Regularity properties for solutions of infinite dimensional Kolmogorov ...

Regularity properties for solutions of infinite dimensional Kolmogorov equations in Hilbert spaces

arXiv:1611.00858v1 [math.AP] 3 Nov 2016

Adam Andersson, Mario Hefter, Arnulf Jentzen, and Ryan Kurniawan November 4, 2016 Abstract In this article we establish regularity properties for solutions of infinite dimensional Kolmogorov equations. We prove that if the nonlinear drift coefficients, the nonlinear diffusion coefficients, and the initial conditions of the considered Kolmogorov equations are n-times continuously Fr´echet differentiable, then so are the generalized solutions at every positive time. In addition, a key contribution of this work is to prove suitable enhanced regularity properties for the derivatives of the generalized solutions of the Kolmogorov equations in the sense that the dominating linear operator in the drift coefficient of the Kolmogorov equation regularizes the higher order derivatives of the solutions. Such enhanced regularity properties are of major importance for establishing weak convergence rates for spatial and temporal numerical approximations of stochastic partial differential equations.

Contents 1 Introduction 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 4

2 Some auxiliary results for the differentiation of random fields

5

3 Regularity of transition semigroups for stochastic evolution equations

10

4 Regularity of transition semigroups for mollified stochastic evolution equations 25

1

Introduction

In this article we establish regularity properties for solutions of infinite dimensional Kolmogorov equations. Infinite dimensional Kolmogorov equations are the Kolmogorov equations associated to stochastic partial differential equations (SPDEs) and such equations have been intensively studied in the literature in the last three decades (cf., e.g., Ma & R¨ockner [17], 1

R¨ockner [19], Zabczyk [27], Cerrai [6], Da Prato & Zabczyk [11], R¨ockner & Sobol [21], Da Prato [9], R¨ockner [20], R¨ockner & Sobol [22], R¨ockner & Sobol [23], Da Prato [10], and the references mentioned therein). In Theorem 1.1 below we summarize some of the main findings of this paper. In our formulation of Theorem 1.1 we employ the following notation. For every n ∈ N = {1, 2, . . . , } and every non-trivial R-Banach space (V, k·kV ) we denote by Cbn (V, R) the set of all n-times continuously Fr´echet differentiable functions f : V → R with globally bounded derivatives, we denote by k·kC n (V,R) the associated norm on Cbn (V, R) (cf. (6) below), we denote b by Lipn (V, R) the set of all functions f : V → R in Cbn (V, R) which have globally Lipschitz continuous derivatives, and we denote by |·|Lipn (V,R) an associated semi-norm on Lipn (V, R) (cf. (7) below). Theorem 1.1. Let (H, k·kH , h·, ·iH ) and (U, k·kU , h·, ·iU ) be non-trivial separable R-Hilbert spaces, let U ⊆ U be an orthonormal basis of U, let A : D(A) ⊆ H → H be a generator of a strongly continuous analytic semigroup, and let T ∈ (0, ∞), n ∈ N, F ∈ Cbn (H, H), B ∈ Cbn (H, HS(U, H)). Then

(i) it holds that there exist unique functions Pt : Cb1 (H, R) → C (H, R), t ∈ [0, T ], such that for every ϕ ∈ Cb1 (H, R) it holds that (Pt ϕ)(x) ∈ R, (t, x) ∈ [0, T ] × H, is a generalized solution of P ∂ 1 (Pt ϕ)′′ (x)(B(x)u, B(x)u) + (Pt ϕ)′ (x)[Ax + F (x)] (P ϕ)(x) = t (1) ∂t 2 u∈U

for (t, x) ∈ (0, T ] × D(A) with (P0 ϕ)(x) = ϕ(x) for x ∈ H (cf., e.g., [11, page 127]),

(ii) it holds for all k ∈ {1, . . . , n}, t ∈ [0, T ] that Pt (Cbk (H, R)) ⊆ Cbk (H, R),

(iii) it holds for all k ∈ {1, . . . , n}, t ∈ [0, T ] with |F |Lipk (H,H) + |B|Lipk (H,HS(U,H)) < ∞ that Pt (Lipk (H, R)) ⊆ Lipk (H, R), P (iv) it holds for all k ∈ {1, . . . , n}, δ1 , . . . , δk ∈ [0, 1/2) with ki=1 δi < 1/2 that # " Pk t i=1 δi |(Pt ϕ)(k) (x)(u1 , . . . , uk )| sup < ∞, sup sup sup (2) Q kϕkCbk (H,R) ki=1 kui kH−δi ϕ∈Cbk (H,R)\{0} x∈H u1 ,...,uk ∈H\{0} t∈(0,T ] and

(v) it holds for all k ∈ {1, . . . , n}, δ1 , . . . , δk ∈ [0, 1/2) with |B|Lipk (H,HS(U,H)) < ∞ that sup

sup

sup

sup

ϕ∈Lipk (H,R)\{0} x,y∈H, u1 ,...,uk t∈(0,T ] x6=y ∈H\{0}

< ∞.

"

Pk

t

i=1 δi

Pk

i=1 δi

< 1/2 and |F |Lipk (H,H) +

|[(Pt ϕ)(k) (x) − (Pt ϕ)(k) (y)](u1, . . . , uk )| Q kϕkLipk (H,R) kv − wkH ki=1 kui kH−δi

2

#

(3)

In the case n = 2, item (ii) in Theorem 1.1 is a generalization of Theorem 6.7 in Zabczyk [27] and Theorem 7.4.3 in Da Prato & Zabczyk [11] (in this paper Cb2 -functions do not necessarily need to be globally bounded; compare the sentence above Lemma 3.4 in [27] and item (ii) on page 31 in [11] with (6) in this paper). Theorem 1.1 is a straightforward consequence of Theorem 3.3 in Section 3 below. In Theorem 3.3 below we also specify for every natural number n ∈ N and every t ∈ [0, T ] an explicit formula for the n-th derivative of the generalized solution H ∋ x 7→ (Pt ϕ)(x) ∈ R of (1) at time t ∈ [0, T ]. Moreover, Theorem 3.3 below provides explicit bounds for the left hand sides of (2) and (3) (see items (vii) and (x) in Theorem 3.3 below). Next we would like to emphasize that Theorem 1.1 and Theorem 3.3, respectively, prove finiteness of (2) and (3) even though the denominators in (2) and (3) contain rather weak norms from negative Sobolev-type spaces for the multilinear arguments of the derivatives of the generalized solution. In particular, item (iv) in Theorem 1.1 above and item (vii) in Theorem 3.3 below, respectively, reveal for every p ∈ [1, ∞), k ∈ {1, 2, . . . , n}, δ1 , δ2 , . . . , δk ∈ [0, 1/2), x ∈ H, t ∈ (0, T ] that the k-th derivative (Pt ϕ)(k) (x) even takes values in the continuously embedded subspace (4) L(⊗ki=1 H−δi , R) of L(H ⊗k , R) provided that the hypothesis Pk

i=1 δi

< 1/2

(5)

is satisfied. In addition, we employ items (iv)–(v) in Theorem 1.1 above and items (vii) and (x) in Theorem 3.3 below, respectively, to establish similar a priori bounds as (2)–(3) for a family of appropriately mollified solutions of (1) which hold uniformly in the mollification parameter; see items (iv)–(v) in Corollary 4.2 below for details. Items (iv)–(v) in Theorem 1.1 above, items (vii) and (x) in Theorem 3.3 below, and, especially, items (iv)–(v) in Corollary 4.2 below, respectively, are of major importance for establishing essentially sharp probabilistically weak convergence rates for numerical approximation processes as the analytically weak norms for the multilinear arguments of the derivatives of the generalized solution (cf. the denominators in (2) and (3) above) 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 [12], Theorem 2.1 in Wang & Gan [26], Theorem 1.1 in Andersson & Larsson [3], Theorem 1.1 in Br´ehier [4], Theorem 5.1 in Br´ehier & Kopec [5], Corollary 1 in Wang [25], Corollary 5.2 in Conus et al. [8], Theorem 6.1 in Kopec [16], and Corollary 8.2 in [15]).

1.1

Notation

In this section we introduce some of the notation which we employ throughout the article (cf., e.g., [1, Section 1.1]). For two sets A and B we denote by M(A, B) the set of all mappings from A to B. For two measurable spaces (A, A) and (B, B) we denote by M(A, B) the set of 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 a Borel measurable set A ∈ B(R) we denote by µA : B(A) → [0, ∞] the Lebesgue-Borel measure on A. We denote by ⌊·⌋ : R → R and ⌈·⌉ : R → 3

R the functions which satisfy for all t ∈ R that ⌊t⌋ = max((−∞, t] ∩ {0, 1, −1, 2, −2, . . . }) and ⌈t⌉ = min([t, ∞) ∩ {0, 1, −1, 2, −2, . . . }). For R-Banach spaces (V, k·kV ) and (W, k·kW ) with #V > 1 and a natural number n ∈ N we denote by |·|C n (V,W ) : C n (V, W ) → [0, ∞] and b k·kC n (V,W ) : C n (V, W ) → [0, ∞] the functions which satisfy for all f ∈ C n (V, W ) that b

|f |C n (V,W ) = supx∈V f (n) (x) L(n) (V,W ) , b

kf kC n (V,W ) = kf (0)kW + b

Pn

k=1 |f |Cbk (V,W )

(6)

and we denote by Cbn (V, W ) the set given by Cbn (V, W ) = {f ∈ C n (V, W ) : kf kC n (V,W ) < ∞}. b For R-Banach spaces (V, k·kV ) and (W, k·kW ) with #V > 1 and a nonnegative integer n ∈ N0 we denote by |·|Lipn (V,W ) : C n (V, W ) → [0, ∞] and k·kLipn (V,W ) : C n (V, W ) → [0, ∞] the functions which satisfy for all f ∈ C n (V, W ) that    kf (x)−f (y)k W  :n=0 supx,y∈V, x6=y kx−ykV   (n) (n) |f |Lipn (V,W ) = , kf (x)−f (y)kL(n) (V,W )  (7) : n ∈ N supx,y∈V, x6=y kx−ykV P kf kLipn (V,W ) = kf (0)kW + nk=0 |f |Lipk (V,W )

and we denote by Lipn (V, W ) the set given by Lipn (V, W ) = {f ∈ C n (V, W ) : kf kLipn (V,W ) < ∞}.

