Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems

arXiv:1305.6903v1 [math.DS] 29 May 2013 PATHWISE SOLUTIONS OF SPDES DRIVEN BY H ¨ OLDER-CONTINUOUS INTEGRATORS WITH EXPONENT LARGER THAN 1/2AND RANDOM DYNAMICAL SYSTEMS Y. CHEN, H. GAO, M. J. GARRIDO-ATIENZA, AND B. SCHMALFUSS Abstract. This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a H¨older continuous function with H¨older exponent in (1/2,1), and with nontrivial multiplicative noise. As a particular situation, we shall consider the case where the equation is driven by a fractional Brownian motion BHwith Hurst parameter H > 1/2. In contrast to the article by Maslowski and Nualart [17], we present here an existence and uniqueness result in the space of H¨older continuous functions with values in a Hilbert space V. If the initial condition is in the latter space this forces us to consider solutions in a different space, which is a generalization of the H¨older continuous functions. That space of functions is appropriate to introduce a non-autonomous dynamical system generated by the corresponding solution to the equation. In fact, when choosing BHas the driving process, we shall prove that the dynamical system will turn out to be a random dynamical system, defined over the ergodic metric dynamical system generated by the infinite dimensional fractional Brownian motion. 1. Introduction During the last years, there has been an increasing interest in solving stochastic partial differential equations (SPDEs) beyond the Brownian motion case. One of these attempts is given by the Rough Path Theory, and we refer to the monographs by Lyons and Qian [16] and Friz and Victoir [5] for a comprehensive presentation of this theory. When the driving process is a H¨older continuous function with H¨older exponent greater than 1/2, a different technique to solve these equations consists of using fractional calculus tools, see for example the papers [6, 13, 14, 17, 19, 21], to name only a few of them. We generalize the fractional calculus methods in order to give a pathwise meaning to the solutions of some general nonlinear infinite-dimensional evolution equations, associated to an analytic semigroup and driven by a H¨older continuous process with H¨older exponent greater than 1/2. The existence of solutions of this type has been already studied for instance in Maslowski and Nualart [17] and in Garrido-Atienza et al. [6]. In both papers the driving process BHis a regular fractional Brownian motion (fBm), in the sense that the Hurst index H∈(1/2,1), and in both of them, the stochastic integral with integrator BHis defined following the theory developed by Z¨ahle [23], which is based on the so-called fractional derivatives (see Section 2). For these integrals the usual adaptedness assumptions are not necessary. The phase spaces taken in those articles are somehow not as natural as the space of H¨older continuous functions with adequate H¨older exponents, the space that we shall consider in this article as natural phase space. However, at that point we need to afford the problem that the semigroup generated by the linear part of the equation is not H¨older continuous on any interval [0, T ]. Nevertheless, in spite of this lack of regularity, we are able to obtain the existence of solutions in a modification of the space of H¨older continuous functions, and to the best of our knowledge this is