1−α,1/2−β}

· |T ∨ 1|

0

k ˜ Lϖ,p(# L . ϖ +2) p(#ϖ +2)

Again H¨ older’s inequality assures that for all l ∈ {0, 1}, p ∈ [2, ∞), ϖ ∈ Πk , j ∈ {1, 2, . . . , #ϖ }, [i] u = (u0 , u1 , . . . , uk+1 ) ∈ ×k+1 i=0 H , t ∈ (0, T ] it holds that #I ϖ ,[θ0k+1 (u)]ϖ  (#ϖ +1) 0,u ( #I ϖ ,[θ0k+1 (u)]ϖ 1 1 2 0 1 2 G (X ) X , X ,..., ∏k+1 t t t l i=1 ∥ui ∥H #I ϖ ,[θ0k+1 (u)]ϖ #ϖ 0,u +u 1,(u ,u ) ) Xt #ϖ , Xt 0 k+1 − Xt0,u0 − Xt 0 k+1  p

L (P;Vl,0 )

#I ϖ ,[θ0k+1 (u)]ϖ ] [# i ϖ ∏ ∥Lp(#ϖ +1) (P;H) ∥Xt i ≤ |Gl |C #ϖ+1 (H,V ) ∏#Iiϖ l,0 b i=1 ϖ q=1 ∥uIi,q ∥H 0,u0 +uk+1

∥Xt

·

≤ |T ∨ 1|

1,(u0 ,uk+1 )

− Xt0,u0 − Xt

(112)

∥Lp(#ϖ +1) (P;H)

∥uk+1 ∥H ⌊k/2⌋ min{1−α,1/2−β} |Gl |C #ϖ +1 (H,V ) l,0 b

0k Lϖ,p(# d˜ (u0 , uk+1 ) ϖ +1) 1,p(#ϖ +1)

and k+1

1 ∏k+1 i=1

∥ui ∥H

#I ϖ ,[θ0 #I ϖ ,[θ0k+1 (u)]ϖ  (#ϖ +1) 0,u ( #I ϖ ,[θ0k+1 (u)]ϖ 1 2 G (Xt 0 ) Xt 1 , Xt 2 , . . . , Xt j−1 l

#I ϖ ,[θ1k+1 (u)]ϖ j j

Xt

#I ϖ

#ϖ

Xt

#I ϖ ,[θ0k+1 (u)]ϖ j j

− Xt

,[θ1k+1 (u)]ϖ #

ϖ

0,u0 +uk+1

#I ϖ

j+1

, Xt

,[θ1k+1 (u)]ϖ j+1

#I ϖ

, Xt

j+2

,[θ1k+1 (u)]ϖ j+2

(u)]ϖ j−1

,

,...,

) − Xt0,u0 

, Xt

Lp (P;Vl,0 )

#I ϖ ,[θ0k+1 (u)]ϖ #I ϖ ,[θ1k+1 (u)]ϖ [j−1 ][ # ] i i ϖ ∏ ∥Xt i ∏ ∥Lp(#ϖ +1) (P;H) ∥Lp(#ϖ +1) (P;H) (113) ∥Xt i ≤ |Gl |C #ϖ +1 (H,V ) ∏#Iiϖ ∏#Iiϖ l,0 b i=1 i=j+1 ϖ ϖ q=1 ∥uIi,q ∥H q=1 ∥uIi,q ∥H

[ ·

#I ϖ ,[θ1k+1 (u)]ϖ j j

∥Xt

#I ϖ ,[θ0k+1 (u)]ϖ j j

− Xt

∏#Ijϖ

≤ |T ∨ 1|

q=1 ⌊k/2⌋ min{1−α,1/2−β}

] 0,u +u ∥Lp(#ϖ +1) (P;H) ∥Xt 0 k+1 − Xt0,u0 ∥Lp(#ϖ +1) (P;H) ∥uk+1 ∥H

ϖ ∥H ∥uIj,q

|Gl |C #ϖ +1 (H,V

l,0 )

b

0

k ˜ Lϖ\{I ϖ },p(# +1) Lp(#ϖ +1) d# ϖ ,p(#ϖ +1) (u0 , u0 + uk+1 ). I ϖ j

j

[i] Combining (108)–(113) yields that for all l ∈ {0, 1}, p ∈ [2, ∞), u = (u0 , u1 , . . . , uk+1 ) ∈ ×k+1 i=0 H , t ∈ (0, T ] it holds that k,θ1k+1 (u)