We denote by Πk , Π∗k ∈ P P P(N) , k ∈ N0 , the sets which satisfy for all k ∈ N that  Π0 = Π∗0 = ∅, Π∗k = Πk \ {{1, 2, . . . , k}} , and  / A] ∧ [∪a∈A a = {1, 2, . . . , k}] ∧ [∀ a, b ∈ A : (a 6= b ⇒ a ∩ b = ∅)] (8) Πk = A ⊆ P(N) : [∅ ∈

(see, e.g., (10) in Andersson et al. [2]). For a natural number k ∈ N and a set ̟ ∈ Πk we denote ̟ ̟ by I1̟ , I2̟ , . . . , I# ∈ ̟ the sets which satisfy that min(I1̟ ) < min(I2̟ ) < · · · < min(I# ). For ̟ ̟ a natural number k ∈ N, a set ̟ ∈ Πk , and a natural number i ∈ {1, 2, . . . , #̟ } we denote by ̟ ̟ ̟ ̟ ̟ ̟ Ii,1 , Ii,2 , . . . , Ii,# ∈ Ii̟ the natural numbers which satisfy that Ii,1 < Ii,2 < · · · < Ii,# . For I̟ I̟ i

i

a measure 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) 6= g(ω)} ⊆ A)} .

1.2

(9)

Setting

Throughout this article the following setting is frequently used. Let T ∈ (0, ∞), η ∈ R, let (H, k·kH , h·, ·iH ) and (U, k·kU , h·, ·iU ) be separable R-Hilbert spaces with #H > 1, let (V, k·kV ) be a separable R-Banach space, 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 , k·kHr , h·, ·iHr ), r ∈ R, be a family interpolation spaces associated to η−A (cf., e.g., [24, Section 3.7]), for every k ∈ N, ̟ ∈ Πk , i ∈ {1, 2, . . . , #̟ } # ̟ +1 k+1 let [·]̟ → H Ii be the mapping which satisfies for all u = (u0 , u1, . . . , uk ) ∈ H k+1 i : H ̟ , uI ̟ , . . . , uI ̟ that [u]̟ ), for every k ∈ N, δ = (δ1 , δ2 , . . . , δk ) ∈ Rk , α ∈ [0, 1), i = (u0 , uIi,1 i,2 i,# ̟ I i

4

P β ∈ [0, 1/2), J ∈ P(R) let ιδ,α,β ∈ R be the real number given by ιδ,α,β = i∈J∩{1,2,...,k} δi − J J 1[2,∞)(#J∩{1,2,...,k} ) min{1 − α, 1/2 − β}, and for every separable R-Banach space (J, k·kJ ) and evRb ery a ∈ R, b ∈ (a, ∞), I ∈ B(R), X ∈ M(B(I)⊗F , B(J)) with (a, b) ⊆ I let a Xs ds ∈ L0 (P; J)  Rb Rb be the set given by a Xs ds = a 1{R b kXu kJ du 1, let (V, k·kV ) and (W, k·kW ) be separable R-Banach spaces, let (Ω, F , P) be a probability space, let X k,u ∈ ∩p∈[1,∞)Lp (P; V ), u ∈
U k+1 , k ∈ {0, 1}, satisfy for all p ∈ [1,
∞), x, u ∈ U that d U ∋ y 7→ [X 0,y ]P,B(V ) ∈ Lp (P; V ) ∈ C 1 (U, Lp (P; V )) and dx [X 0,x ]P,B(V ) u = [X 1,(x,u) ]P,B(V ) ,

and let ϕ ∈ C 1 (V, W ) satisfy that lim suppր∞ supx∈V

kϕ′ (x)kL(V,W ) |max{1,kxkV }|p

< ∞. Then

(i) it holds for all x, u ∈ U that E[kϕ(X 0,x )kW + kϕ′ (X 0,x )X 1,(x,u) kW ] < ∞,
(ii) it holds that U ∋ x 7→ E[ϕ(X 0,x )] ∈ W ∈ C 1 (U, W ), and
d (iii) it holds for all x, u ∈ U that dx E[ϕ(X 0,x )] u = E[ϕ′ (X 0,x )X 1,(x,u) ].

Proof. Throughout this proof let ck,r ∈ [0, ∞], r ∈ (0, ∞), k ∈ {0, 1}, be the extended real numbers which satisfy for all r ∈ (0, ∞) that     kϕ′ (x)kL(V,W ) kϕ(x)kW and c1,r = sup (10) c0,r = sup r r x∈V |max{1, kxkV }| x∈V |max{1, kxkV }| and let p ∈ [1, ∞) be a real number which satisfies that c1,p < ∞. We note that the fundamental theorem of calculus implies that for all x ∈ V it holds that

Z 1

Z 1

′ kϕ(x) − ϕ(0)kW = kϕ′ (ρx)kL(V,W ) kxkV dρ ϕ (ρx)x dρ

≤ 0

0

W

p

≤ c1,p kxkV supρ∈[0,1] |max{1, kρxkV }| = c1,p kxkV |max{1, kxkV }|p

(11)

≤ c1,p |max{1, kxkV }|(p+1) .

This ensures that c0,p+1 < ∞. H¨older’s inequality and the fact that c1,p < ∞ therefore show that for all x, u ∈ U it holds that E[kϕ′ (X 0,x )X 1,(x,u) kW ] ≤ c1,p E[|max{1, kX 0,x kV }|p kX 1,(x,u) kV ] ≤ c1,p kmax{1, kX 0,xkV }kpL2p (P;R) kX 1,(x,u) kL2 (P;V ) < ∞

(12)

and E[kϕ(X 0,x )kW ] ≤ c0,p+1 E[|max{1, kX 0,xkV }|(p+1) ] < ∞.

(13)

This proves item (i). Next (12) and the fact that ∀ q ∈ [1, ∞), x ∈ U : U ∋ u 7→
note that 1,(x,u) q q [X ]P,B(V ) ∈ L (P; V ) ∈ L(U, L (P; V )) ensure that for every x ∈ U it holds 5

a) that sup u∈U, kukU =1

kE[ϕ′ (X 0,x )X 1,(x,u)]kW

≤ c1,p kmax{1, kX 0,xkV }kpL2p (P;R)

sup u∈U, kukU =1

kX 1,(x,u) kL2 (P;V ) < ∞

(14)

and
b) that the function U ∋ u 7→ E[ϕ′ (X 0,x )X 1,(x,u) ] ∈ W is linear. Hence, we obtain that


U ∋ u 7→ E[ϕ′ (X 0,x )X 1,(x,u) ] ∈ W ∈ L(U, W ).

In the next step we demonstrate that for all x ∈ U it holds that   kE[ϕ(X 0,x+u )] − E[ϕ(X 0,x )] − E[ϕ′ (X 0,x )X 1,(x,u) ]kW = 0. lim sup kukU U \{0}∋u→0

(15)

(16)

For this we first observe that for all x, u ∈ U it holds that kE[ϕ(X 0,x+u )] − E[ϕ(X 0,x )] − E[ϕ′ (X 0,x )X 1,(x,u) ]kW ≤ kE[ϕ(X 0,x+u ) − ϕ(X 0,x ) − ϕ′ (X 0,x )(X 0,x+u − X 0,x )]kW

(17)

+ kE[ϕ′ (X 0,x )(X 0,x+u − X 0,x − X 1,(x,u) )]kW .

Moreover, we note that H¨older’s inequality and the fact that c1,p < ∞ ensure that for all x ∈ U it holds that   kE[ϕ′ (X 0,x )(X 0,x+u − X 0,x − X 1,(x,u) )]kW lim sup kukU U \{0}∋u→0   0,x+u kX − X 0,x − X 1,(x,u)kL2 (P;V ) ′ 0,x ≤ kϕ (X )kL2 (P;L(V,W )) lim sup (18) kukU U \{0}∋u→0  0,x+u  kX − X 0,x − X 1,(x,u)kL2 (P;V ) p 0,x = 0. ≤ c1,p kmax{1, kX kV }kL2p (P;R) lim sup kukU U \{0}∋u→0 Furthermore, we observe that the fundamental theorem of calculus shows that for all x, u ∈ U it holds that kϕ(X 0,x+u ) − ϕ(X 0,x ) − ϕ′ (X 0,x )(X 0,x+u − X 0,x )kW

Z 1

 ′ 0,x  0,x+u 0,x+u 0,x ′ 0,x 0,x

= ϕ (X + ρ[X − X ]) − ϕ (X ) (X − X ) dρ

0 W Z 1 ≤ kX 0,x+u − X 0,x kV kϕ′ (X 0,x + ρ[X 0,x+u − X 0,x ]) − ϕ′ (X 0,x )kL(V,W ) dρ. 0

6

(19)

H¨older’s inequality and Jensen’s inequality therefore imply that for all x, u ∈ U it holds that kE[ϕ(X 0,x+u ) − ϕ(X 0,x ) − ϕ′ (X 0,x )(X 0,x+u − X 0,x )]kW  Z 1 1/2 ′ 0,x 0,x+u 0,x ′ 0,x 2 ≤ E kϕ (X + ρ[X − X ]) − ϕ (X )kL(V,W ) dρ · kX

0 0,x+u

(20)

− X 0,x kL2 (P;V ) .

