Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters H ∈ (1/3, 1/2]

arXiv:1411.5237v1 [math.AP] 19 Nov 2014 LOCAL PATHWISE SOLUTIONS TO STOCHASTIC EVOLUTION EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS WITH HURST PARAMETERS H∈(1/3,1/2] MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS Abstract. In this article we are concerned with the study of the existence and uniqueness of pathwise mild solutions to evolutions equations driven by a H¨older continuous function with H¨older exponent in (1/3,1/2). Our stochastic integral is a generalization of the well-known Young integral. To be more precise, the integral is defined by using a fractional integration by parts formula and it involves a tensor for which we need to formulate a new equation. From this it turns out that we have to solve a system consisting in a path and an area equations. In this paper we prove the existence of a unique local solution of the system of equations. The results can be applied to stochastic evolution equations with a non-linear diffusion coefficient driven by a fractional Brownian motion with Hurst parameter in (1/3,1/2], which is particular includes white noise. September 2, 2016 1. Introduction In this article, we shall focus on the study of a local solution for the following kind of stochastic evolution equations du(t) = Au(t)dt +G(u(t))dω(t), u(0) = u0, (1) in a Hilbert–space V, where the noise input ωis a H¨older continuous function with H¨older exponent in the interval (1/3,1/2), Ais the infinitesimal generator of an analytic semigroup S(·) on Vand Gis a nonlinear term satisfying certain assumptions which will be described in the next sections. As a particular case of driving noises we can consider a fractional Brownian motion BHwith Hurst parameter H∈(1/3,1/2]. To be more precise, we will study (1) in the sense of mild solutions given by (2) u(t) = S(t)u0+Zt 0 S(t−r)G(u(r))dω. Our interpretation of pathwise is that we obtain a solution of these stochastic equations which does not produce exceptional sets depending on the initial conditions. In the classical theory of stochastic evolution equations, i.e., stochastic evolution 2000 Mathematics Subject Classification. Primary: 60H15; Secondary: 60H05, 60G22, 26A33, 26A42. Key words and phrases. Stochastic PDEs, Hilbert-valuedfractional Brownian motion, pathwise solutions. This work was partially supported by MTM2011-22411, FEDER founding (M.J. Garrido-Atienza and B. Schmalfuß), and by NSF0909400 (K. Lu). 1 2 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS equations (SEEs) driven by Brownian motion B1/2, stochastic Ito integrals are constructed to be a limit in probability of particular random variables defined only almost surely, where the exceptional sets may depend on the initial conditions, which is in contradiction with the cocycle property needed to define a random dynamical system. Pathwise results for that classical theory are only available for the white noise case (G= id) and a few special cases when u7→ G(u) is linear. During the last two decades different integration theories have been developed to treat more general noise inputs, and in particular, for tackling the fractional Brownian motion BH. One of these attempts is given by the Rough Path Theory, and we refer to Lyons and Qian [18] and Friz and Victoir [8] for a comprehensive presentation of this theory. Some interesting papers dealing with the study of SEEs by using the rough path theory are [2], [3], [7], [13], [15], [6] and [14] among others. In particular, in this last paper the authors proved the existence of local mild solutions of stochastic SEEs driven by rough paths for β-H¨older–continuous paths (β∈(1/3,1/2]) with a special quadratic nonlinearity. A