the first time that such a modification of the H¨older continuous space is considered to overcome such a difficulty. However, this modification forces us to focus very precisely on checking that the fractional derivatives for the corresponding kind of functions are well-defined. After taking into account all these facts, we are able to prove the existence of a unique solution, by using the Banach fixed point theorem applied pathwise with respect to 1991 Mathematics Subject Classification. Primary: 37L55; Secondary: 60H15, 60G22, 37H05, 35R60. Key words and phrases. Stochastic PDEs, fractional Brownian motion, pathwise solutions, random dynamical systems. Y. Chen and H. Gao: Supported by a China NSF Grant 11171158, National Basic Research Program of China (973 Program) No. 2013CB834100, the Natural Science Foundation of Jiangsu Province (BK2011777), Qing Lan and ”333” Project of Jiangsu Province and the NSF of the Jiangsu Higher Education Committee of China (11KJA110001). M.J. Garrido-Atienza and B. Schmalfuß: Partially supported by the European Funds for Regional Development and Ministerio de Economia y Competitividad (Spain) under grant MTM2011-22411. 1 2 Y. CHEN, H. GAO, M. J. GARRIDO-ATIENZA, AND B. SCHMALFUSS an adequate equivalent norm of the function spaces mentioned above. This paper can be seen in fact as the base for other projects whose main aim is to study asymptotical properties of SPDEs driven by any fBm, by exploiting the pathwise sense of their corresponding solutions. On the one hand, the results presented in this article cover the cases H > 1/2 but do not the white noise case, i.e., when H= 1/2. For the existence of pathwise solutions for this case (and more general for the cases in which H∈(1/3,1/2]) we refer to Garrido-Atienza et al. [8], [9] and the forthcoming paper [7]. We want to emphasize here that the techniques to obtain such a pathwise solution are much more involved and qualitatively different from the methods that we shall present in this article. On the other hand, we wish to analyze the asymptotic behavior of these pathwise solutions by means of the random dynamical systems theory, and therefore one may wonder whether these solutions could generate what is known as a cocycle. This question is course difficult to answer at a first glance, because it is well-known that a large class of partial differential equations with stationary random coefficients and Itˆo stochastic ordinary differential equations generate random dynamical systems (see Arnold [2]), but for the stochastic partial differential equations driven by the standard Brownian motion the problem is rather unsolved. The main obstacle is that the stochastic Itˆo integral is only defined almost surely where the exceptional sets may depend on the initial state. But as far as the cocycle property is concerned this fact contradicts the definition of a random dynamical system, since initial state dependent exceptional sets are not permitted. As pointed out in [6], the main advantage of the pathwise integration with respect to the classical Itˆo integration theory in this context is that we can avoid this dependence on exceptional sets, and this means that we will be able to study the random dynamical system associated to the corresponding evolution equation. This article is therefore the starting point to some other projects in which we aim at analyzing