Moreover, note that for all q ∈ (2, ∞), ρ ∈ [0, 1], x, y ∈ U it holds that   E kϕ′ (X 0,x + ρ[X 0,y − X 0,x ])kqL(V,W )   ≤ |c1,p |q E |max{1, kX 0,x + ρ[X 0,y − X 0,x ]kpV }|q (21)   ≤ |c1,p |q E |max{1, kX 0,x kV , kX 0,y kV }|pq    0,y pq 
≤ |c1,p |q 1 + E kX 0,x kpq kV . V + E kX
This and the fact that ∀ q ∈ [1, ∞) : U ∋ x 7→ [X 0,x ]P,B(V ) ∈ Lq (P; V ) ∈ C (U, Lq (P; V )) ensure that for all q ∈ (2, ∞), x ∈ U it holds that Z 1    
lim sup E kϕ′ (X 0,x + ρ[X 0,x+u − X 0,x ])kqL(V,W ) dρ ≤ |c1,p |q 1 + 2 E kX 0,x kpq V (22) U ∋u→0 0 < ∞.
In addition, observe that the fact that ∀ q ∈ [1, ∞) : U ∋ x 7→ [X 0,x ]P,B(V ) ∈ Lq (P; V ) ∈ C (U, Lq (P; V )) shows that for all x ∈ U it holds that   lim supU ∋y→x E min{1, kX 0,x − X 0,y kV } = 0. (23) This implies that for all ρ ∈ [0, 1], x ∈ U it holds that   lim supU ∋y→x E min{1, k(X 0,x + ρ[X 0,y − X 0,x ]) − X 0,x kV } = 0.

(24)

The fact that ϕ′ ∈ C (V, L(V, W )) hence ensures that for all ρ ∈ [0, 1], x ∈ U it holds that   lim supU ∋y→x E min{1, kϕ′(X 0,x + ρ[X 0,y − X 0,x ]) − ϕ′ (X 0,x )kL(V,W ) } = 0. (25)

This and Lebesgue’s theorem of dominated convergence imply that for all x ∈ U it holds that Z 1   lim sup E min{1, kϕ′(X 0,x + ρ[X 0,x+u − X 0,x ]) − ϕ′ (X 0,x )kL(V,W )} dρ = 0. (26) U ∋u→0

0

Combining this and, e.g., Lemma 4.2 in Hutzenthaler et al. [14] (with I = {∅}, (Ω, F , P) = ([0, 1] × Ω, B([0, 1]) ⊗ F , µ[0,1] ⊗ P), c = 1, X n (∅, (ρ, ω)) = kϕ′ (X 0,x (ω) + ρ[X 0,x+un (ω) − X 0,x (ω)]) − ϕ′ (X 0,x (ω))kL(V,W ) for (ρ, ω) ∈ [0, 1] × Ω, n ∈ N, x ∈ U, (um )m∈N ∈ {v ∈

7

M(N, U) : lim supm→∞ kvm kU = 0} in the notation of Lemma 4.2 in Hutzenthaler et al. [14]) establishes that for all ε ∈ (0, ∞), x ∈ U and all sequences (un )n∈N ⊆ U with lim supn→∞ kun kU = 0 it holds that  lim supn→∞ (µ[0,1] ⊗ P) (ρ, ω) ∈ [0, 1] × Ω : kϕ′ (X 0,x (ω) + ρ[X 0,x+un (ω) − X 0,x (ω)])
− ϕ′ (X 0,x (ω))kL(V,W ) ≥ ε = 0. (27)

This, (22), and, e.g., Proposition 4.5 in Hutzenthaler et al. [14] (with I = {∅}, (Ω, F , P) = ([0, 1] × Ω, B([0, 1]) ⊗ F , µ[0,1] ⊗ P), p = q, V = R, X n (∅, (ρ, ω)) = kϕ′ (X 0,x (ω) + ρ[X 0,x+un (ω) − X 0,x (ω)])−ϕ′ (X 0,x (ω))kL(V,W ) for (ρ, ω) ∈ [0, 1]×Ω, n ∈ N0 , x ∈ U, q ∈ (2, ∞), (um )m∈N0 ∈ {v ∈ M(N0 , U) : lim supm→∞ kvm kU = kv0 kU = 0} in the notation of Proposition 4.5 in Hutzenthaler et al. [14]) yield that for all x ∈ U and all sequences (un )n∈N0 ⊆ U with lim supn→∞ kun kU = ku0kU = 0 it holds that Z 1   lim sup E kϕ′ (X 0,x + ρ[X 0,x+un − X 0,x ]) − ϕ′ (X 0,x )k2L(V,W ) dρ = 0. (28) n→∞

0

Moreover, observe that the
triangle qinequality and the fact that ∀ q ∈ [1, ∞), x ∈ U : U ∋ u 7→ 1,(x,u) q [X ]P,B(V ) ∈ L (P; V ) ∈ L(U, L (P; V )) assure that for all x ∈ U it holds that   kX 0,x+u − X 0,x kL2 (P;V ) lim sup kukU U \{0}∋u→0     kX 1,(x,u)kL2 (P;V ) kX 0,x+u − X 0,x − X 1,(x,u) kL2 (P;V ) + sup ≤ lim sup (29) kukU kukU u∈U \{0} U \{0}∋u→0   1,(x,u) kX kL2 (P;V ) < ∞. = sup kukU u∈U \{0}

Putting (28)–(29) into (20) yields that for all x ∈ U it holds that   kE[ϕ(X 0,x+u ) − ϕ(X 0,x ) − ϕ′ (X 0,x )(X 0,x+u − X 0,x )]kW lim sup kukU U \{0}∋u→0  0,x+u  kX − X 0,x kL2 (P;V ) ≤ lim sup kukU U \{0}∋u→0 " #1/2 Z 1  ′ 0,x  · lim sup E kϕ (X + ρ[X 0,x+u − X 0,x ]) − ϕ′ (X 0,x )k2L(V,W ) dρ = 0. U \{0}∋u→0

(30)

0

Combining (17), (18), and (30) proves (16). In the next step we demonstrate that

U ∋ x 7→ U ∋ u 7→ E[ϕ′ (X 0,x )X 1,(x,u) ] ∈ W ∈ L(U, W ) ∈ C (U, L(U, W )).

(31)

Observe that (21) and the fact that ∀ q ∈ [1, ∞) : lim supU ∋y→x E[kX 0,y kqV ] = E[kX 0,x kqV ] < ∞ ensure that for all q ∈ (2, ∞), ρ ∈ [0, 1], x ∈ U it holds that   lim supU ∋y→x E kϕ′ (X 0,x + ρ[X 0,y − X 0,x ])kqL(V,W )   0,y pq 
 (32) + lim sup E kX kV ≤ |c1,p |q 1 + E kX 0,x kpq U ∋y→x  0,xV pq 
q = |c1,p | 1 + 2 E kX kV < ∞. 8

Hence, we obtain that for all q ∈ (2, ∞), x ∈ U it holds that   lim sup E kϕ′ (X 0,x ) − ϕ′ (X 0,y )kqL(V,W ) U ∋y→x   ≤ 2q lim sup max E kϕ′ (X 0,x + ρ[X 0,y − X 0,x ])kqL(V,W ) < ∞.

(33)

U ∋y→x ρ∈{0,1}

Moreover, note that (25) (with ρ = 1 in the notation of (25)) and, e.g., Lemma 4.2 in Hutzenthaler et al. [14] (with I = {∅}, (Ω, F , P) = (Ω, F , P), c = 1, X n (∅, ω) = kϕ′ (X 0,un (ω)) − ϕ′ (X 0,u0 (ω))kL(V,W ) for ω ∈ Ω, n ∈ N, (um )m∈N0 ∈ {v ∈ M(N0 , U) : lim supm→∞ kvm −v0 kU = 0} in the notation of Lemma 4.2 in Hutzenthaler et al. [14]) establishes that for all ε ∈ (0, ∞) and all sequences (un )n∈N0 ⊆ U with lim supn→∞ kun − u0 kU = 0 it holds that
lim supn→∞ P kϕ′ (X 0,un ) − ϕ′ (X 0,u0 )kL(V,W ) ≥ ε = 0. (34)

Combining this, (33), and, e.g., Proposition 4.5 in Hutzenthaler et al. [14] (with I = {∅}, (Ω, F , P) = (Ω, F , P), p = q, V = R, X n (∅, ω) = kϕ′ (X 0,un (ω)) − ϕ′ (X 0,u0 (ω))kL(V,W ) for ω ∈ Ω, q ∈ (2, ∞), n ∈ N0 , (um )m∈N0 ∈ {v ∈ M(N0 , U) : lim supm→∞ kvm − v0 kU = 0} in the notation of Proposition 4.5 in Hutzenthaler et al. [14]) yields that for all sequences (un )n∈N0 ⊆ U with lim supn→∞ kun − u0 kU = 0 it holds that   lim supn→∞ E kϕ′ (X 0,un ) − ϕ′ (X 0,u0 )k2L(V,W ) = 0. (35) Next observe that the fact that
for everyq q ∈ [1, ∞) it holds that the function U ∋ x 7→ U ∋ 1,(x,u) q u 7→ [X ]P,B(V ) ∈ L (P; V ) ∈ L(U, L (P; V )) is continuous shows that for all x ∈ U it holds that lim sup sup kX 1,(y,u) kL2 (P;V ) = sup kX 1,(x,u) kL2(P;V ) < ∞ (36) U ∋y→x u∈U, kukU =1

u∈U, kukU =1

and lim sup

sup

U ∋y→x u∈U, kukU =1

kX 1,(x,u) − X 1,(y,u) kL2 (P;V ) = 0.

(37)

H¨older’s inequality and (33) hence ensure that for all x ∈ U it holds that

E[ϕ′ (X 0,x )X 1,(x,u) ] − E[ϕ′ (X 0,y )X 1,(y,u) ] lim sup sup W U ∋y→x u∈U, kukU =1

≤ lim sup

sup

+ lim sup

sup

U ∋y→x u∈U, kukU =1

U ∋y→x u∈U, kukU =1

  E kϕ′ (X 0,x )(X 1,(x,u) − X 1,(y,u) )kW

  E k[ϕ′ (X 0,x ) − ϕ′ (X 0,y )]X 1,(y,u) kW

≤ kϕ′ (X 0,x )kL2 (P;L(V,W )) lim sup sup kX 1,(x,u) − X 1,(y,u) kL2 (P;V ) U ∋y→x u∈U, kukU =1   ′ 0,x ′ 0,y + lim sup kϕ (X ) − ϕ (X )kL2(P;L(V,W )) lim sup sup kX 1,(y,u) kL2 (P;V ) U ∋y→x

≤ c1,p kmax{1, kX 

′

+ lim sup kϕ (X U ∋y→x

0,x

0,x

kV }kpL2p (P;R) ′

) − ϕ (X

0,y

U ∋y→x u∈U, kukU =1

lim sup

sup

U ∋y→x u∈U, kukU =1

)kL2(P;L(V,W )) 9



kX 1,(x,u) − X 1,(y,u) kL2 (P;V )

sup u∈U, kukU =1

kX 1,(x,u) kL2 (P;V ) = 0.

