Lévy-areas of Ornstein-Uhlenbeck processes in Hilbert-spaces

arXiv:1411.4765v1 [math.DS] 18 Nov 2014 L´ EVY–AREAS OF ORNSTEIN–UHLENBECK PROCESSES IN HILBERT–SPACES MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS Abstract. In this paper we investigate the existence and some useful properties of the L´evy areas of Ornstein–Uhlenbeck processes associated to Hilbert– space-valued fractional Brownian–motions with Hurst parameter H∈(1/3,1/2]. We prove that this stochastic area has a H¨older–continuous version with sufficiently large H¨older–exponent and that can be approximated by smooth areas. In addition, we prove the stationarity of this area. 1. Introduction During the last decades new techniques have been developed generalizing the well known Ito–or Stratonovich–integration. For fundamental publications in this area see for instance Lyons an Qian [18] for the so called Rough–Paths theory and Z¨ahle [22] for the Fractional Calculus1theory. In particular, these techniques allow to have stochastic integrators which are more general than the Brownian motion. A candidate for such an integrator is for instance the fractional Brownian motion. This stochastic process does not have in general the semi–martingale property, which would allow to define the stochastic integral as a limit in probability if the integrator satisfies special measurability and integrability conditions, see e.g. Karatzas and Shreve [14]. Another advantage of these new techniques is to treat stochastic integrals in a path–wise way. In particular, for any sufficient regular integrand and integrator with more general properties than the bounded variation property, an integral can be defined. This kind of integrals goes back to Young [21] which allows to consider integrals for H¨older–continuous integrands and integrators fulfilling special conditions with respect to the H¨older–exponents. In practice, this integral cannot be used to replace the well known Ito–integral by a Young–integral if the integrator is a Brownian motion and the integrand has a H¨older–exponent less than or equal to 1/2. Our reason for dealing with this type of new integral is to take advantage of the path–wise property in order to further introduce Random Dynamical Systems (RDS) for infinite dimensional differential equations, namely, for stochastic evolution equations and stochastic partial differential equations as well. The random driver of this kind of equation will be in general a trace class fractional Brownian– motion with Hurst–parameter H∈(1/3,1/2]. To introduce an RDS one needs at first a model for a noise which is called a metric dynamical system: Consider the quadruple (Ω,F,P, θ) where (Ω,F,P) is a probability space and θis a measurable 1991 Mathematics Subject Classification. Primary: ; Secondary: Key words and phrases. Stochastic PDEs, Hilbert-valued fractional Brownian motion, path– wise solution. 1The name fractional calculus theory goes back to D. Nualart. 