asymptotical properties of solutions to stochastic evolutions equations, as for instance, investigating the existence and structure of the random attractor associated to those equations. Actually, we have already given some previous considerations in that direction when considering stochastic ordinary equations driven by a fBm with H > 1/2, see [10]. This article is organized as follows. In Section 2 we introduce the definition and important properties of the integral having a H¨older continuous function as integrator. Section 3 is devoted to show that evolution equations driven by such a functions have a pathwise mild solution, with the property of generating a nonautonomous dynamical system. In the last section of the article we consider the fractional Brownian motion case. Having already obtained a non-autonomous dynamical system in the previous section, we shall focus on the measurability properties needed to claim that this dynamical system is also a random dynamical system, provided that the ergodic metric dynamical system defined by the fBm is considered. 2. Preliminaries 2.1. Dynamical systems. Let (V, |·|) be a Banach space and let T+=R+or Z+. A mapping ϕ:T+×V→V having the semigroup property ϕ(t, ·)◦ϕ(τ, u0) = ϕ(t+τ, u0), ϕ(0, u0) = u0for t, τ ∈T+and u0∈V is called an autonomous dynamical system. We want to consider a generalization of the concept of an autonomous dynamical system to non-autonomous and random dynamical systems. As a first ingredient, for the time set T=Ror Z, we introduce the flow (θt)t∈Ton the set Ω of non-autonomous perturbations by θ:T×Ω→Ω θt◦θτ=θt+τ, θ0ω=ωfor t, τ ∈T, ω ∈Ω. As a generalization of the semigroup property we consider a cocycle, which is given by a mapping ϕ:T+×Ω×V→V such that ϕ(0, ω, u0) = u0, ϕ(t+τ, ω, u0) = ϕ(t, θτω, ·)◦ϕ(τ, ω, u0),for all t, τ ∈T+, u0∈V, ω ∈Ω. (1) PATHWISE SOLUTIONS OF SPDES AND RANDOM DYNAMICAL SYSTEMS 3 ϕis also called a non-autonomous dynamical system. Let us now equip (Ω, θ) with a measurable structure. In order to do that, we consider the probability space (Ω,F,P), where Fis a σ-algebra on Ω and Pis a measure, assumed to be invariant and ergodic with respect to θ. Then (Ω,F,P, θ) is called a metric dynamical system. AB(T+)⊗F⊗B(V),B(V) measurable mapping ϕhaving the cocycle property (1) is called a random dynamical system (RDS) with respect to the metric dynamical (Ω,F,P, θ). 2.2. Integrals in Hilbert spaces for H¨older continuous integrators with H¨older exponent >1/2. Assume (V,|·|) is a separable Hilbert space. In addition, let −Abe a strictly positive and symmetric operator with a compact inverse which is the generator of an analytic exponential decreasing semigroup Son V. We also introduce the spaces Vδ:= D((−A)δ) with norm |·|Vδfor δ≥0 such that V=V0. The spaces Vδ, δ > 0 are continuously embedded in V. Let (ei)i∈Nbe the complete orthonormal base in Vgenerated by the eigenelements of −Awith associated eigenvalues (λi)i∈N. Let L(Vδ, Vγ) denote the space of continuous linear operators from Vδinto Vγ. Then there exists a constant c > 0 such that we have the estimates |S(t)|L(V,Vγ)=|(−A)γS(t)|L(V)≤cS,γ tγe−λt for γ > 0,(2) |S(t)−id|L(Vσ+µ,Vθ+µ)≤ctσ−θ,for θ≥0, σ ∈[θ, 1 + θ], µ ∈R.