¯ u (t, X ∥G t k+1,l

k,θ0k+1 (u)

− Xt

( ·

[ ∑ ϖ∈Πk∗

|Gl |C #ϖ (H,V b

l,0 )

∑

0

I∈ϖ

∑

[ ϖ∈Πk

)∥Lp (P;Vl,0 )

k ˜ Lϖ\{I},p #ϖ d#I +1,p #ϖ (u0 , uk+1 )

J∈ϖ, min(J)>min(I)

+

k,θ0k+1 (u)

− Xt

]

0k+1 + L{I∪{k+1}},p #ϖ

∑

k,θ1k+1 (u)

) − Gu k+1,l (t, Xt ∏k+1 i=1 ∥ui ∥H

0k ˜ Lϖ,p(# L ϖ +2) p(#ϖ +2)

0k Lϖ\{I,J},p #ϖ

d#J ,p #ϖ (u0 , u0 + uk+1 )

≤ |T ∨ 1|

k

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

158

1

∫

 (#ϖ +1) ( 0,u ) 0,u +u (# +1) ( 0,u0 ) G  p(# +2) Xt 0 + ρ[Xt 0 k+1 − Xt0,u0 ] − Gl ϖ Xt dρ ϖ l L (P;L(#ϖ +1) (H,Vl,0 )) 0 ( 0k + |Gl |C #ϖ+1 (H,V ) Lϖ,p(# d˜ (u0 , uk+1 ) (114) ϖ +1) 1,p(#ϖ +1) l,0 b )]) ∑ 0 ˜ p(# +1) d# ,p(# +1) (u0 , u0 + uk+1 ) + L k L . ·

ϖ\{I},p(#ϖ +1)

ϖ

ϖ

I

I∈ϖ

[i] This and Minkowski’s inequality imply that for all l ∈ {0, 1}, p ∈ [2, ∞), u = (u0 , u1 , . . . , uk+1 ) ∈ ×k+1 i=0 H it holds that

⎡ ∫ ⎢ ⎢ ⎣

⎛

k,θ1k+1 (u) ¯u T ⎜ ∥Gk+1,l (t, Xt

−

k,θ k+1 (u) Xt 0 )

k,θ1k+1 (u) − Gu k+1,l (t, Xt ∏k+1 i=1 ∥ui ∥H

⎝

0

≤ |T ∨ 1|

k

[∫

(

T

−

⎤1/(l+1)

⎞(l+1) k,θ k+1 (u) Xt 0 )∥Lp (P;Vl,0 ) ⎟

⎥ dt⎥ ⎦

⎠

[ ∑

0

ϖ∈Πk∗

|Gl |C #ϖ (H,V b

0k ˜ Lϖ\{I},p #ϖ d#I +1,p #ϖ (u0 , uk+1 )

∑ l,0 )

I∈ϖ

] 0k+1 + L{I∪{k+1}},p #ϖ

0k Lϖ\{I,J},p #ϖ

∑

d#J ,p #ϖ (u0 , u0 + uk+1 )

J∈ϖ, min(J)>min(I)

[ 0

∑

+

k ˜ Lϖ,p(# L ϖ +2) p(#ϖ +2)

ϖ∈Πk 1

∫ · 0

 (#ϖ +1) ( 0,u ) 0,u +u (# +1) ( 0,u0 )  p(# +2) G Xt 0 + ρ[Xt 0 k+1 − Xt0,u0 ] − Gl ϖ Xt ϖ l L (P;L(#ϖ +1) (H,V

l,0 ))

dρ

( + |Gl |C #ϖ+1 (H,V

l,0 )

b

0k Lϖ,p(# d˜ (u0 , uk+1 ) ϖ +1) 1,p(#ϖ +1)

]1/(l+1)

)])(l+1) +

0k Lϖ\{I},p(# ϖ +1)

∑

˜ p(# +1) d# ,p(# +1) (u0 , u0 + uk+1 ) L ϖ ϖ I

(115)

dt

I∈ϖ

≤ |T ∨ 1|

k

{[ ∫

T

(

[ ∑

0

∑

|Gl |C #ϖ (H,V

l,0 )

b

ϖ∈Πk∗

0k ˜ Lϖ\{I},p #ϖ d#I +1,p #ϖ (u0 , uk+1 )

I∈ϖ

])(l+1) 0k+1 + L{I∪{k+1}},p #ϖ

0k Lϖ\{I,J},p #ϖ

∑

d#J ,p #ϖ (u0 , u0 + uk+1 )