1 2 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS flow on Ω: θ: (R×Ω,B(R)⊗ F)→(Ω,F) θt◦θτ=θtθτ=θt+τ, θ0= idΩ. An Hilbert–(or topological) space RDS is a measurable mapping ϕ: (R+×Ω×V, B(R+)⊗ F ⊗ B(V)) →(V, B(V)) satisfying the cocycle property ϕ(t+τ, ω, x) = ϕ(t, θτω, ϕ(τ, ω, x)), ϕ(0, ω, x) = x, for all t, τ ∈R+,x∈Vand for all ω∈Ω or at least for all ωof a θ–invariant set ˜ Ω∈ F of full P–measure which is independent of x, t, τ. This is different to the fact that a property holds almost surely for any x∈V, which is often used in Stochastic Analysis, since exceptional sets depending on t, τ and xare in most cases forbidden when dealing with the cocycle property. An example of a metric dynamical system is for instance (C0,B(C0),PH, θ), where the probability space (C0,B(C0),PH) is the canonical probability space such that C0is the space of continuous functions on Rwith values in a separable Hilbert– space V,B(C0) is its Borel–σ–algebra, PHis the distribution on B(C0) of a trace class fractional Brownian–motion ωwith Hurst–parameter H∈(0,1), and θ: (R×C0,B(R)⊗ B(C0)) →(C0,B(C0)) is given by (1) θτ(t) = ω(t+τ)−ω(τ),for t, τ ∈R, see Arnold [1] or Maslowski and Schmalfuss [19]. It is known that under typical conditions on the coefficients an ordinary Ito– equation generates an RDS, see Arnold [1]. The main instrument to obtain this property is the Kolmogorov–theorem on the existence of a H¨older–continuous version of a random field. Such a random field is derived from an Ito–equation where the parameters of this field are the time and the non-random initial condition. The so called perfection technique then can be used to conclude the existence of a version of the ordinary Ito–equation which defines an RDS. However, there is no appropriate version of the Kolmogorov–theorem for infinite dimensional random fields that could be applied to show that solutions of stochastic evolution equations generate an RDS, and therefore this property is rather an open problem, although there are partial results for particular cases, see e.g. the recent papers [3], [5], [7], [9] and [10]. Consider a stochastic evolution equation du =Audt +G(u)dω where Ais the generator of an analytic stable semigroup Son the separable Hilbert– space Vand ωis a β–H¨older–continuous fractional–Brownian motion in Vwith Hurst–parameter H∈(1/3,1/2], so that β∈(1/3,1/2). This equation has the mild interpretation (2) u(t) = S(t)u0+Zt 0 S(t−r)G(u(r))dω(r) where u0is a non-random initial condition in V. The integral in the above equation has to be interpreted in a fractional sense. For a good understanding of that integral we refer to [12] and [11]. L´ EVY AREAS OF ORNSTEIN–UHLENBECK PROCESSES IN HILBERT–SPACES 3 For the following let ¯ ∆a,b ⊂R2be the set of the pairs (s, t) such that −∞ < a ≤ s≤t≤b < ∞. Let V⊗Vbe the Hilbert–tensor space of Vwith tensor product ⊗V. We consider now functions ¯ ∆0,T ∋(s, t)7→ (u⊗ω)(s, t). The reason to introduce these elements is to interpret the integral of (2) in a fractional sense. We need to consider the tensor product of a possible solution u and a noise path ω. In particular, for a smooth ωthis tensor product is given by (u⊗ω)(s, t) = Zt s (S(ξ−s)−id)u(s)⊗Vω′(ξ)dξ +Zt sZξ s S(ξ−r)G(u(r))ω′(r)dr ⊗Vω′(ξ)dξ, (3) and, exchanging the order of integration, the last integral of (3) can be written as Zt s G(u(r))D1(ω⊗Sω)(r, t)dr2 where (ω⊗Sω)(s, t) is defined by L2(V, ˆ V)∋E7→ E(ω⊗Sω)(s, t) = Zt sZξ s S(ξ−r)Eω′(r)⊗Vω′(ξ)drdξ =Zt sZt r S(ξ−r)Eω′(r)⊗Vω′(ξ)dξdr (4) for (s, t)∈¯ ∆0,T . Here ˆ Vis