(3) In (2) notice that λis a positive constant. We also note that from these inequalities, for 0 ≤q≤r≤s≤t, we can derive that |S(t−r)−S(t−q)|L(Vδ,Vγ)≤c(r−q)α(t−r)−α−γ+δ, |S(t−r)−S(s−r)−S(t−q) + S(s−q)|L(V) ≤c(t−s)β(r−q)γ(s−r)−(β+γ). (4) Next we introduce some proper spaces where later on we shall investigate the existence of pathwise solutions to stochastic evolution systems. Let Cβ([T1, T2]; V) be the Banach space of H¨older continuous functions with exponent β > 0 having values in V. A norm on this space is given by kukβ=kukβ,T1,T2= sup s∈[T1,T2]|u(s)|+|||u|||β,T1,T2, with |||u|||β,T1,T2= sup T1≤s<t≤T2 |u(t)−u(s)| |t−s|β. C([T1, T2]; V) denotes the space of continuous functions on [T1, T2] with values in Vwith finite supremum norm, and let Cβ,∼([T1, T2]; V)⊂C([T1, T2]; V) equipped with the norm kukβ,∼=kukβ,∼,T1,T2= sup s∈[T1,T2]|u(s)|+ sup T1<s<t≤T2 (s−T1)β|u(t)−u(s)| |t−s|β. For every ρ > 0 we can consider the equivalent norm kukβ,ρ,∼=kukβ,ρ,∼,T1,T2= sup s∈[T1,T2] e−ρ(s−T1)|u(s)| + sup T1<s<t≤T2 (s−T1)βe−ρ(t−T1)|u(t)−u(s)| |t−s|β. Lemma 1. Cβ,∼([T1, T2]; V)is a Banach space. Proof. Let (un)n∈Nbe a Cauchy sequence in Cβ,∼([T1, T2]; V). Then it is straightforward that (un)n∈Ntends to u0in C([T1, T2]; V). Let us prove that (un)n∈Ntends also to u0∈Cβ,∼([T1, T2]; V) which follows easily from the convergence of (un)n∈Nin C([T1, T2]; V) and by the boundedness of this sequence in Cβ,∼([T1, T2]; V). Indeed, for every s < t ∈(T1, T2] 4 Y. CHEN, H. GAO, M. J. GARRIDO-ATIENZA, AND B. SCHMALFUSS (s−T1)β|(un−u0)(t)−(un−u0)(s)| |t−s|β= lim m→∞(s−T1)β|(um−un)(t)−(um−un)(s)| |t−s|β. The elements of the sequence on the right hand side are uniformly bounded by kum−unkβ,∼and hence kun−u0kβ,∼≤lim inf m→∞ kum−unkβ,∼=: Yn where (Yn)n∈Ntends to zero for n→ ∞. In the following we wish to define the integral ZT2 T1 Zdω where ωis a H¨older continuous path. Let us assume that ˜ V , ˆ Vare separable Hilbert spaces, then for 0 < α < 1 and general measurable functions Z: [T1, T2]7→ ˆ Vand ω: [T1, T2]7→ ˜ V, we define their Weyl fractional derivatives by Dα T1+Z[r] = 1 Γ(1 −α)Z(r) (r−T1)α+αZr T1 Z(r)−Z(q) (r−q)1+αdq∈ˆ V , D1−α T2−ωT2−[r] = (−1)1−α Γ(α)ω(r)−ω(T2−) (T2−r)1−α+ (1 −α)ZT2 r ω(r)−ω(q) (q−r)2−αdq∈˜ V , where ωT2−(r) = ω(r)−ω(T2−), being ω(T2−) the left side limit of ωat T2. Suppose that z(T1+), ζ(T1+), ζ(T2−) exist, being respectively the right side limit of zat T1and the right and left side limits of ζat T1, T2, and that zT1+∈Iα T1+(Lp((T1, T2); R)), ζT2−∈Iα T2−(Lp′((T1, T2); R)) with 1/p + 1/p′≤1 (see the definition of these spaces in Samko et al. [20]). Then following Z¨ahle [23] we define ZT2 T1 zdζ = (−1)αZT2 T1 Dα T1+zT1+[r]D1−α T2−ζT2−[r]dr +z(T1+)(ζ(T2−)−ζ(T1+)), where zT1+(r) = z(r)−z(T1+) and ζT2−(r) = ζ(r)−ζ(T2−), for r∈(T1, T2). Under the previous conditions, if in addition αp < 1, then the above integral can be rewritten in a shorter way by (5) ZT2 T1 zdζ = (−1)αZT2 T1 Dα T1+z[r]D1−α T2−ζT2−[r]dr. In fact, under the above assumptions, the appearing fractional derivatives in the integrals are well defined taking ˜ V=ˆ V=R. Consider now the separable Hilbert space L2(V) of Hilbert-Schmidt operators from Vinto Vwith the usual norm k·kL2(V)and inner product (·,·)L2(V). A base in this space is given by (6) Eijek=0 : j6=k ei:j=k. Let us consider now mappings Z: [0, T ]→L2(V) and ω: [0, T ]→V. Suppose that zji = (Z, Eji)L2(V)∈ Iα T1+(Lp((T1, T2); R)) and zji(T1+) exists and αp < 1. Moreover, ζiT2−= (ωT2−(t), ei)∈I1−α T2−(Lp′((T1, T2); R)) such that 1/p + 1/p′≤1. In addition, [T1, T2]∋r7→ kDα T1+Z[r]kL2(V)|D1−α T2−ωT2−[r]| ∈ L1((0, T ); R). We