(38)

This proves (31). Combining (15), (16), and (31) establishes item (ii) and item (iii). The proof of Lemma 2.1 is thus completed. Lemma 2.2 (Pointwise differentiation). Let (V, k·kV ) and (W, k·kW ) be R-Banach spaces with #V > 1 and let n ∈ N, f ∈ C n (V, W ), g ∈ C (V, L(n+1) (V, W )) satisfy for all u = (u1, u2 , . . . , un ) ∈ V n , x ∈ V that  (n)  kf (x + h)u − f (n) (x)u − g(x)(u1, u2 , . . . , un , h)kW lim sup = 0. (39) khkV V \{0}∋h→0 Then it holds that f ∈ C n+1 (V, W ) and f (n+1) = g. Proof. We first
note that (39) and the fact that ∀ x, u1 , u2 , . . . , un ∈ V : V ∋ h 7→
g(x)(u1 , u2 ,
. . . , un , h) ∈ W ∈ L(V, W ) and V ∋ y 7→ V ∋ h 7→ g(y)(u1, u2 , . . . , un , h) ∈ W ∈ L(V, W ) ∈ C (V, L(V, W ))
imply that for all u = (u1 , u2, .

. . , un ) ∈ V n , x, h ∈ V it holds that V ∋ y 7→ d f (n) (y)u ∈ W ∈ C 1 (V, W ) and dx f (n) (x)u h = g(x)(u1 , u2 , . . . , un , h). This and the fundamental theorem of calculus imply that for all u = (u1 , u2 , . . . , un ) ∈ V n , x, h ∈ V it holds that kf (n) (x + h)u − f (n) (x)u − g(x)(u1, u2 , . . . , un , h)kW

R1

 = 0 g(x + ρh) − g(x) (u1 , u2, . . . , un , h) dρ W  Qn R1 ≤ khkV kg(x + ρh) − g(x)kL(n+1) (V,W ) dρ. i=1 kui kV 0

(40)

In addition, observe that the assumption that g ∈ C (V, L(n+1) (V, W )) ensures that for all x ∈ V it holds that lim sup sup kg(x + ρh)kL(n+1) (V,W ) < ∞. (41) V ∋h→0 ρ∈[0,1]

Lebesgue’s theorem of dominated convergence therefore ensures that for all x ∈ V it holds that R1 lim supV ∋h→0 0 kg(x + ρh) − g(x)kL(n+1) (V,W ) dρ = 0. (42)

Combining (40) with (42) yields that for all x ∈ V it holds that   (n) kf (x + h)u − f (n) (x)u − g(x)(u1, u2 , . . . , un , h)kW Q = 0. lim sup sup khkV ni=1 kui kV V \{0}∋h→0 u=(u1 ,u2 ,...,un )∈(V \{0})n (43) (n+1) This and the assumption that g ∈ C (V, L (V, W )) complete the proof of Lemma 2.2.

3

Regularity of transition semigroups for stochastic evolution equations

This section establishes regularity properties of the transition semigroup.

10

Lemma 3.1. Let (V, k·kV ) and (W, k·kW ) be R-Banach spaces with #V > 1, let n ∈ N, ϕ ∈ C n+1 (V, W ), and let Φ : V n+1 → W be the function which satisfies for all v = (v1 , v2 , . . . , vn+1 ) ∈ V n+1 that Φ(v) = ϕ(n) (vn+1 )(v1 , v2 , . . . , vn ). Then it holds for all v = (v1 , v2 , . . . , vn+1 ), h = (h1 , h2 , . . . , hn+1 ) ∈ V n+1 that Φ ∈ C 1 (V n+1 , W ) and Φ′ (v)h =ϕ(n+1) (vn+1 )(v1 , v2 , . . . , vn , hn+1 ) P + ni=1 ϕ(n) (vn+1 )(v1 , v2 , . . . , vi−1 , hi , vi+1 , vi+2 , . . . , vn ).

(44)

Proof. Throughout this proof let P : V n+1 → L(n) (V, W ) × V n , β : L(n) (V, W ) × V n → W , and φ : V 2 → L(n) (V, W ) be the functions which satisfy for all A ∈ L(n) (V, W ), v = (v1 , v2 , . . . , vn ) ∈ V n , v, h ∈ V that P (v1 , v2 , . . . , vn , v) = (ϕ(n) (v), v),

β(A, v) = A(v1 , v2 , . . . , vn ),

(45)

and φ(v, h)v = ϕ(n+1) (v)(v1 , v2 , . . . , vn , h).

(46)

We note that for all v = (v1 , v2 , . . . , vn ) ∈ V n , v ∈ V it holds that Φ(v1 , v2 , . . . , vn , v) = β(P (v1, v2 , . . . , vn , v)).

(47)

Furthermore, observe that the assumption that ϕ ∈ C n+1 (V, W ) ensures that for all v = (v1 , v2 , . . . , vn ), h = (h1 , h2 , . . . , hn ) ∈ V n , v, h ∈ V it holds that P ∈ C 1 (V n+1 , L(n) (V, W ) × V n ) and P ′ (v1 , v2 , . . . , vn , v)(h1 , h2 , . . . , hn , h) = (φ(v, h), h). (48) Moreover, the fact that β is an (n+1)-multilinear and continuous function and, e.g., Theorem 3.7 in Coleman [7] assure that for all A, A˜ ∈ L(n) (V, W ), v = (v1 , v2 , . . . , vn ), h = (h1 , h2 , . . . , hn ) ∈ V n it holds that β ∈ C 1 (L(n) (V, W ) × V n , W ) and ˜ h) = A(v ˜ 1 , v2 , . . . , vn ) + β (A, v)(A, ′

n X

A(v1 , v2 , . . . , vi−1 , hi , vi+1 , vi+2 , . . . , vn ).

(49)

i=1

Combining (47)–(49) with the chain rule yields that for all v = (v1 , v2 , . . . , vn ), h = (h1 , h2 , . . . , hn ) ∈ V n , v, h ∈ V it holds that Φ ∈ C 1 (V n+1 , W ) and Φ′ (v1 , v2 , . . . , vn , v)(h1 , h2 , . . . , hn , h) = β ′ (P (v1 , v2 , . . . , vn , v))P ′(v1 , v2 , . . . , vn , v)(h1 , h2 , . . . , hn , h) = β ′ (P (v1 , v2 , . . . , vn , v))(φ(v, h), h) = β ′ (ϕ(n) (v), v)(φ(v, h), h) n X = φ(v, h)v + ϕ(n) (v)(v1 , v2 , . . . , vi−1 , hi , vi+1 , vi+2 , . . . , vn ) i=1

=ϕ

(n+1)

(v)(v1 , v2 , . . . , vn , h) +

n X

ϕ(n) (v)(v1 , v2 , . . . , vi−1 , hi , vi+1 , vi+2 , . . . , vn ).

i=1

This implies (44). The proof of Lemma 3.1 is thus completed. 11

(50)

Lemma 3.2. Assume the setting in Section 1.2, let n ∈ N, ϕ ∈ Cbn (H, V ), F ∈ Cbn (H, H), B ∈ Cbn (H, HS(U, H)), 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 satisfy all k ∈ {0, 1, . . . , n}, u = (u0 , u1 , . . . , uk ) ∈  for k+1 k,u p H , p ∈ (0, ∞), t ∈ [0, T ] that sups∈[0,T ] E kXs kH < ∞ and [Xtk,u − etA 1{0,1} (k) uk ]P,B(H) " Z t = e(t−s)A 1{0} (k) F (Xs0,u0 ) 0

+

X

F (#̟ ) (Xs0,u0 )

̟∈Πk

+

Z

t

ds

#

dWs ,

(51)

e(t−s)A 1{0} (k) B(Xs0,u0 )

0

+

"

#

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

X

̟∈Πk

B (#̟ ) (Xs0,u0 )

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

and let φ : [0, T ] × H → V be the function which satisfies for all t ∈ [0, T ], x ∈ H that φ(t, x) = E[ϕ(Xt0,x )]. Then (i) it holds for all k ∈ {1, 2, . . . , n}, u = (u0 , u1, . . . , uk ) ∈ H k+1, t ∈ [0, T ] that X

̟∈Πk

i h #I ̟ ,[u]̟ #I ̟ ,[u]̟ #I ̟ ,[u]̟ #̟
2 1

< ∞, , . . . , Xt # ̟ , Xt 2 E ϕ(#̟ ) (Xt0,u0 ) Xt 1 V

(52)


(ii) it holds for all t ∈ [0, T ] that H ∋ x 7→ φ(t, x) ∈ V ∈ Cbn (H, V ),

(iii) it holds for all k ∈ {1, 2, . . . , n}, u ∈ H k , x ∈ H, t ∈ [0, T ] that ∂k φ ∂xk




(t, x)u i X h #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #̟
2 1 0,x #̟ (#̟ ) 2 1 , , . . . , Xt , Xt E ϕ (Xt ) Xt =

(53)

̟∈Πk

(iv) it holds for all pP∈ (0, ∞), k ∈ {1, 2, . . . , n}, δ = (δ1 , δ2 , . . . , δk ) ∈ [0, 1/2)k , α ∈ [0, 1), β ∈ [0, 1/2) with ki=1 δi < 1/2 that " δ,α,β # k,(x,u) tιN kXt kLp (P;H) sup sup sup < ∞, (54) Qk x∈H u=(u1 ,u2 ,...,uk )∈(H\{0})k t∈(0,T ] i=1 kui kH−δi

12

(v) P it holds for all k ∈ {1, 2, . . . , n}, δ = (δ1 , δ2 , . . . , δk ) ∈ [0, 1/2)k , α ∈ [0, 1), β ∈ [0, 1/2) with k 1 i=1 δi < /2 that " Pk