another Hilbert–space to be determined later and Eis an element of the Hilbert–Schmidt space L2(V, ˆ V). Let C2β(¯ ∆a,b, L2(L2(V, ˆ V), V ⊗V)) the space of 2β–H¨older–continuous fields on ¯ ∆a,b with values in V⊗Vwith norm (5) kvk2β=kvk2β,a,b = sup (s,t)∈¯ ∆a,b kvkL2(L2(V, ˆ V),V ⊗V) |t−s|2β<∞. We know that ω∈Cβ([0, T ]; V) for every β < H. We also denote by ωna piecewise linear (continuous) approximation of ωwith respect to an equidistant partition of length 2−nT:= δsuch that ωn(t) = ω(t) for the partition points t. For these ωn we can define (ωn⊗Sωn) by the right hand side of (4). The main purpose of this paper is to prove the following result, a property which is needed to establish the existence of solutions to (2). For a detailed description on the construction of solutions to (2) we refer the reader to the paper [11]. Theorem 1. Let (ωn)n∈Nbe the sequence of piecewise linear approximations of some ωintroduced above such that ((ωn⊗Sωn))n∈Nis defined by (4). Then for any β < H the sequence ((ωn,(ωn⊗Sωn)))n∈Nconverges to (ω, (ω⊗Sω)) in Cβ([0, T ]; V)×C2β(¯ ∆0,T ;L2(L2(V, ˆ V), V ⊗V)) on a set of full measure. In particular (ω⊗Sω)is continuous. In the following we consider only the statements of this theorem with respect to (ω⊗Sω). The convergence properties of (ωn)n∈Nfollow in a similar and simpler manner. We omit here. 2D1means the first derivative w.r.t. the first variable of (ω⊗Sω)(·,·). 4 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS The object (ω⊗Sω) will be called L´evy area of the Ornstein-Uhlembeck process. Since a priori it is not clear whether (ω⊗Sω) is well defined, in what follows we are going to give an appropriate meaning to this term by presenting two proofs of the above theorem. The first one is related to β-H¨older continuous paths covering the case of a fractional Brownian–motion for an appropriate Hilbert–space ˆ V. The second proof deals with the case of a Brownian-motion for a more general space ˆ V. As we have said, Ais the generator of the analytic semigroup Son the separable Hilbert–space V. We also suppose that −Ais positive and symmetric such that its inverse is compact. Then −Ahas a positive point spectrum (λi)i∈Nof finite multiplicity such that limi→∞ λi=∞. We denote by (ei)i∈Nthe associated eigenelements which form a complete orthonormal system in V. Furthermore, −A generates for κ∈Rthe separable Hilbert–spaces D((−A)κ) =: Vκ={u=X i ˆuiei:kuk2 Vκ=X i |ˆui|2λ2κ i<∞}, with V=V0. 2. The construction of (ω⊗Sω)for a fractional Brownian–motion Let us begin this section by introducing a V-valued fractional Brownian–motion. For brevity we suppose that ωcan be presented by ω(t) = X i q 1 2 iωi(t)ei where (ωi)i∈Nis a sequence of one-dimensional independent β–H¨older continuous standard fractional Brownian–motions for any β < H such that Piqi<∞. Then ωcan be interpreted as a β–H¨older–continuous fractional Brownian–motion in V with Hurst–parameter H∈(0,1). The covariance is given by the operator Qwith is a diagonal operator in our standard basis with diagonal elements qi. Let us also denote by ωn, ωn ithe piecewise linear approximations with respect to the equidistant partition {tn i}of [0, T ] of length 2−nT=δ. Throughout this section we assume that ωhas a Hurst–parameter