then introduce PATHWISE SOLUTIONS OF SPDES AND RANDOM DYNAMICAL SYSTEMS 5 (7) ZT2 T1 Zdω := (−1)αZT2 T1 Dα T1+Z[r]D1−α T2−ωT2−[r]dr. Due to Pettis’ theorem and the separability of Vthe integrand is weakly measurable and hence measurable. In addition, we can present this integral by (8) ZT2 T1 Zdω =X jX iZT2 T1 Dα T1+zji[r]D1−α T2−ζiT2−[r]drej, with norm given by ZT2 T1 Zdω =X jX iZT2 T1 Dα T1+zji[r]D1−α T2−ζiT2−[r]dr 21 2 ≤ZT2 T1kDα T1+Z[r]kL2(V)|D1−α T2−ωT2−[r]|dr. Since these one dimensional integrals under the sums are generalizations of the classical integral, i.e. if ζiwere in C1, we can interpret (7) to be an extension to classical separable Hilbert space valued integrals. Remark 2. Later on ωwill be given by a fractional Brownian motion BHwith Hurst index H > 1/2, in the way BH(t) = ∞ X i=1 √qieiβH i(t), t ∈R, where (βH i(t))i∈Nis a sequence of stochastically independent one-dimensional fBm and P∞ i=1 qi<∞, see Section 4 for a detailed introduction of this process. In that situation, under the extra condition that ∞ X i=1 √qi<∞, the definition of the corresponding integral, i.e. ZT2 T1 ZdBH was given in [17]. In the current article we do not need to require such a strong regularity condition for the noise, but the price we have to pay is that we need to consider a bit more regular integrand functions Z, which belong to the Hilbert-Schmidt space L2(V). In what follows, for H > 1/21such that in fact 1/2< β < β′< H, let Ω be the (θt)t∈R-invariant set of paths ω:R→Vwhich are β′-H¨older continuous on any compact subinterval of R, being zero at zero. For the flow (θt)t∈Ron Ω of non-autonomous perturbations we consider the so-called Wiener shifts given by θ:R×Ω→Ω, θtω(·) = ω(t+·)−ω(t).(9) Lemma 3. Suppose that Z∈Cβ([T1, T2]; L2(V)) and ω∈Ωsuch that 1−β′< α < β. Then ZT2 T1 Zdω ∈V 1This value Hwill be in Section 4 the Hurst parameter of a fractional Brownian motion. 6 Y. CHEN, H. GAO, M. J. GARRIDO-ATIENZA, AND B. SCHMALFUSS is well-defined in the sense of (7). In addition, there exists a constant cdepending only on T2, β, β′such that ZT2 T1 Zdω≤ckZkβ|||θT2ω|||β′,T1−T2,0(T2−T1)β′ =ckZkβ|||θT1ω|||β′,0,T2−T1(T2−T1)β′ =ckZkβ|||ω|||β′,T1,T2(T2−T1)β′. Proof. Remembering that zji = (Z, Eji)L2(V)where Eij denotes the element of the basis in L2(V) given by (6), we have X ij |Dα T1+z(·)ji[r]|21 2 =X ij 1 Γ(1 −α)zji(r) (r−T1)α+αZr T1 zji(r)−zji(q) (r−q)1+αdq21 2 ≤√2c(Pji zji(r)2)1 2 (r−T1)α+X ij Zr T1 zji(r)−zji(q) (r−q)1+αdq21 2 ≤√2ckZ(r)kL2(V) (r−T1)α+Zr T1 kZ(r)−Z(q)kL2(V) (r−q)1+αdq, where we use that    Zr T1 Z(r)−Z(q) (r−q)1+αdq   L2(V)≤Zr T1 kZ(r)−Z(q)kL2(V) (r−q)1+αdq and therefore, since Z∈Cβ([T1, T2]; L2(V)), (10) kDα T1+Z[r]kL2(V)≤ckZkβ((r−T1)−α+ (r−T1)β−α). Similarly, for ω∈Ω, it is straightforward to obtain that (11) |D1−α T2−ωT2−[r]| ≤ c|||ω|||β′,T1,T2(T2−r)α+β′−1. In fact, thanks to the definition of (θt)t∈Rgiven by (9), the following property follows: D1−α T2−ωT2−[r] = (−1)1−α Γ(α)ω(r)−ω(T2) (T2−r)1−α+ (1 −α)ZT2 r ω(r)−ω(q) (q−r)2−αdq =(−1)1−α Γ(α)ω(τ+T2)−ω(T2) (−τ)1−α+ (1 −α)ZT2 τ+T2 ω(τ+T2)−ω(q) (q−τ−T2)2−αdq =(−1)1−α Γ(α)θT2ω(τ) (−τ)1−α+ (1 −α)Z0 τ θT2ω(τ)−θT2ω(q) (q−τ)2−αdq =D1−α 0−(θT2ω)0−[τ], where τ=r−T2∈[T1−T2,0]. From (11), since θT2ω∈Cβ′([T1−T2,0]; V) we also know that |D1−α 0−(θT2ω)0−[τ]| ≤ c|||θT2ω|||β′,T1−T2,0(T2−r)α+β′−1and therefore, for any ω∈Ω we obtain (12) |D1−α T2−ωT2−[r]| ≤ c|||θT2ω|||β′,T1−T2,0(T2−r)α+β′−1. PATHWISE SOLUTIONS OF SPDES AND RANDOM DYNAMICAL SYSTEMS 7 Thus, combining (10) and (12), this leads to ZT2 T1 Zdω ≤ckZkβ|||θT2ω|||β′,T1−T2,0ZT2 T1 ((r−T1)−α+ (r−T1)−α+β)(T2−r)α+β′−1dr ≤ckZkβ|||θT2ω|||β′,T1−T2,0((T2−T1)β′+ (T2−T1)β+β′) ≤ckZkβ|||θT2ω|||β′,T1−T2,0(T2−T1)β′. The constant cappearing above depends on βand β′. Note that the left hand side is independent of the choice of αcontained in an appropriate interval.  