#

∂k


t i=1 δi ∂x k φ (t, v)u V sup sup sup Qk v∈H u=(u1 ,u2 ,...,uk )∈(H\{0})k t∈(0,T ] i=1 kui kH−δi ≤ |T ∨ 1|⌊k/2⌋ min{1−α, /2−β} kϕkCbk (H,V ) # " δ,α,β # ,(x,u) X Y tιI kXt I kL#̟ (P;H) Q < ∞, sup sup sup · x∈H u=(ui )i∈I ∈(H\{0})#I t∈(0,T ] i∈I kui kH−δi ̟∈Π I∈̟ 1

(55)

k

(vi) it holds for all p ∈ (0, ∞) that

sup sup x,y∈H, t∈(0,T ] x6=y

"

# kXt0,x − Xt0,y kLp (P;H) < ∞, kx − ykH

(56)

(vii) it holds for all pP∈ (0, ∞), k ∈ {1, 2, . . . , n}, δ = (δ1 , δ2 , . . . , δk ) ∈ [0, 1/2)k , α ∈ [0, 1), β ∈ [0, 1/2) with ki=1 δi < 1/2 and |F |Lipk (H,H−α ) + |B|Lipk (H,HS(U,H−β )) < ∞ that " (δ,0),α,β # k,(x,u) k,(y,u) kXt − Xt kLp (P;H) tιN sup sup sup < ∞, (57) Q x,y∈H, u=(u1 ,u2 ,...,uk )∈(H\{0})k t∈(0,T ] kx − ykH ki=1 kui kH−δi x6=y

and

(viii) P it holds for all k ∈ {1, 2, . . . , n}, δ = (δ1 , δ2 , . . . , δk ) ∈ [0, 1/2)k , α ∈ [0, 1), β ∈ [0, 1/2) with k 1 i=1 δi < /2 and |F |Lipk (H,H−α ) + |B|Lipk (H,HS(U,H−β )) + |ϕ|Lipk (H,V ) < ∞ that " Pk  k

 # ∂k ∂ φ (t, w) u V t i=1 δi ∂x k φ (t, v) − k ∂x sup sup sup Q v,w∈H, u=(u1 ,u2 ,...,uk )∈(H\{0})k t∈(0,T ] kv − wkH ki=1 kui kH−δi v6=w

≤ |T ∨ 1|⌈k/2⌉ min{1−α, /2−β} kϕkLipk (H,V ) " 0,x ( # X kXt − Xt0,y kL#̟ +1 (P;H) sup sup · kx − ykH x,y∈H, t∈(0,T ] ̟∈Πk x6=y " δ,α,β # # ,(x,u) Y tιI kXt I kL#̟ +1 (P;H) Q sup · sup sup x∈H u=(ui )i∈I ∈(H\{0})#I t∈(0,T ] i∈I kui kH−δi I∈̟ " ι(δ,0),α,β # # ,(x,u) # ,(y,u) X t I∪{k+1} kXt I − Xt I kL#̟ (P;H) Q sup + sup sup kx − yk x,y∈H, u=(ui )i∈I ∈(H\{0})#I t∈(0,T ] H i∈I kui kH−δi I∈̟ x6=y #) " δ,α,β # ,(x,u) Y tιJ kXt J kL#̟ (P;H) Q < ∞. · sup sup sup x∈H u=(ui )i∈J ∈(H\{0})#J t∈(0,T ] i∈J kui kH−δi 1

J∈̟\{I}

13

(58)

Proof. Throughout this proof let α ∈ [0, 1), β ∈ [0, 1/2) and let Dk ∈ P(Rk ), k ∈ N, be the P k 1 sets which satisfy for all k ∈ N that Dk = {(δ1 , δ2 , . . . , δk ) ∈ [0, 1/2)k : i=1 δi < /2}. Note that k+1 H¨older’s inequality shows that for all k ∈ {1, 2, . . . , n}, u = (u0 , u1 , . . . , uk ) ∈ H , t ∈ [0, T ] it holds that i X h #I ̟ ,[u]̟ #I ̟ ,[u]̟ #I ̟ ,[u]̟ #̟
2 1

, . . . , Xt # ̟ , Xt 2 E ϕ(#̟ ) (Xt0,u0 ) Xt 1 V ̟∈Πk

≤

X

̟∈Πk

|ϕ|C #̟ (H,V ) b

#̟ Y i=1

#I ̟ ,[u]̟ i kL#̟ (P;H) . kXt i

(59)

This, the assumption that ϕ ∈ Cbn (H, V ), and the assumption that ∀ k ∈ {1, 2, . . . , n}, u ∈  H k+1, p ∈ (0, ∞) : supt∈[0,T ] E kXtk,ukpH < ∞ establish item (i). Moreover, note that (59) implies that for all k ∈ {1, 2, . . . , n}, δ1 , δ2 , . . . , δk ∈ [0, ∞), u = (u1 , u2, . . . , uk ) ∈ (H \ {0})k , x ∈ H, t ∈ [0, T ] it holds that 1 Qk

i=1

≤

kui kH−δi

X

̟∈Πk

X

̟∈Πk

i h #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #̟
2 1 0,x #̟ (#̟ ) 2 1

, . . . , Xt , Xt E ϕ (Xt ) Xt V #

̟ ,[(x,u)]̟ i

I #̟ Y kXt i |ϕ|C #̟ (H,V ) Q#Ii̟ b

i=1

j=1

kL#̟ (P;H)

(60)

.

̟ kH kuIi,j −δI ̟

i,j

In addition, (51) and item (ii) of Theorem 2.1 in [2] (with T = T , η = η, H = H, U = U, W = W , A = A, n = n, F = F , B = B, α = 0, β = 0, k = k, p = p, δ = (0, 0, . . . , 0) ∈ Rk for p ∈ [2, ∞), k ∈ {1, 2, . . . , n} in the notation of Theorem 2.1 in [2]) ensure that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), t ∈ [0, T ] it holds that " # kXtk,ukLp (P;H) sup < ∞. (61) Qk ku k u=(u0 ,u1 ,...,uk )∈H×(H\{0})k i H i=1

This, Jensen’s inequality, and (60) (with k = k, δ1 = 0, δ2 = 0, . . ., δk = 0 for k ∈ {1, 2, . . . , n} in the notation of (60)) imply that for all k ∈ {1, 2, . . . , n}, t ∈ [0, T ] it holds that  1 Qk sup sup ku k x∈H u=(u1 ,u2 ,...,uk )∈(H\{0})k

i=1

i H

i h #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #̟
2 1 0,x #̟ (#̟ ) 2 1

, . . . , Xt , Xt E ϕ (Xt ) Xt · V ̟∈Πk ! # ,(x,u) X Y kXt I kL#̟ (P;H) Q |ϕ|C #̟ (H,V ) ≤ sup sup < ∞. b x∈H u=(ui )i∈I ∈(H\{0})#I i∈I kui kH ̟∈Π I∈̟ P

(62)

k

Furthermore, (51) and item (iii) of Theorem 2.1 in [2] (with T = T , η = η, H = H, U = U, W = W , A = A, n = n, F = F , B = B, α = 0, β = 0, k = k, p = p, x = x for 14

x ∈ H, p ∈ [2, ∞), k ∈ {1, 2, . . . , n} in the notation of Theorem 2.1 in [2]) assure that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), x ∈ H, t ∈ [0, T ] it holds that
k,(x,u) H k ∋ u 7→ [Xt ]P,B(H) ∈ Lp (P; H) ∈ L(k) (H, Lp (P; H)). (63) This and (62) establish that for all k ∈ {1, 2, . . . , n}, x ∈ H, t ∈ [0, T ] it holds that

i   h #I ̟ ,[(x,u)]̟ P #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #̟
2 1 ∈V , . . . , Xt # ̟ H k ∋ u 7→ ̟∈Πk E ϕ(#̟ ) (Xt0,x ) Xt 1 , Xt 2

∈ L(k) (H, V ). (64)

In the next step we prove that for all k ∈ {1, 2, . . . , n}, t ∈ [0, T ] it holds that   h P #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ 2 1 ,..., H ∋ x 7→ H k ∋ u 7→ ̟∈Πk E ϕ(#̟ ) (Xt0,x ) Xt 1 , Xt 2 i   #I ̟ ,[(x,u)]̟ #̟
∈ V ∈ L(k) (H, V ) ∈ C (H, L(k) (H, V )). (65) Xt # ̟

For this we observe that the triangle inequality and H¨older’s inequality imply that for all k ∈ {1, 2, . . . , n}, δ1 , δ2 , . . . , δk ∈ [0, ∞), u = (u1 , u2 , . . . , uk ) ∈ (H \ {0})k , x, y ∈ H, t ∈ [0, T ] it holds that

#I ̟ ,[(x,u)]̟

P h (# ) 0,x #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #̟
2 1 #̟ 1 2 1 ̟

Qk , . . . , X , X E ϕ (X ) X t t t t

i=1 kui kH−δi ̟∈Πk i #I ̟ ,[(y,u)]̟ #I ̟ ,[(y,u)]̟ #I ̟ ,[(y,u)]̟ #̟
2 1 0,y #̟ (#̟ ) 2 1

, . . . , Xt , Xt −ϕ (Xt ) Xt

V  h   ̟ ,[(x,u)]̟ # #I ̟ ,[(x,u)]̟ P I 1 2 ≤ Qk ku1 k E ϕ(#̟ ) (Xt0,x ) − ϕ(#̟ ) (Xt0,y ) Xt 1 , Xt 2 ,..., i=1

i H−δ i

̟∈Πk

h ̟ ̟ # ̟ ,[(x,u)]̟ ̟ ̟ #̟


ϕ(#̟ ) (Xt0,y ) Xt#I1 ,[(x,u)]1 , Xt#I2 ,[(x,u)]2 , . . . , Xt I#̟ + E V i #I ̟ ,[(y,u)]̟ #I ̟ ,[(y,u)]̟ #I ̟ ,[(y,u)]̟ #̟
1 2

− ϕ(#̟ ) (Xt0,y ) Xt 1 , Xt 2 , . . . , Xt # ̟ V (66) ( ̟ ,[(x,u)]̟ # I #̟ i i X Y kX k # +1 ̟ L (P;H) t kϕ(#̟ ) (Xt0,x ) − ϕ(#̟ ) (Xt0,y )kL#̟ +1 (P;L(#̟ ) (H,V )) ≤ Q#Ii̟ ̟ kH i=1 ̟∈Πk j=1 kuIi,j −δI ̟ #I ̟ ,[(x,u)]̟ #̟


Xt

#̟

i

i,j

#̟

+ |ϕ|C #̟ (H,V ) b

X i=1

#I ̟ ,[(x,u)]̟ i kXt i − Q#I ̟ i

j=1

#I ̟ ,[(y,u)]̟ i Xt i kL#̟ (P;H)

̟ kH kuIi,j −δI ̟

i,j

" i−1 #" # #) #I ̟ ,[(y,u)]̟ #I ̟ ,[(x,u)]̟ j j ̟ Y kXt j Y kL#̟ (P;H) kXt j kL#̟ (P;H) · . Q#Ij̟ Q#Ij̟ ̟ ̟ j=1 j=i+1 l=1 kuIj,l kH−δI ̟ l=1 kuIj,l kH−δI ̟ j,l

j,l

Next we note that (51) and item (iii) 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, p = p, δ = 0 for p ∈ [2, ∞) in the 15

notation of Corollary 2.10 in [1]) ensure that for all p ∈ [2, ∞) it holds that sup x,y∈H, x6=y

kXt0,x − Xt0,y kLp (P;H) sup < ∞. kx − ykH t∈[0,T ]

(67)

This implies that for all x ∈ H, t ∈ [0, T ] it holds that lim supH∋y→x E[min{1, kXt0,x − Xt0,y kH }] = 0.