H∈(1/3,1/2]. Since ωnis smooth we can define (ωn⊗Sωn)(s, t) as a Bochner–integral with respect to the Lebesgue–measure: E(ωn⊗Sωn)(s, t) = Zt sZξ s S(ξ−r)Edωn(r)⊗Vdωn(ξ) for E∈L2(V, ˆ V). By an integration by parts argument this integral can be rewritten as Zt s E(ωn(ξ)−ωn(s)) ⊗Vdωn(ξ) + Zt s AZξ s S(ξ−r)E(ωn(r)−ωn(s))dr ⊗Vdωn(ξ). This motivates to interpret (ω⊗Sω)(s, t) as (6) Zt s E(ω(ξ)−ω(s)) ⊗Vdω(ξ) + Zt s AZξ s S(ξ−r)E(ω(r)−ω(s))dr ⊗Vdω(ξ) where we have to give an appropriate meaning to both integrals. We start with the first one which is abbreviated in the following by E(ω⊗ω)(s, t). L´ EVY AREAS OF ORNSTEIN–UHLENBECK PROCESSES IN HILBERT–SPACES 5 Lemma 2. Let ˆ V=Vκfor κ > 0such that Piλ−2κ i<∞. The sequence ((ωn⊗ ωn))n∈N, which elements can be represented component–wise by q 1 2 jq 1 2 kZt s (ωn j(ξ)−ωn j(s))dωn k(ξ) converges on a set of full measure in C2β(¯ ∆0,T ;L2(L2(V, Vκ), V ⊗V)) to (ω⊗ω). Proof. Consider the orthonormal basis (Eij)i,j∈Nof L2(V, Vκ) given by Eijek=0 : j6=k ei λκ i:j=k, and (el⊗Vek)l,k∈N, the orthonormal basis of V⊗V. First note that, for (s, t)∈¯ ∆0,T , k(ωn⊗ωn)(s, t)k2 L2(L2(V,Vκ),V ⊗V)=X i,j X l,k (Eij(ωn⊗ωn)(s, t), el⊗Vek)2 V⊗V ≤X i λ−2κ iX i,j qjqkZt s (ωn j(ξ)−ωn j(s))dωn k(ξ)2 , and hence we have to study the behavior of the last sum. Let us denote An j,k(s, t)3: = Zt s (ωj(ξ)−ωj(s))dωk(ξ)−Zt s (ωn j(ξ)−ωn j(s))dωn k(ξ). By symmetry we can assume j≤k. In fact we assume that j < k since the case j=kis easier, see a comment at the end of the proof. To estimate the continuous element An j,k(s, t) we apply the Lemma 3.7 in Deya et al. [6], which claims that for p≥1 there exists Kβ,p such that kAn j,kk2β≤Kβ,p(Rj,k n,p +kωj−ωn jkβkωkkβ+kωk−ωn kkβkωn jkβ),(7) where Rj,k n,p := ZT 0ZT 0 |An j,k(s, t)|2p |t−s|4βp+2 dsdt1/(2p) . In particular, from the proof of Lemma 3.7 in [6] we know that E(Rj,k n,p)2p≤cn−4p(H−β′)<∞, for β < β′< H, being β′close enough to Hand plarge enough. Indeed, let us take plarge enough such that 4p(H−β′)>1, and thus P(X j,k qjqk(Rj,k p,n)2> o(n)2)≤(trVQ)2(p−1) o(n)2pX j,k qjqkE(Rj,k n,p)2p≤cn−4p(H−β′) o(n)2p. For an appropriate sequence (o(n))n∈Nwith limit zero, the right-hand side has a finite sum. Then by the Borel–Cantelli–lemma, (Pj,k qjqk(Rj,k p,n)2)n∈Ntends to zero almost surely. In a similar manner we obtain the convergence of the last terms in (7). It suffices to take into account that, for β < β′< H, kωj−ωn jkβ≤Gβ′(j, ω)nβ−β′,kωjkβ≤Gβ′(j, ω),kωn jkβ≤Gβ′(j, ω) (8) 3In [6], An j,k(s, t) are considered over the square [0, T ]2. However, since they are symmetric w.r.t. the diagonal of [0, T ] it is sufficient to consider these elements over ¯ ∆0,T . 6 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS where Gβ′(j, ω)≥ kωjkβ′and Gβ′(j, ω)∈Lp(Ω) for any p∈Nare iid random variables, see Kunita [16] Theorem 1.4.1. We then have P(X j,k qjqkkωn j−ωjk2 βkωkk2 β> o(n)2) ≤(trVQ)2(p−1) o(n)2pX j,k qjqk(EGβ′(j, ω)4p)1 2(EGβ′(k, ω)4p)1 2n2p(β−β′)≤cn2p(β−β′) o(n)2p. For pchosen sufficiently large and an appropriate zero–sequence (o(n))n∈Nwe obtain the almost sure convergence of (Pj,k qjqkkωn j−ωjk2 βkωkk2 β)n∈N. Similarly we can treat the last term of (7), that is, (Pj,k qjqkkωn j−ωjk2 βkωn kk2 β)n∈N. Finally, An j,j(s, t) = 1 2(ωj(t)−ωj(s))2−1 2(ωn j(t)−ωn j(s))2 and thanks to (8) kAn j,jk2β≤Gβ′(j, ω)2n2(β−β′), which completes the proof.  