]1/(l+1) dt

J∈ϖ, min(J)>min(I)

+

∑

{[ ∫

T

0

ϖ∈Πk

(

0k Lϖ,p(# ϖ +2)

˜ p(# +2) L ϖ

∫ 0

1

 (#ϖ +1) ( 0,u ) 0,u +u G Xt 0 + ρ[Xt 0 k+1 − Xt0,u0 ] l

(# +1) ( 0,u0 )  p(# +2) − Gl ϖ Xt ϖ L (P;L(#ϖ +1) (H,Vl,0 ))

[∫

T

+ 0

(

)(l+1) ]1/(l+1) dρ dt

(

|Gl |C #ϖ+1 (H,V b

l,0 )

0

k d˜ (u0 , uk+1 ) Lϖ,p(# ϖ +1) 1,p(#ϖ +1)

))(l+1) +

∑ I∈ϖ

0k Lϖ\{I},p(# ϖ +1)

˜ p(# +1) d# ,p(# +1) (u0 , u0 + uk+1 ) L ϖ ϖ I

]1/(l+1) }} dt

.

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

159

[i] Jensen’s inequality hence shows that for all l ∈ {0, 1}, p ∈ [2, ∞), u = (u0 , u1 , . . . , uk+1 ) ∈ ×k+1 it holds i=0 H that

⎡ ∫ ⎢ ⎢ ⎣

0

⎛

k,θ1k+1 (u) ¯u T ⎜ ∥Gk+1,l (t, Xt

−

k,θ k+1 (u) Xt 0 )

k,θ1k+1 (u) − Gu k+1,l (t, Xt ∏k+1 i=1 ∥ui ∥H

⎝

{ ≤ |T ∨ 1|

T

1/(l+1)

l,0 )

b

⎥ dt⎥ ⎦

0

k ˜ Lϖ\{I},p #ϖ d#I +1,p #ϖ (u0 , uk+1 )

I∈ϖ

]

0k+1 + L{I∪{k+1}},p #ϖ

0k Lϖ\{I,J},p #ϖ

∑

J∈ϖ, min(J)>min(I) [∫ T

{ +

∑

|Gl |C #ϖ (H,V

ϖ∈Πk∗

0k ˜ Lϖ,p(# L ϖ +2) p(#ϖ +2)

∑

⎠

⎤1/(l+1)

[ ∑

k

−

⎞(l+1) k,θ0k+1 (u) Xt )∥Lp (P;Vl,0 ) ⎟

ϖ∈Πk

1

(∫

0

0

d#J ,p #ϖ (u0 , u0 + uk+1 )

 (#ϖ +1) ( 0,u ) 0,u +u G Xt 0 + ρ[Xt 0 k+1 − Xt0,u0 ] l

(# +1) ( 0,u0 )  p(# +2) Xt − Gl ϖ ϖ L (P;L(#ϖ +1) (H,Vl,0 ))

)(l+1) ]1/(l+1) dρ dt

( +T

1/(l+1)

0

|Gl |C #ϖ +1 (H,V

l,0 )

b

k Lϖ,p(# d˜ (u0 , uk+1 ) ϖ +1) 1,p(#ϖ +1) )}}

0k ˜ Lϖ\{I},p(# L d#I ,p(#ϖ +1) (u0 , u0 + uk+1 ) ϖ +1) p(#ϖ +1) I∈ϖ { [ ∑ 1 ∑ 0k k /(l+1) ˜ ≤ |T ∨ 1| T |Gl |C #ϖ (H,V ) Lϖ\{I},p #ϖ d#I +1,p #ϖ (u0 , uk+1 )

+

∑

+

0k Lϖ\{I,J},p #ϖ

∑

J∈ϖ, min(J)>min(I) [∫ T

{ 0

I∈ϖ

]

0k+1 + L{I∪{k+1}},p #ϖ

∑

l,0

b

ϖ∈Πk∗

k ˜ Lϖ,p(# L ϖ +2) p(#ϖ +2)

ϖ∈Πk

0

∫ 0

1

d#J ,p #ϖ (u0 , u0 + uk+1 )

 (#ϖ +1) ( 0,u ) 0,u +u G Xt 0 + ρ[Xt 0 k+1 − Xt0,u0 ] l

(# +1) ( 0,u0 ) (l+1) − Gl ϖ Xt Lp(#ϖ +2) (P;L(#ϖ +1) (H,Vl,0 ))

]1/(l+1) dρ dt

( +T

1/(l+1)

|Gl |C #ϖ +1 (H,V

l,0 )

b

+

∑

0

k Lϖ,p(# d˜ (u0 , uk+1 ) ϖ +1) 1,p(#ϖ +1) )}}

0k ˜ Lϖ\{I},p(# L d#I ,p(#ϖ +1) (u0 , u0 + uk+1 ) ϖ +1) p(#ϖ +1)

.