Remark 4. As a generalization of Z¨ahle [23] Theorem 2.5 we have the additivity of the integrals: ZT2 T1 Zdω +ZT3 T2 Zdω =ZT3 T1 Zdω for T1< T2< T3. Furthermore, for the set Ω introduced above and the flow θdefined by (9), we can also establish the behavior of the stochastic integral when performing a change of variable, which is a generalization of Lemma 5 in [6]. Nevertheless, for the sake of completeness, we include here a short proof of it. Remark 5. For any τ∈Ryields ZT2 T1 Z(r)dω(r) = ZT2−τ T1−τ Z(r+τ)dθτω(r). Proof. We know that ZT2 T1 Z(r)dω(r) = X jX iZT2 T1 Dα T1+zji[r]D1−α T2−ζiT2−[r]drej, where zji = (Z, Eji)L2(V)and ζiT2−= (ωT2−(t), ei) have been introduced previously in the construction of the integral. Taking into account the definition of the fractional derivatives and the expression of the Wiener shift, making the change of variables s=r−τand afterwards renaming sas r, we have ZT2 T1 Dα T1+zji[r]D1−α T2−ζiT2−[r]dr =ZT2−τ T1−τ Dα (T1−τ)+zji(τ+·)[r]D1−α (T2−τ)−θτζi(T2−τ)−[r]dr. Therefore ZT2 T1 Z(r)dω(r) = X jX iZT2−τ T1−τ Dα (T1−τ)+zji(τ+·)[r]D1−α (T2−τ)−θτζi(T2−τ)−[r]drej =ZT2−τ T1−τ Z(r+τ)dθτω(r).  In the following we consider non-linear operators G:V→L2(V) with appropriate regularity assumptions, which allows to establish the next collection of estimates: Lemma 6. Let G:V→L2(V)be a twice continuously Fr´echet–differentiable operator with bounded first and second derivatives. Let us denote, respectively, by cDG, cD2Gthe bounds for these derivatives and set 8 Y. CHEN, H. GAO, M. J. GARRIDO-ATIENZA, AND B. SCHMALFUSS cG=kG(0)kL2(V). Then, for u1, u2, v1, v2∈V, we have kG(u1)kL2(V)≤cG+cDG|u1|, kG(u1)−G(v1)kL2(V)≤cDG|u1−v1|, kG(u1)−G(v1)−(G(u2)−G(v2))kL2(V) ≤cDG|u1−v1−(u2−v2)|+cD2G|u1−u2|(|u1−v1|+|u2−v2|). The proof of this Lemma is straightforward, see for instance Maslowski and Nualart [17]. In the next lemma we estimate the fractional derivative of a very specific term which will appear later. Lemma 7. Let Sbe the semigroup generated by −Aintroduced at the beginning of this subsection and assume that Gsatisfies the assumptions of Lemma 6. Assume that u∈Cβ,∼([T1, T2]; V)for T1≥0. Then for the orthonormal base (ei)i∈Nof Vintroduced at the beginning of this section, for any i, j ∈N, (ei, S(t−·)G(u(·))ej)∈Iα T1+(Lp((T1, t); R)). In addition, the mapping r7→ Dα T1+S(t− ·)G(u(·))[r]is measurable on [T1, t]for t≤T2and satisfies the estimate ||Dα T1+S(t−·)G(u(·))[r]||L2(V)≤c(1 + kukβ,∼)(r−T1)−α1 + (r−T1)β (t−r)β. Proof. We restrict ourselves to the case T1= 0. We want to prove that e−λi(t−·)(ei, G(u(·))ej)∈Iα 0+(Lp((0, t); R)), p > 1. Since e−λi(t−·)is Lipschitz on [0, t] it is enough to consider a Lipschitz function g:V→R, where the Lipschitz constant is denoted by Lg, and ask whether g(u(·)) ∈Iα 0+(Lp((0, t); R)). To see this property we apply Samko et al. [20] Theorem 13.2. Trivially g(u(·)) ∈Lp((0, t); R) and g(u(0)) exists since uis continuous on [0, t). Stated as before β > α (see Lemma 3), choosing p > 1 such