Lemma 3. Suppose that there exists γsuch that γ+β > 1and X i λ2γ−2κ i<∞. Then the mapping ((s, t), E)∈¯ ∆0,T ×L2(V, Vκ)7→ Zt sAZξ s S(ξ−r)E(ω(r)−ω(s))dr⊗Vdω(ξ) is in C2β(¯ ∆0,T ;L2(L2(V, Vκ), V ⊗V)). Proof. Thanks to Pazy [20] Theorem 4.3.5 (iii),  AZξ s S(ξ−r)Eij(ω(r)−ω(s))dr =q 1 2 j λκ iZξ s AS(ξ−r)ei(ωj(r)−ωj(s))dr ≤cq 1 2 j λκ i kωjkβ(ξ−s)β, (9) and applying Bensoussan and Frehse [4] Corollary 2.1 we also have  AZξ s S(ξ−r)Eij (ω(r)−ω(s))dr −AZξ′ s S(ξ′−r)Eij (ω(r)−ω(s))dr ≤q 1 2 j λκ−γ iZξ s (−A)1−γS(ξ−r)ei(ωj(r)−ωj(s))dr −Zξ′ s (−A)1−γS(ξ′−r)ei(ωj(r)−ωj(s))dr ≤cq 1 2 j λκ−γ i kωjkβ|ξ−ξ′|γ. (10) Therefore, for an α < γ such that β > 1−α, we can define the integral Zt sAZξ s S(ξ−r)ei(ωj(r)−ωj(s))drdωk(ξ) =(−1)αZt s Dα s+AZ· s S(· − r)ei(ωj(r)−ωj(s))dr[ξ]D1−α t−(ωk)t−[ξ]dξ. L´ EVY AREAS OF ORNSTEIN–UHLENBECK PROCESSES IN HILBERT–SPACES 7 For the definition of the so-called fractional derivatives Dα s+and D1−α t−and the definition of the stochastic integral in terms of these expressions we refer to [22]. Note that as a consequence of (9) and (10), and because γ > β,  Dα s+AZ· s S(· − r)ei(ωj(r)−ωj(s))dr[ξ] ≤ckωjkβ(ξ−s)β−α. Since β+α > 1, |D1−α t−(ωk)t−[ξ]| ≤ ckωkkβ(t−ξ)α+β−1, and then X i,j,k Zt sq 1 2 j λκ iZξ s AS(ξ−r)ei(ωj(r)−ωj(s))dr⊗Vq 1 2 kdωk(ξ)21 2 ≤X i λ2γ−2κ i1 2X j,k qjqkkωjk2 βkωkk2 β1 2Zt s (ξ−s)β−α(t−ξ)α+β−1dξ, see [11] to find the estimate of the integral in terms of the norms of the fractional derivatives. Now it suffices to take into account that the last integral can be estimated by c(t−s)2β, which follows from the definition of the Beta function.  Remark 4. Replacing in the above proof ωby ωn−ωwe obtain lim n→∞ Zt s AZξ s S(ξ−r)E(ωn(r)−ωn(s))dr ⊗Vdωn(ξ) =Zt s AZξ s S(ξ−r)E(ω(r)−ω(s))dr ⊗Vdω(ξ) in C2β(¯ ∆0,T ;L2(L2(V, Vκ), V ⊗V)). Indeed, in the proof of Lemma 2 we have shown lim n→∞ X i,j qiqi(kωi−ωn ik2 βkωjk2 β+kωj−ωn jk2 βkωn ik2 β) = 0 a.s. In view of Lemma 2 and Remark 4, we conclude with the proof of Theorem 1. Corollary 5. The mapping ¯ ∆0,T ∋(s, t)→(ω⊗Sω)(s, t)is continuous. Indeed, from the above estimate we have the convergence lim n→∞ sup (s,t)∈¯ ∆0,T k(ω⊗Sω)(s, t)−(ωn⊗Sωn)(s, t)kL2(L2(V,Vκ),V ⊗V)= 0 and straightforwardly ¯ ∆0,T ∋(s, t)→(ωn⊗Sωn)(s, t) is continuous. 3. The construction of (ω⊗Sω)for a Brownian–motion We consider ωto be a trace class Brownian–motion with a positive symmetric trace–class covariance operator Q. Therefore, in that case ωis a trace class fractional Brownian–motion with Hurst–parameter H= 1/2. We want to formulate a weaker condition for the existence of (ω⊗Sω) than the assumption of Lemma 3. Again we will take ˆ V=Vκand will determine which conditions κmust satisfy. 8 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS Now let us split (ω⊗Sω) into its components. Taking into account that e−λitgives the decomposition of S(t) with respect to the base (ei)i∈N, we characterize (ω⊗Sω) for l=iby (Eij(ω⊗Sω)(s, t), el⊗Vek)V⊗V=q 1 2 jq 1 2 k λκ iZt sZξ s e−λi(ξ−r)dωj(r)◦dωk(ξ) =q 1 2 jq 1 2 k λκ iZt s (ωj(ξ)−ωj(s)) ◦dωk(ξ) −λiZt sZξ s e−λi(ξ−r)(ωj(r)−ωj(s))drdωk(ξ) =q 1 2 jq 1 2 k λκ iZt sZξ s e−λi(ξ−r)dωj(r)dωk(ξ)−1 2δjk q 1 2 jq 1 2 k λκ i (t−s) (11) and for