different technique called Fractional Calculus was developed by Z¨ahle [24], who considered for a fractional Brownian motion with H > 1/2 the well-known Young integral. In contrast to the Ito-or Stratonovich integral, that integral can be defined in a pathwise sense, given by fractional derivatives, which allows a pathwise estimate of the integrals in terms of the integrand and the integrator using special norms. In [20] it is shown the existence and uniqueness of the solution of a finite-dimensional stochastic differential equation driven by a fractional Brownian motion for H > 1/2. These results were extended in [19] to show the existence of mild solutions for SEEs driven by fractional Brownian motion for H > 1/2. Recently Hu and Nualart [16] have proved an existence and uniqueness result for finite-dimensional stochastic differential equations having coefficients which are sufficiently smooth and driven by a fractional Brownian motion BHwith H∈ (1/3,1/2], for which they needed to formulate a second equation for the so-called area in the space of tensors. In our article we adapt the techniques in [16] to obtain a mild solution for (1). However, there are significant differences between our setting and the one in [16], as for instance that in order to define the area equation in the infinite-dimensional setting we have to construct an area object ω⊗Sω, depending on the noise path ωas well as on the semigroup S, satisfying useful properties as the Chen–equality. Under general hypothesis on the nonlinearity Gwe derive the existence and uniqueness of a local pathwise mild solution uto (1). However, global existence is missing in this general context. Under some more restrictive conditions on G, in a forthcoming paper we will obtain the existence and uniqueness of a global mild pathwise solution, which in particular will guarantee that stochastic evolution equations like (1) and driven by an fBm BHwith H∈(1/3,1/2] generate random dynamical systems, a challenging and rather open problem to the best of our knowledge. The article is organized as follows. In Section 2 we give the analytical background to present our theory. In Section 3 we present the so called fractional integration by parts method. Using this technique we can introduce pathwise stochastic integrals allowing us to formulate pathwise stochastic differential equations. In Section 4 we introduce mild path–area solutions. In addition, we formulate and solve a fixedpoint equation having two components, a path- and an area-component. The role of the semigroup Sin the area equation will be given in terms of a particular tensor LOCAL PATHWISE SOLUTIONS TO SEES 3 object ω⊗Sω. We also present an example to show a nonlinearity Gthat matches the abstract theory. The appendix section contains the proofs of some technical results. Finally, we want to stress that two different constructions of the key tensor object ω⊗Sωby using an approximation of the noise path by smooth paths can be found in [12]. One construction considers as driving noise an infinite-dimensional fractional Brownian motion BHwith H∈(1/3,1/2], while, in a less restrictive setting, the second one considers a Hilbert-valued trace-class Brownian motion B1/2. Furthermore, we refer to [10] for a short and recent announcement of our results. 2. Preliminaries Let V= (V, (·,·),| ·|) be a separable Hilbert–space. On Vwe define Ato be the negative and symmetric generator of an analytic semigroup S. We suppose that −A has a point spectrum 0 < λ1≤λ2≤ ··· tending to infinite where the associated eigenelements (ei)i∈Nform a complete orthonormal system on V.D((−A)κ) = Vκ denotes the domain of (−A)κfor κ∈R, and as usual, L(Vκ, Vζ) denotes the space of continuous linear operators from Vκinto Vζ, for κ, ζ ∈R. We then have the following estimates for the semigroup S: kS(t)kL(Vκ,Vγ)=k(−A)γS(t)kL(Vκ,V )≤ctκ−γ,for γ≥κ,(3) kS(t)−idkL(Vσ,Vθ)≤ctσ−θ,for σ−θ∈[0,1].