(68)

The fact that ∀ k ∈ {1, 2, . . . , n} : ϕ(k) ∈ C (H, L(k) (H, V )) therefore assures that for all k ∈ {1, 2, . . . , n}, x ∈ H, t ∈ [0, T ] it holds that lim supH∋y→x E[min{1, kϕ(k) (Xt0,x ) − ϕ(k) (Xt0,y )kL(k) (H,V ) }] = 0.

(69)

Combining this and, e.g., Lemma 4.2 in Hutzenthaler et al. [14] (with I = {∅}, (Ω, F , P) = (Ω, F , P), c = 1, X m (∅, ω) = kϕ(k) (Xt0,ym (ω)) − ϕ(k) (Xt0,y0 (ω))kL(k) (H,V ) for ω ∈ Ω, t ∈ [0, T ], m ∈ N, k ∈ {1, 2, . . . , n}, (yl )l∈N0 ∈ {v ∈ M(N0 , H) : lim supl→∞ kvl − v0 kH = 0} in the notation of Lemma 4.2 in Hutzenthaler et al. [14]) establishes that for all k ∈ {1, 2, . . . , n}, ε ∈ (0, ∞), t ∈ [0, T ] and all sequences (ym )m∈N0 ⊆ H with lim supm→∞ kym − y0 kH = 0 it holds that
(70) lim supm→∞ P kϕ(k) (Xt0,ym ) − ϕ(k) (Xt0,y0 )kL(k) (H,V ) ≥ ε = 0. Combining this, the fact that ∀ k ∈ {1, 2, . . . , n} : supx∈H kϕ(k) (x)kL(k) (H,V ) < ∞, and, e.g., Proposition 4.5 in Hutzenthaler et al. [14] (with I = {∅}, (Ω, F , P) = (Ω, F , P), p = p, V = R, X m (∅, ω) = kϕ(k) (Xt0,ym (ω)) − ϕ(k) (Xt0,y0 (ω))kL(k) (H,V ) for ω ∈ Ω, t ∈ [0, T ], p ∈ [2, ∞), k ∈ {1, 2, . . . , n}, m ∈ N0 , (yl )l∈N0 ∈ {v ∈ M(N0 , H) : lim supl→∞ kvl − v0 kH = 0} in the notation of Proposition 4.5 in Hutzenthaler et al. [14]) therefore shows that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), t ∈ [0, T ] and all sequences (ym )m∈N0 ⊆ H with lim supm→∞ kym − y0 kH = 0 it holds that i h (71) lim supm→∞ E kϕ(k) (Xt0,ym ) − ϕ(k) (Xt0,y0 )kpL(k) (H,V ) = 0. Furthermore, (51) and item (v) of Theorem 2.1 in [2] (with T = T , η = η, H = H, U = U, W = W , A = A, n = n, F = F , B = B, α = 0, β = 0, k = k, p = p for p ∈ [2, ∞), k ∈ {1, 2, . . . , n} in the notation of Theorem 2.1 in [2]) ensure that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), t ∈ [0, T ] it holds that k,(x,u)

H ∋ x 7→ H k ∋ u 7→ [Xt



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

∈ C (H, L(k) (H, Lp (P; H))). (72)

Combining (66) (with k = k, δ1 = 0, δ2 = 0, . . . , δk = 0 for k ∈ {1, 2, . . . , n} in the notation of (66)) with (61), (71), (72), and Jensen’s inequality yields that for all k ∈ {1, 2, . . . , n}, x ∈ H,

16

t ∈ [0, T ] it holds that 

1 Qk lim sup sup i=1 kui kH H∋y→x u=(u1 ,u2 ,...,uk )∈(H\{0})k

#I ̟ ,[(x,u)]̟

P h (# ) 0,x #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #̟
2 1 2 1 ̟

, . . . , Xt # ̟ , Xt E ϕ (Xt ) Xt · ̟∈Πk i  #I ̟ ,[(y,u)]̟ #I ̟ ,[(y,u)]̟ #I ̟ ,[(y,u)]̟ #̟
2 1 0,y #̟ (#̟ ) 2 1

, . . . , Xt , Xt −ϕ (Xt ) Xt

V ( i h X 0,x 0,y (#̟ ) (#̟ ) lim sup kϕ (Xt ) − ϕ (Xt )kL#̟ +1 (P;L(#̟ ) (H,V )) ≤ ̟∈Πk

H∋y→x

"

#I ,(x,u)

kL#̟ +1 (P;H) # i∈I kui kH I∈̟ u=(ui )i∈I ∈(H\{0}) I " # ,(x,u) #! # ,(y,u) X kXt I − Xt I kL#̟ (P;H) Q + |ϕ|C #̟ (H,V ) lim sup sup b H∋y→x u=(ui )i∈I ∈(H\{0})#I i∈I kui kH I∈̟ #!) " # ,(y,u) Y kXt J kL#̟ (P;H) Q sup · sup = 0. y∈H u=(ui )i∈J ∈(H\{0})#J i∈J kui kH ·

Y

sup

kXt

(73)

#!

Q

J∈̟\{I}

This proves (65). Next we
claim that for all k ∈ {1, 2, . . . , n}, u ∈ H k , x ∈ H, t ∈ [0, T ] it holds that H ∋ y → 7 φ(t, y) ∈ V ∈ Cbk (H, V ) and ∂k φ ∂xk

i #I ̟ ,[(x,u)]̟
P h (#̟ ) 0,x #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #̟
1 2 . E ϕ (Xt ) Xt 1 , Xt 2 , . . . , Xt # ̟ (t, x)u =

(74)

̟∈Πk

We now prove (74) by induction on k ∈ {1, 2, . . . , n}. For the base case k = 1 we note that (51), Jensen’s inequality, and items (ix)–(x) of Theorem 2.1 in [2] (with T = T , η = η, H = H, U = U, W = W , A = A, n = n, F = F , B = B, α = 0, β = 0, p = p, t = t for t ∈ [0, T ], p ∈ [2, ∞) in the notation of Theorem 2.1 in [2]) ensure that for all p ∈ [1, ∞),
0,y p x, u1 ∈ H, t ∈ [0, T ] it holds that H ∋ y 7→ [Xt ]P,B(H) ∈ L (P; H) ∈ C 1 (H, Lp (P; H)) and
1,(x,u1 ) 0,x d [X ] u1 = [Xt ]P,B(H) . (75) P,B(H) t dx

Lemma 2.1 (with U = H, V = H, W = V , (Ω, F , P) = (Ω, F , P), X m,u = Xtm,u , ϕ = ϕ for t ∈ [0, T ], u ∈ H m+1 , m ∈ {0, 1} in the notation of Lemma 2.1) therefore implies that for all x, u ∈ H, t ∈ [0, T ] it holds that
H ∋ y 7→ φ(t, y) = E[ϕ(Xt0,y )] ∈ V ∈ C 1 (H, V ) (76) and

∂ φ ∂x




1,(x,u)

(t, x)u = E[ϕ′ (Xt0,x )Xt

].

(77)

This and (62) prove (74) 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} 17

such that (74) holds for k = 1, k = 2, . . . , k = k, let Φm : H m+1 → V , m ∈ {1, 2, . . . , k}, be the functions which satisfy for all m ∈ {1, 2, . . . , k}, u = (u1 , u2, . . . , um+1 ) ∈ H m+1 that Φm (u) = ϕ(m) (um+1 )(u1 , u2, . . . , um ), and let Y m,v,̟,u : [0, T ] × Ω → H #̟ +1 , u ∈ H k , ̟ ∈ Πk , v ∈ H m+1 , m ∈ {0, 1}, be the stochastic processes which satisfy for all ̟ ∈ Πk , u ∈ H k , x, h ∈ H, t ∈ [0, T ] that #I ̟ ,[(x,u)]̟ 1

Yt0,x,̟,u = Xt

1

#I ̟ ,[(x,u)]̟ 2

, Xt

2

#I ̟ ,[(x,u)]̟ #̟

, . . . , Xt

#̟

, Xt0,x

and 1,(x,h),̟,u

Yt

#I ̟ +1,([(x,u)]̟ 1 ,h)

= Xt

1

#I ̟ +1,([(x,u)]̟ 2 ,h)

, Xt

2

#I ̟ +1,([(x,u)]̟ #̟ ,h)

, . . . , Xt

#̟




(78)

1,(x,h)


.