l6=iby 0. Here ◦means Stratonovich-integration where for the inner stochastic integration Ito– and Stratonovich–integrals are the same. On the right hand side we have Ito–integration, and the last term there is the Ito–correction which only appears for j=k, expressed by the Kronnecker–symbol δjk. Note that the second line in (11) is obtained by stochastic integration by parts. Let us abbreviate Zt sZξ s e−λi(ξ−r)dωj(r)◦dωk(ξ) =: (ω⊗Sω)ijk. In this section, ωn, ωn iare piecewise linear approximations of the Brownian motions ω, ωiwith respect to the equidistant partition {tn i}of [0, T ] of length 2−nT=δ. We now deal with the convergence of (ωn⊗Sωn)ijk to (ω⊗Sω)ijk. At first we present some lemmata which will be needed for this purpose. As a preparatory result for the following we need (see Karatzas and Shreve [14], Exercise 3.25, Page 163): Lemma 6. Let ωdenote an one-dimensional Brownian motion and xbe a measurable, adapted process satisfying EZT 0 |x(ξ)|2pdξ < ∞ for some real numbers T > 0and p≥1, then EZT 0 x(ξ)dω(ξ)2p ≤(p(2p−1))pTp−1EZT 0 |x(ξ)|2pdξ. Lemma 7. For any p∈Nthere exists a cp>0such that for any M∈N X k1+···+kM=p (2p)! (2k1)!(2k2)! ···(2kM)! ≤cpMp. Let (xi)i=1,···Ma sequence of independent random variables in L2pwhere the odd moments are zero. If in addition Ex2p i≤c,Ex0 i= 1 for xi6≡ 0then EM X i=1 xi2p ≤cpcMp, assuming that all terms containing at least one odd power disappear. L´ EVY AREAS OF ORNSTEIN–UHLENBECK PROCESSES IN HILBERT–SPACES 9 Proof. We have X k1+···+kM=p 1≤Mp. On the other hand sup k1+···+kM=p (2p)! (2k1)!(2k2)! ···(2kM)! ≤cp≤(2p)!. Hence a bound for this expression can be chosen independently of M. To conclude the proof it suffices to apply the multinomial theorem, which reduces to the following situation since the terms containing odd powers are neglected: EM X i=1 xi2p =X a1+···+aM=2p2p a1,··· , aMExa1 1···ExaM M =X k1+···+kM=p2p 2k1,··· ,2kMEx2k1 1···Ex2kM M ≤X k1+···+kM=p2p 2k1,··· ,2kMY m=1,··· ,M,km>0 (Ex2p m)2km 2p ≤cpcMp. If some of the km= 0 then Ex0 m= 1 and the corresponding terms would be removed from the above product.  Lemma 8. For every T > 0,p∈Nand a sufficiently small ǫ > 0there exists cp>0such that for (s, t)∈¯ ∆0,T ,i, j, k, n ∈N E(ω⊗Sω)ijk −(ωn⊗Sωn)ijk 2p≤cpδǫλi(t−s)2p−ǫ. (12) Proof. We here only study the case j=ksince the case j6=kcan be studied similarly, see also Friz and Hairer [8] Page 33f.. For the following we assume that λ1≥1 without lost of generality. For a partition interval [tn m−1, tn m) we use the notation ∆δ k(m) = ωk(tn m)−ωk(tn m−1) and when s < t ∈[tn m−1, tn m) we will also use the notation ∆t−s k(m) = ωk(t)−ωk(s). We divide the proof in several cases: (i) We consider at first the case that s, t ∈[tn m−1, tn m]. In that situation, the difference of double integrals we want to estimate is given for any iby ∆δ k(m)2 δ2λ2 i (e−λi(t−s)+λi(t−s)−1) −∆t−s k(m)2 2 +λiZt sZξ s e−λi(ξ−r)(ωk(r)−ωk(s))drdωk(ξ) (13) which follows by the second part of (11), and where the first expression corresponds to the integral with respect to the piecewise linear approximated Brownian–motions. 