I∈ϖ

Combining (102) with (116) ensures that for all p ∈ [2, ∞), x ∈ H, uk+1 ∈ H \ {0} it holds that  k,(x+uk+1 ,u) k+1,(x,u,uk+1 )  k,(x,u) Xt  p − Xt − Xt L (P;H) sup sup ∏k+1 ∥u ∥ u=(u1 ,u2 ,...,uk )∈(H\{0})k t∈[0,T ] i H i=1 ) k 0,0,0 ( ≤ |T ∨ 1| χ0,T Θ |F | , |B| 1 1 A,η A,η,p,T Cb (H,H) Cb (H,HS(U,H)) { √ ] ∑ [ · T |F |C #ϖ (H,H) + p (p−1) T |B| # ϖ 2 C (H,HS(U,H)) ϖ∈Πk∗

b

[

·

∑ I∈ϖ

0

k ˜ Lϖ\{I},p #ϖ d#I +1,p #ϖ (x, uk+1 )

b

(116)

A. Andersson et al. / Nonlinear Analysis 162 (2017) 128–161

160

] 0k+1 + L{I∪{k+1}},p #ϖ

[ 0

∑

+

0k Lϖ\{I,J},p #ϖ

∑

J∈ϖ, min(J)>min(I) (∫ T

k ˜ Lϖ,p(# L ϖ +2) p(#ϖ +2)

ϖ∈Πk (#ϖ +1)

0

∫

d#J ,p #ϖ (x, x + uk+1 )

1

0,x+uk+1

∥F (#ϖ +1) (Xs0,x + ρ[Xs

0

−F (Xs0,x )∥Lp(#ϖ +2) (P;L(#ϖ +1) (H,H)) dρ ds [ ∫ T∫ 1 0,x+uk+1 ∥B (#ϖ +1) (Xs0,x + ρ[Xs + p (p−1) 2 0 0

− Xs0,x ]) ]1/2 ) (#ϖ +1) 0,x 2 −B (Xs )∥Lp(#ϖ +2) (P;L(#ϖ +1) (H,HS(U,H))) dρ ds √ [ ] + T |F |C #ϖ +1 (H,H) + p (p−1) T |B|C #ϖ +1 (H,HS(U,H)) 2 b b ( 0 · L k d˜1,p(# +1) (x, uk+1 ) ϖ,p(#ϖ +1)

∑

+

− Xs0,x ])

(117)

ϖ

0k Lϖ\{I},p(# ϖ +1)

)]} ˜ Lp(#ϖ +1) d#I ,p(#ϖ +1) (x, x + uk+1 ) .

I∈ϖ

This, (94), and (95) establish item (vi) in the case k + 1. Induction thus completes the proof of item (vi). Combining item (iii), item (v), and item (vi) with item (ii) establishes item (vii) and item (viii). Next we note that (43) and item (ii) ensure that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), x ∈ H, t ∈ [0, T ] it holds that (

k,(x,u)

H k ∋ u ↦→ [Xt

) ]P,B(H) ∈ Lp (P; H) ∈ L(k) (H, Lp (P; H)).

(118)

In addition, (87) shows that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), t ∈ [0, T ] it holds that (

( ) ) k,(x,u) H ∋ x ↦→ H k ∋ u ↦→ [Xt ]P,B(H) ∈ Lp (P; H) ∈ L(k) (H, Lp (P; H)) ∈ C(H, L(k) (H, Lp (P; H))). (119)

Combining (118) and (119) with item (ii) and item (vi) proves item (ix) and item (x). The proof of Theorem 2.1 is thus completed. □ Acknowledgments Stig Larsson and Christoph Schwab are gratefully acknowledged for a number of useful comments. This project has been supported through the SNSF-Research project 200021 156603 “Numerical approximations of nonlinear stochastic ordinary and partial differential equations” (Grant No. 156603) and the ETH Research Grant ETH-47 15-2 “Mild stochastic calculus and numerical approximations for nonlinear stochastic evolution equations with L´evy noise”. References

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