that pβ < 1 (and hence pα < 1) and p(1 + α−β)<1, and considering ψǫ(r) =   Rr−ǫ 0 g(u(r))−g(u(q)) (r−q)1+αdq :r > ǫ g(u(r)) α1 ǫα−1 rα: 0 ≤r≤ǫ we want to show that the function ψǫconverges to ψ0given by the first part of the definition of ψǫfor ǫ= 0 in Lp((0, t); R) for ǫ→0. Take at first ǫ < r, then Zt ǫ|ψǫ(r)−ψ0(r)|pdr ≤Lgkukβ,∼Zt ǫZr r−ǫ (r−q)β (r−q)1+αqβdqp dr ≤Lgkukβ,∼Zt ǫZr r−ǫ (r−q)pβ (r−q)p(1+α)qpβ dqǫp/p′dr where p′is the conjugate exponent of p. Since the interior integral can be enlarged to an integral from 0 to r which is finite if pis close to 1, we have that Zt ǫ|ψǫ(r)−ψ0(r)|pdr ≤cǫp/p′Zt 0 r1−(1+α)pdr that converges to zero for ǫ→0. Furthermore, for r∈[0, ǫ], the convergence of ψǫto 0 in Lp((0, t); R) follows thanks to the boundedness of g(u(·)) over that interval for pclose to 1. To see the a priori estimate for Dα T1+S(t−·)G(u(·))[r] we refer to (18) where a similar estimate is derived.  We also can state that ζiT2−= (ωT2−(t), ei) is in I1−α T2−(Lp′((T1, T2); R)) for any p′>1 because this function is β′–H¨older continuous. Note that ζiT2−∈Lp′((T1, T2); R)) and in addition r7→ ζiT2−(r) (T2−r)1−α PATHWISE SOLUTIONS OF SPDES AND RANDOM DYNAMICAL SYSTEMS 9 is β′+α−1–H¨older-continuous when we augment this definition by 0 at r=T2, which is based on the fact that ζiT2−(T2) = 0, which follows from ωT2−(T2) = 0. Hence this function is in Lp′((T1, T2); R)) for any p′>1. We can also check the other conditions in Samko [20] Theorem 13.2 such that we get that ζiT2−∈I1−α T2−(Lp′((T1, T2); R)). Hence the integral (13) ZT2 T1 Dα T1+S(t−·)G(u(·))[r]D1−α T2−ωT2[r]dr is well defined in the sense of (8). In particular by Lemma 7, by the fact that (ei, S(t)G(u(0))ej) is bounded and the previous discussion, it make sense to define the integral (13) by means of its components. 3. Non–autonomous dynamical systems of evolution equations driven by an integral with H¨ older continuous integrator We now consider the following evolution equation on [0, T ]: (14) du =Au dt +F(u)dt +G(u)dω, u(0) = u0∈V driven by a H¨older continuous path ωwith H¨older exponent greater than 1/2. This equation is interpreted in the mild sense such that for t∈[0, T ] we have to solve (15) u(t) = S(t)u0+Zt 0 S(t−r)F(u(r))dr +Zt 0 S(t−r)G(u(r))dω. The non-linear term Gsatisfies the assumptions of Lemma 6. The mapping F:V→Vis supposed to be Lipschitz continuous, but in fact we assume that F≡0; this is a simplification that we make for brevity, and of course that we would achieve the same existence results as we obtain below assuming that Fwere Lipschitz. The integral with respect to dω is interpreted in the sense of the previous section. Let uin Cβ,∼([0, T ]; V) and denote by T(·, ω, u0) the operator defined on Cβ,∼([0, T ]; V) given by the right hand side of (15). Note that Talso depends on the time interval [0, T ], but to make simpler the notation we do not indicate this dependence in the notation of T. We also note that we can consider the system (14) on other general time intervals [T1, T2]. Existence of this kind of equation has been investigated by Maslowski and Nualart [17] when considering as integrators of the stochastic integrals a fractional Brownian motion with Hurst parameter in (1/2,1). However, as we pointed out in the Introduction and in Remark 2, in this article we want to present the existence theory in other function spaces, namely the space of H¨older