16 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS on a set of probability one. On this set we have for every m∈Nthat (ω⊗Sω) is 2βH¨older–continuous in the sense of (5) on ¯ ∆−m,m. In addition, for every m∈N lim n→∞ k(ω⊗Sω)−(ωn⊗Sωn)k2β,−m,m = 0. Outside this set let us define (ωn⊗Sωn)≡0 such that those relations hold for all ω. Theorem 13. There exists a set of full measure where we have the Chen–equality: On a set of probability one for −∞ < s ≤τ≤t < ∞ (ω⊗Sω)(s, t)−(ω⊗Sω)(τ, t)−(ω⊗Sω)(s, τ) = (−1)−αωS(r, t)Sω(s, r). It is a simple exercise to check this Chen–equality for (ωn⊗Sωn) which follows easily by the semigroup properties and by the additivity of the integral with respect to the integral bounds. To see this inequality for (ω⊗Sω) we can apply the convergence conclusion of Theorem 1. Finally, we show that (ω⊗Sω) is strongly stationary. To do this we recall the definition of strong stationarity of the metric dynamical system (Ω,F,P, θ). Definition 14. The stochastic process X= (Xt)t∈Ris called strongly stationary if there exists a θ–invariant set ˜ Ω∈ F of full measure such that for any τ∈Rand ω∈˜ Ωwe have that (19) X0(θτω) = Xτ(ω). Such a process is called weakly stationary if for any τ∈Rthere exists a set Ωτ∈F of full P–measure such that for ω∈Ωτwe have (19). To establish the last result of this paper we need to use the following theorem, see Lederer [17]. Similar results can be found in Arnold and Scheutzow [2], Imkeller and Lederer [13], and K¨ummel [15]. Theorem 15. Let Sbe a Polish–space and let X= (Xt)t∈Rbe a stochastic process over the metric dynamical system (Ω,F,P, θ). Suppose that Xis weakly stationary. Then there exists a continuous stochastic process ˆ X= ( ˆ X)t∈Rsuch that Xand ˆ X are indistinguishable, and ˆ Xis strongly stationary. Now we apply this theorem. Theorem 16. The random fields (ω⊗Sω)defined in Section 2 and Section 3 have a strongly stationary version satisfying the convergence property of Theorem 1 with respect to ¯ ∆−m,m. Proof. We know that (ω⊗Sω) is continuous. Since ¯ ∆−m,m is compact (ω⊗Sω) restricted to this set is uniformly continuous and hence τ7→ (ω⊗Sω)(·+τ, ·+τ)∈ C(¯ ∆−m,m, L2(L2(V, Vκ), V ⊗V)). In addition, shifting the integrals of the random field (ω⊗Sω) we easily obtain that τ7→ (ω⊗Sω)(·,·) is weakly stationary: For every τ∈Rwe have that almost surely (ω⊗Sω)(·+τ, ·+τ) = (θτω⊗Sθτω)(·,·). In particular, the weak stationarity of (ω⊗ω) of Section 2 follows by Deya et al. [6] Section 2, where this area is presented by a double integral of smooth processes, while for the second expression of (6) we use that D1−α ·− and θtcommute in some sense. Therefore, taking S:= C(∆, L2(L2(V, Vκ), V ⊗V)) we can apply Theorem L´ EVY AREAS OF ORNSTEIN–UHLENBECK PROCESSES IN HILBERT–SPACES 17 15 to τ7→ Xτ(ω) := (ω⊗Sω)(·+τ, ·+τ)∈S, obtaining a version which is strongly stationary. However this version is in addition indistinguishable such that we obtain the regularity, the convergence properties and the Chen–equality for this new version of (ω⊗Sω).  References [1] L. Arnold. Random dynamical systems. Springer Monographs in Mathematics. Springer- Verlag, Berlin, 1998. [2] L. Arnold and M. Scheutzow. Perfect Cocycles through stochastic Differential equations. Probab. Theory Relat. Fields. 