, Xt

(79)

Next note that Lemma 3.1 (with V = H, W = V , n = m, ϕ = ϕ, Φ = Φm for m ∈ {1, 2, . . . , k} in the notation of Lemma 3.1) shows that for all m ∈ {1, 2, . . . , k}, u = (u1 , u2 , . . . , um+1 ), u ˜ = (˜ u1 , u˜2 , . . . , u˜m+1 ) ∈ H m+1 it holds that Φm ∈ C 1 (H m+1 , V ) and Φ′m (u)˜ u = ϕ(m+1) (um+1 )(u1 , u2 , . . . , um, u˜m+1 ) P (m) (um+1 )(u1 , u2 , . . . , ui−1 , u˜i , ui+1 , ui+2 , . . . , um ). + m i=1 ϕ

(80)

This and H¨older’s inequality imply that for all m ∈ {1, 2, . . . , k}, u = (u1 , u2 , . . . , um+1 ), u ˜ = (˜ u1, u˜2 , . . . , u˜m+1 ) ∈ H m+1 it holds that kΦ′m (u)˜ ukV Q Pm Q m ui kH j∈{1,2,...,m}\{i} kuj kH ≤ |ϕ|C m+1 (H,V ) k˜ um+1 kH m i=1 |ϕ|Cb (H,V ) k˜ i=1 kui kH + b   Pm Q Q 2 1/2 2 2 ku k . ≤ k˜ ukH m+1 |ϕ|2C m+1 (H,V ) m ku k + |ϕ| m j i H H i=1 j∈{1,2,...,m}\{i} i=1 C (H,V )

(81)

b

b

Hence, we obtain that for all m ∈ {1, 2, . . . , k}, u = (u1 , u2 , . . . , um+1 ) ∈ H m+1 it holds that kΦ′m (u)kL(H m+1 ,V ) ≤ kϕkC m+1 (H,V ) b

 Qm

i=1

kui k2H +

Pm Q i=1

j∈{1,2,...,m}\{i}

kuj k2H

1/2

≤ kϕkC m+1 (H,V ) b   Qm Pm Q 2 1/2 2 m+1 }| m+1 }| + |max{1, kuk · |max{1, kuk H H i=1 j∈{1,2,...,m}\{i} i=1 √ ≤ m + 1 kϕkC m+1 (H,V ) |max{1, kukH m+1 }|m .

(82)

b

This shows that for all m ∈ {1, 2, . . . , k} it holds that sup u∈H m+1

kΦ′m (u)kL(H m+1 ,V ) < ∞. |max{1, kukH m+1 }|m

Next we note that for all m ∈ N, p ∈ [1, ∞), Y1 , Y2 , . . . , Ym ∈ L0 (P; H) it holds that

P

P 2 1/2

≤ m k(Y1 , Y2 , . . . , Ym)kLp (P;H m ) = [ m i=1 kYi kH Lp (P;R) i=1 kYi kH ] Lp (P;R) P ≤ m i=1 kYi kLp (P;H) . 18

(83)

(84)

This shows that for all m ∈ {0, 1}, p ∈ [1, ∞), ̟ ∈ Πk , u ∈ H k , v ∈ H m+1 , t ∈ [0, T ] it holds that Ytm,v,̟,u ∈ Lp (P; H #̟ +1 ). (85)

Next observe that (63), (84), and Jensen’s inequality imply that for all p ∈ [1, ∞), ̟ ∈ Πk , u ∈ H k , x ∈ H, t ∈ [0, T ] it holds that
 1,(x,h),̟,u ∈ Lp (P; H #̟ +1 ) ∈ L(H, Lp (P; H #̟ +1 )). (86) H ∋ h 7→ Yt P,B(H #̟ +1 )

Furthermore, we note that (51) and item (vi) of Theorem 2.1 in [2] (with T = T , η = η, H = H, U = U, W = W , A = A, n = n, F = F , B = B, α = 0, β = 0, k = m, p = p, x = x for x ∈ H, p ∈ [2, ∞), m ∈ {2, 3, . . . , k + 1} in the notation of Theorem 2.1 in [2]) ensure that for all m ∈ {2, 3, . . . , k + 1}, p ∈ [2, ∞), x ∈ H, t ∈ [0, T ] it holds that " m−1,(x+u ,u) # m−1,(x,u) m,(x,u,um ) m kXt − Xt − Xt kLp (P;H) Qm lim sup sup = 0. H\{0}∋um →0 u=(u1 ,u2 ,...,um−1 )∈(H\{0})m−1 i=1 kui kH (87) Combining (75) and (87) with (84) and Jensen’s inequality yields that for all p ∈ [1, ∞), ̟ ∈ Πk , u ∈ H k , x ∈ H, t ∈ [0, T ] it holds that # " 0,x+h,̟,u 1,(x,h),̟,u

Yt

p − Yt0,x,̟,u − Yt L (P;H #̟ +1 ) lim sup khkH H\{0}∋h→0 " # 1,(x,h) kXt0,x+h − Xt0,x − Xt kLp (P;H) ≤ lim sup (88) khkH H\{0}∋h→0

# " #Ii̟ ,[(x+h,u)]̟ #I ̟ ,[(x,u)]̟ #I ̟ +1,([(x,u)]̟ i i i ,h) #̟

Xt

p X − Xt i − Xt i L (P;H) = 0. lim sup + khkH i=1 H\{0}∋h→0 In addition, combining (72) with (84) and Jensen’s inequality yields that for all p ∈ [1, ∞), ̟ ∈ Πk , u ∈ H k , x ∈ H, t ∈ [0, T ] it holds that " 1,(x,h),̟,u # 1,(y,h),̟,u kYt − Yt kLp (P;H #̟ +1 ) lim sup sup khkH H∋y→x h∈H\{0} # " 1,(x,h) 1,(y,h) kXt − Xt kLp (P;H) ≤ lim sup sup (89) khkH H∋y→x h∈H\{0}   #I ̟ +1,([x,u]̟ #I ̟ +1,([y,u]̟ #̟ i ,h) i ,h) X kXt i − Xt i kLp (P;H)  = 0. lim sup sup  + khk H∋y→x h∈H\{0} H i=1

Combining (86) and (88) hence yields that for all p ∈ [1, ∞), ̟ ∈ Πk , u ∈ H k , x, h ∈ H, t ∈ [0, T ] it holds that 
 (90) H ∋ y 7→ Yt0,y,̟,u P,B(H #̟ +1 ) ∈ Lp (P; H #̟ +1 ) ∈ C 1 (H, Lp (P; H #̟ +1 )) 19

and ∂ ∂x


 1,(x,h),̟,u  0,x,̟,u . h = Yt Yt # +1 P,B(H #̟ +1 ) P,B(H ̟ )

(91)

This, (80), (83), and Lemma 2.1 (with U = H, V = H #̟ +1 , W = V , (Ω, F , P) = (Ω, F , P), 1,(x,h),̟,u X 0,x = Yt0,x,̟,u, X 1,(x,h) = Yt , ϕ = Φ#̟ for t ∈ [0, T ], x, h ∈ H, u ∈ H k , ̟ ∈ Πk in the notation of Lemma 2.1) assure that (a) it holds for all ̟ ∈ Πk , u ∈ H k , t ∈ [0, T ] that  h
i 0,x,̟,u H ∋ x 7→ E Φ#̟ Yt i  h #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #̟
2 1 0,x #̟ (#̟ ) 2 1 ∈ V ∈ C 1 (H, V ) (92) , . . . , Xt , Xt =E ϕ (Xt ) Xt and

(b) it holds for all ̟ ∈ Πk , u ∈ H k , x, uk+1 ∈ H, t ∈ [0, T ] that  i h #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #̟
2 1 0,x #̟ (#̟ ) ∂ 2 1 , . . . , X uk+1 , X E ϕ (X ) X t t t t ∂x i h 
∂ uk+1 E Φ#̟ Yt0,x,̟,u = ∂x h
1,(x,uk+1 ),̟,ui 0,x,̟,u ′ Yt = E Φ#̟ Yt i h #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #̟ 1 2 1,(x,uk+1 )
0,x #̟ (#̟ +1) 1 2 =E ϕ (Xt ) Xt , Xt , . . . , Xt , Xt h #I ̟ ,[(x,u)]̟ P ̟ #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ i−1 1 2 0,x i−1 (#̟ ) 1 2 E ϕ (X + # ) X , X , . . . , X , t t t t i=1 ̟ #I ̟ ,[(x,u)]#̟
i #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #I ̟ +1,([(x,u)]̟ i+1 i+2 i ,uk+1 ) i+1 i+2 i Xt . , Xt , Xt , . . . , Xt # ̟

(93)

Combining item (a) with the induction hypothesis shows that for all u ∈ H k , t ∈ [0, T ] it holds that i  #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ P h (#̟ ) 0,x #̟
1 2 E ϕ (Xt ) Xt 1 , Xt 2 , . . . , Xt # ̟ H ∋ x 7→ ̟∈Πk 
∂k = ∂x φ (t, x)u ∈ V ∈ C 1 (H, V ). (94) k Item (b) hence proves that for all u ∈ H k , x, h ∈ H, t ∈ [0, T ] it holds that  ∂k

d φ (t, x)u h k dx ∂x i X  h #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #̟ 2 1 1,(x,h)
, . . . , Xt # ̟ , Xt 2 E ϕ(#̟ +1) (Xt0,x ) Xt 1 , Xt = ̟∈Πk

#I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ P#̟ h (#̟ ) 0,x i−1 2 1 i−1 2 1 , . . . , X , , X E ϕ (X ) X t t t t i=1 i #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #I ̟ +1,([(x,u)]̟ ,h) #̟
i+1 i+2 i # i+1 i+2 Xt i , Xt , Xt . , . . . , Xt ̟

+

20

(95)

In addition, note that n o  Πk+1 = ̟ ∪ {k + 1} : ̟ ∈ Πk o ] n ̟ ̟ ̟ ̟ : i ∈ {1, 2, . . . , # }, ̟ ∈ Π I1̟ , I2̟ , . . . , Ii−1 , Ii̟ ∪ {k + 1}, Ii+1 , Ii+2 , . . . , I# ̟ k . ̟

(96)

This ensures that for all u = (u0 , u1 , . . . , uk ) ∈ H k+1, h ∈ H, t ∈ [0, T ] it holds that P

(#̟ )

(Xt0,u0 )

#I ̟ ,[(u,h)]̟ #I ̟ ,[(u,h)]̟ #I ̟ ,[(u,h)]̟ #̟
1 2 1 2 Xt , Xt , . . . , Xt # ̟

ϕ h #I ̟ ,[u]̟ P #I ̟ ,[u]̟ #I ̟ ,[u]̟ #̟ 2 1 1,(u ,h)
= ̟∈Πk ϕ(#̟ +1) (Xt0,u0 ) Xt 1 , . . . , Xt # ̟ , Xt 2 , Xt 0 ̟∈Πk+1

P#̟

(#̟ )

i=1 ϕ #I ̟ ,[u]̟ i+1

+

Xt

i+1

#I ̟ ,[u]̟ #I ̟ +1,([u]̟ #I ̟ ,[u]̟ #I ̟ ,[u]̟ i−i 1 2 i ,h) , . . . , Xt i−1 Xt 1 , Xt 2 , Xt i , i ̟ ̟ #I ̟ ,[u]#̟
,[u]i+2

(97)

(Xt0,u0 ) #I ̟

, Xt

i+2

, . . . , Xt

#̟

.