continuous functions for appropriate exponents. We need the following technical lemma. Lemma 8. Let a > −1, b > −1and a+b≥ −1, d > 0and t∈[0, T ]. If for ρ > 0we define K(ρ) := sup t∈[0,T ] tdZ1 0 e−ρt(1−v)va(1 −v)bdv, then we have that limρ→∞ K(ρ) = 0. Proof. The result follows easily since tdZ1 0 e−ρt(1−v)va(1 −v)bdv ≤(ρ−d 2R1 0va(1 −v)bdv : for t≤ρ−1 2 tdR1 0e−ρ1 2(1−v)va(1 −v)bdv : for t > ρ−1 2 which means that 16 Y. CHEN, H. GAO, M. J. GARRIDO-ATIENZA, AND B. SCHMALFUSS The (θt)t∈R-invariance of Ω is straightforward.  In what follows we consider the ergodic metric dynamical system introduced above restricted to the set Ω: let ˜ Fbe the trace-σ-algebra of Fwith respect to Ω, let ˜ Pthe restriction of Pto this σ-algebra, and θrepresents the restriction of the Wiener shifts to Ω ×R. Then (Ω,˜ F,˜ P, θ) forms a metric dynamical system such that for every ˜ A∈˜ Fand every A∈ F with ˜ A=ATΩ we have that ˜ P(˜ A) = P(A) independent of the representation by A. In addition, the ergodicity of (C0(R, V ),B(C0(R, V )),P, θ) is transferred to (Ω,˜ F,˜ P, θ), see [3] for details. Once we have the adequate ergodic metric dynamical system given by (Ω,˜ F,˜ P, θ), we can prove the main result of this section: Theorem 16. Assume that the driving process ωof (15) is a fractional Brownian motion with Hurst index greater than 1/2. Under the conditions of Theorem 12, for every u0∈Vthere exists a unique mild solution u∈Cβ,∼([0, T ]; V), which generates a random dynamical system ϕ:R+×Ω×V→Vdefined by ϕ(t, ω, u0) = S(t)u0+Zt 0 S(t−s)G(u(s))dω. Proof. Having identified ωas an fBm with Hurst index greater than H > 1/2, the existence of a unique pathwise mild solution to (15) follows by the β′-H¨older regularity of ωand Theorem 12. Furthermore, the pathwise definition of the integral just gave us the non–autonomous dynamical system ϕ studied in Section 3. Therefore, all we have to do is to establish the proper measurability conditions for the mapping ϕ, in that case the B(R+)⊗˜ F ⊗ B(V),B(V) measurability, according to the definition introduced in Subsection 2.1. It suffices to observe that, when starting the iteration procedure of the Banach fixed point theorem with a measurable initial function u0, then ϕ(t, ω, u0) is a pointwise limit of measurable mappings. Moreover, the parameter version of the fixed point theorem for fixed (t, ω) ensures that the fixed point depends continuously on u0∈V. These last two considerations together with Lemma III.14 in [4] allows to claim that ϕ(t, ω, u0) is measurable with respect to its three variables.  Remark 17. Let us briefly comment what we could do in case of having a more regular initial condition, assuming u0∈Vβ. 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Young, An integration of H¨oder type, connected with Stieltjes integration, Acta Math., 67: 251–282, 1936. [23] M. Z¨ahle. Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields, 111(3): 333–374, 1998. (Yong Chen) Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, China, E-mail address, Yong Chen: [email protected] (Hongjun Gao) Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, China, E-mail address, Hongjun Gao: [email protected] (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] (Bj¨orn Schmalfuß) Institut f¨ ur Mathematik, Institut f¨ ur Stochastik, Ernst Abbe Platz 2, 07737, Jena,Germany, E-mail address, Bj¨orn Schmalfuß: [email protected]