110 (1998), 559–588. [3] J. B´artek, M.J. Garrido-Atienza and B. Maslowski, Stochastic porous media equation driven by fractional Brownian motion, Stoch. Dyn. 13 (2013), no. 4, 1350010. [4] A. Bensoussan and J. Frehse. Local Solutions for Stochastic Navier Stokes Equations. Mathematical Modelling and Numerical Analysis, 34 (2000), no. 2, 241–273. [5] T. Caraballo, P. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optimization, 50 (2004), 183–207. [6] A. Deya, A. Neuenkirch and S. Tindel, A Milstein-type scheme without L´evy area terms for SDES driven by fractional Brownian motion, Ann. Inst. H. Poincar´e Probab. Statist., 48 (2012), no. 2, 518–550. [7] F. Flandoli and H. Lisei, Stationary conjugation of flows for parabolic SPDEs with multiplicative noise and some applications, Stochastic Anal. Appl., 22 (2004), 1385–1420. [8] P. Friz and M. Hairer. A course on Rough Paths. 2014. To appear. [9] H. Gao, M.J. Garrido-Atienza and B. Schmalfuss, Random attractors for stochastic evolution equations driven by fractional Brownian motion, SIAM J. Math. Anal. 46 (2014), no. 4, 2281– 2309 [10] M.J. Garrido-Atienza, K. Lu and B. Schmalfuss, Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), no. 2, 473–493. [11] M. J. Garrido-Atienza, K. Lu and B. Schmalfuss, Pathwise local solutions to stochastic partial differential equations driven by fractional Brownian motions with Hurst parameters in (1/3,1/2] In preparation. [12] Y. Hu, D. Nualart. Rough path analysis via fractional calculus. Trans. Amer. Math. Soc., 361 (2009), no. 5, 2689–2718. [13] P. Imkeller and C. Lederer. The Cohomology of Stochastic and Random Differential Equations, and local linearization of stochastic flows, Stoch. Dyn. 2 (2002), no. 2, 131–159. [14] I. Karatzas and S.E. Shreve. Brownian Motion and Stochastic Calculus, second edition. Springer-Verlag, 1991. [15] K. K¨ummel. Dynamics of Marcus type SDEs. In preparation. [16] H. Kunita. Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, 1990. [17] C. Lederer, Konjugation stochastischer und zuf¨alliger station¨arer Differentialgleichungen und eine Version des lokalen Satzes von Hartman-Grobman f¨ur stochastische Differentialgleichungen. PhD Thesis. HU Berlin, 2001. [18] T. Lyons and Z. Qian, System control and rough paths. Oxford Mathematical Monographs, Oxford Science Publications. Oxford University Press, Oxford, (2002). [19] B. Maslowski and B. Schmalfuß, Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion, Stochastic Anal. Appl., 22 (2004), 15771607. [20] A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer Applied Mathematical Series. Springer-Verlag, Berlin, 1983. [21] L.C. Young, An integration of H¨older type, connected with Stieltjes integration, Acta Math., 67 (1936), pp. 251–282. [22] M. Z¨ahle. Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields, 111(3):333–374, 1998. 18 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS (Mar´ıa J. Garrido-Atienza) Dpto. Ecuaciones Diferenciales y An´ alisis Num´ erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain E-mail address, Mar´ıa J. Garrido-Atienza: [email protected] (Kening Lu) 346 TMCB, Brigham Young University, Provo, UT 84602, USA E-mail address, Kening Lu: [email protected]u.edu (Bj¨orn Schmalfuß) Institut f¨ ur Stochastik, Friedrich Schiller Universit¨ at Jena, Ernst Abbe Platz 2, 77043, Jena, Germany E-mail address, Bj¨orn Schmalfuß: [email protected]