Combining this with (95) establishes that for all u ∈ H k , x, h ∈ H, t ∈ [0, T ] it holds that 
 ∂k
d φ (t, x)u h dx ∂xk i X h #I ̟ ,[(x,u,h)]̟ #I ̟ ,[(x,u,h)]̟ #I ̟ ,[(x,u,h)]̟ #̟
2 1 (98) , . . . , Xt # ̟ , Xt 2 E ϕ(#̟ ) (Xt0,u0 ) Xt 1 , . = ̟∈Πk+1

Hence, we obtain that for all u = (u1 , u2 , . . . , uk ) ∈ H k , x ∈ H, t ∈ [0, T ] it holds that 
1

∂k
∂k lim sup

∂xk φ (t, x + h)u − ∂x k φ (t, x)u khkH H\{0}∋h→0 (99) i  h #I ̟ ,[(x,u,h)]̟ P #I ̟ ,[(x,u,h)]̟ #I ̟ ,[(x,u,h)]̟ #̟
2 1 0,x # = 0. , . . . , Xt ̟ − ̟∈Πk+1 E ϕ(#̟ ) (Xt ) Xt 1 , Xt 2

V

Combining (65) and Lemma 2.2 (with V = H, W = V , n = k, f = (H ∋ x 7→ φ(t, x) ∈ P #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ 1 2 (#̟ ) V ), g = (H ∋ x 7→ (H k+1 ∋ u 7→ (Xt0,x )(Xt 1 , Xt 2 ,..., ̟∈Πk+1 E[ϕ #I ̟ ,[(x,u)]̟ #̟

Xt # ̟ )] ∈ V ) ∈ L(k+1) (H, V )) for t ∈ [0, T ] in the notation of Lemma 2.2) therefore shows that for all u ∈ H k+1, x ∈ H, t ∈ [0, T ] it holds that (H ∋ y 7→ φ(t, y) ∈ V ) ∈ C k+1(H, V ) and
∂ k+1 φ (t, x)u ∂xk+1 i X h #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #I ̟ ,[(x,u)]̟ #̟
2 1 (100) 0,x #̟ (#̟ ) 2 1 . , . . . , Xt , Xt E ϕ (Xt ) Xt = ̟∈Πk+1

This and (62) prove (74) in the case k + 1. Induction thus completes the proof of (74). Next observe that item (ii) and item (iii) follow immediately from (74). It thus remains to prove items (iv)–(viii). To prove item (iv) we first note that (51) and item (ii) of Theorem 2.1 in [2] (with T = T , η = η, H = H, U = U, W = W , A = A, n = n, F = F , B = B, 21

α = α, β = β, k = k, p = p, δ = δ for δ ∈ Dk , p ∈ [2, ∞), k ∈ {1, 2, . . . , n} in the notation of Theorem 2.1 in [2]) ensure that for all k ∈ {1, 2, . . . , n}, p ∈ [2, ∞), δ = (δ1 , δ2 , . . . , δk ) ∈ Dk it holds that " δ,α,β # tιN kXtk,ukLp (P;H) sup sup < ∞. (101) Qk u=(u0 ,u1 ,...,uk )∈H×(H\{0})k t∈(0,T ] i=1 kui kH−δi

This and Jensen’s inequality establish item (iv). Moreover, observe that for all k ∈ N, δ = (δ1 , δ2 , . . . , δk ) ∈ Rk , ̟ ∈ Πk it holds that Y P δ,α,β P 1 sup t(−ιI + i∈I δi ) = sup tmin{1−α, /2−β} I∈̟ 1[2,∞) (#I ) t∈(0,T ] I∈̟ t∈(0,T ] (102) P min{1−α,1/2−β} I∈̟ 1[2,∞) (#I ) ⌊k/2⌋ min{1−α,1/2−β} =T ≤ |T ∨ 1| . Combining (60) with item (iii), (101), (102), and Jensen’s inequality yields that for all k ∈ {1, 2, . . . , n}, δ = (δ1 , δ2 , . . . , δk ) ∈ Dk it holds that " Pk

∂k


#

t i=1 δi ∂x k φ (t, v)u V sup sup sup Qk v∈H u=(u1 ,u2 ,...,uk )∈(H\{0})k t∈(0,T ] i=1 kui kH−δi #! " Y X P + δ ) (−ιδ,α,β i∈I i t I |ϕ|C #̟ (H,V ) sup ≤ b

t∈(0,T ]

̟∈Πk

·

Y

I∈̟

sup

I∈̟

sup

sup

x∈H u=(ui )i∈I ∈(H\{0})#I t∈(0,T ]

"

≤ |T ∨ 1|⌊k/2⌋ min{1−α,1/2−β} kϕkCbk (H,V ) ·

X Y

̟∈Πk I∈̟

sup

sup

δ,α,β

tιI

sup

x∈H u=(ui )i∈I ∈(H\{0})#I t∈(0,T ]

#I ,(x,u)

kXt Q

i∈I

"

δ,α,β

tιI

kL#̟ (P;H)

kui kH−δi #I ,(x,u)

kX Qt

i∈I

#!

kL#̟ (P;H)

kui kH−δi

#

(103)

< ∞.

This proves item (v). Next we observe that (51) and item (iv) of Theorem 2.1 in [2] (with T = T , η = η, H = H, U = U, W = W , A = A, n = n, F = F , B = B, α = α, β = β, k = k, p = p, δ = δ for δ ∈ Dk , p ∈ [2, ∞), k ∈ {l ∈ {1, 2, . . . , n} : |F |Lipl (H,H−α ) + |B|Lipl (H,HS(U,H−β )) < ∞} in the notation of Theorem 2.1 in [2]) ensure 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=(u1 ,u2 ,...,uk )∈(H\{0})k t∈(0,T ] x6=y

"

(δ,0),α,β

tιN

# k,(y,u) − Xt kLp (P;H) < ∞. Q kx − ykH ki=1 kui kH−δi k,(x,u)

kXt

(104)

Combining this and (67) with Jensen’s inequality establish items (vi) and (vii). Moreover, note

22

that for all k ∈ N, δ = (δ1 , δ2 , . . . , δk ) ∈ Rk , ̟ ∈ Πk , I ∈ ̟ it holds that # " Y P (δ,0),α,β P δ,α,β (−ι + δ ) sup t I∪{k+1} i∈I i t(−ιJ + i∈J δi ) t∈(0,T ]

min{1−α,1/2−β} [1+

= sup t t∈(0,T ]

= T min{1−α, /2−β} [1+ 1

J∈̟\{I} P

J ∈̟\{I}

P

J ∈̟\{I}

1[2,∞) (#J )]

1[2,∞) (#J )]

(105)

≤ |T ∨ 1|⌈k/2⌉ min{1−α,1/2−β} .

Furthermore, note that item (iii) and (66) imply that for all k ∈ {1, 2, . . . , n}, δ = (δ1 , δ2 , . . . , δk ) ∈ Dk with |ϕ|Lipk (H,V ) < ∞ it holds that Pk

∂k φ ∂xk



 # ∂k

(t, v) − ∂x k φ (t, w) u V sup sup sup Qk v,w∈H, u=(u1 ,u2 ,...,uk )∈(H\{0})k t∈(0,T ] kv − wkH i=1 kui kH−δi v6=w " #! " 0,x ( #! 0,y Y X kX − X k # +1 δ,α,β P ̟ L (P;H) t t sup sup |ϕ|Lip#̟ (H,V ) sup t(−ιI + i∈I δi ) ≤ kx − ykH x,y∈H, t∈(0,T ] t∈(0,T ] I∈̟ ̟∈Πk x6=y " δ,α,β #! # ,(x,u) Y tιI kXt I kL#̟ +1 (P;H) Q · sup sup sup x∈H u=(ui )i∈I ∈(H\{0})#I t∈(0,T ] i∈I kui kH−δi I∈̟ #! " X Y (δ,0),α,β P δ,α,β P (−ιI∪{k+1} + i∈I δi ) + |ϕ|C #̟ (H,V ) sup t t(−ιJ + i∈J δi ) "

b

I∈̟

· ·

i=1 δi

t



t∈(0,T ]

J∈̟\{I}

"

(δ,0),α,β ιI∪{k+1}

# ,(x,u)

# ,(y,u)

kXt I −X I kL#̟ (P;H) Q t sup sup sup kx − ykH i∈I kui kH−δi x,y∈H, u=(ui )i∈I ∈(H\{0})#I t∈(0,T ] x6=y " δ,α,β #!) # ,(x,u) Y tιJ kXt J kL#̟ (P;H) Q sup sup sup . x∈H u=(ui )i∈J ∈(H\{0})#J t∈(0,T ] i∈J kui kH−δi t

#!

J∈̟\{I}

(106)

Combining (106) with (67), (101), (102), (104), (105), and Jensen’s inequality establishes item (viii). The proof of Lemma 3.2 is thus completed. Theorem 3.3. Assume the setting in Section 1.2 and let n ∈ N, ϕ ∈ Cbn (H, V ), F ∈ Cbn (H, H), B ∈ Cbn (H, HS(U, H)). Then (i) it holds that 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 satisfy for all k ∈ {0, 1, . . . , n}, u = (u0, u1 , . . . , uk ) ∈ H k+1 , p ∈ (0, ∞), t ∈ [0, T ] that sups∈[0,T ] E kXsk,ukpH