(4) From these properties we can derive easily the following result: Lemma 1. For any ν, η, µ ∈[0,1],κ, γ, ρ ∈Rsuch that κ≤γ+µ, there exists a constant c > 0such that for 0< q < r < s < t we have that kS(t−r)−S(t−q)kL(Vκ,Vγ)≤c(r−q)µ(t−r)−µ−γ+κ, kS(t−r)−S(s−r)−S(t−q) + S(s−q)kL(Vρ,Vρ) ≤c(t−s)η(r−q)ν(s−r)−(ν+η). Throughout the whole paper we will write very often a constant c. This constant can change from line to line. However this constant is always chosen independent of time parameters contained in a fixed interval [0, T ]. Let V×Vand V⊗Vbe the cartesian product and the tensor product of V, see [17]. The norm of V⊗Vis denoted by k·k. For x, y ∈Vwe denote by x⊗Vythe rank-one tensor of V⊗V. Then (ei⊗Vej)i,j∈Nis a complete orthonormal system of V⊗Vwhere (ei)i∈Ncan be any complete orthonormal system for V, although for the following we choose the orthonormal system given at the beginning of this section. Let ( ˆ V , | · |ˆ V,(·,·)ˆ V) be another separable Hilbert–space. By L2(V, ˆ V) (L2(V×V, ˆ V)) we denote the Hilbert-Schmidt operators from V(V×V) to ˆ V. In particular G∈L2(V×V, ˆ V) if and only if X i,j |G(ei, ej)|2 ˆ V<∞. We note that G∈L2(V×V, ˆ V) can be extended to a linear operator ˆ Gdefined on V⊗Vsuch that ˆ G∈L2(V⊗V, ˆ V), see [17] Chapter 2.6. More precisely, we can construct a weak Hilbert-Schmidt mapping p:V×V→V⊗Vwhere 4 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS p(ei, ej) = ei⊗Vejfor i, j ∈N. Then ˆ Gon V⊗Vis determined by factorization such that G=ˆ Gp. In addition, we have kˆ Gk2 L2(V⊗V, ˆ V):= X i,j |ˆ G(ei⊗Vej)|2 ˆ V=X i,j |G(ei, ej)|2 ˆ V=kGk2 L2(V×V, ˆ V). In the following we will write for ˆ Galso the symbol G. Let us now describe the coefficient of the evolution equation that we have in mind. Lemma 2. Let ˆ Vbe a subspace of V. Assume that the mapping G:V→L2(V, ˆ V) is three times continuously Fr´echet–differentiable with bounded first, second and third derivatives DG(u),D2G(u)and D3G(u), for u∈V. Let us denote, respectively, by cDG, cD2Gand cD3Gthe bounds for DG,D2Gand D3G, and let cG=kG(0)kL2(V, ˆ V). Then, for u1, u2, v1, v2∈V, we have • kG(u1)kL2(V, ˆ V)≤cG+cDG|u1|, • kG(u1)−G(v1)kL2(V, ˆ V)≤cDG|u1−v1|, • kDG(u1)−DG(v1)kL2(V×V, ˆ V)≤cD2G|u1−v1|, • kG(u1)−G(u2)−DG(u2)(u1−u2)kL2(V, ˆ V)≤cD2G|u1−u2|2, • kG(u1)−G(v1)−(G(u2)−G(v2))kL2(V, ˆ V)≤cDG|u1−v1−(u2−v2)| +cD2G|u1−u2|(|u1−v1|+|u2−v2|), • kDG(u1)−DG(v1)−(DG(u2)−DG(v2))kL2(V×V, ˆ V) ≤cD2G|u1−v1−(u2−v2)|+cD3G|u1−u2|(|u1−v1|+|u2−v2|). • kG(u1)−G(u2)−DG(u2)(u1−u2)−(G(v1)−G(v2)−DG(v2)(v1−v2))kL2(V, ˆ V) ≤cD2G(|u1−u2|+|v1−v2|)|u1−v1−(u2−v2)| +cD3G|v1−v2||u2−v2|(|u1−u2|+|u1−v1−(u2−v2)|). These estimates follow by the mean value theorem; for a proof of the last one see [20]. Notice that, in particular, DG :V→L2(V, L2(V, ˆ V)) (or equivalently, DG :V→ L2(V×V, ˆ V)) is a bilinear map, that can be extended to DG :V→L2(V⊗V, ˆ V), and D2G(u) is a trilinear map. Next we introduce some function spaces. Let T > 0. For β∈(0,1], we consider the Banach–space of β–H¨older–continuous functions on [0, T ] with values in V, denoted by Cβ([0, T ]; V), with the seminorm kukβ= sup 0≤t≤T|u(t)|+|||u|||β,|||u|||β= sup 0≤s<t≤T |u(t)−u(s)| (t−s)β. If β= 1 we call these functions Lipschitz–continuous. Let Cβ,∼([0, T ]; V) be the space of functions on [0, T ] with values in Vand with norm kukβ,∼= sup 0≤t≤T|u(t)|+ sup 0<s<t≤T sβ|u(t)−u(s)| (t−s)β. LOCAL PATHWISE SOLUTIONS TO SEES 5 Lemma 3. Cβ,∼([0, T ], V )is a Banach–space. The proof of this result can be found in Chen et al. [4]. Let ∆0,T be the triangle {(s, t) : 0 < s ≤t≤T}. For β+β′<1, β ≤β′we introduce the space Cβ+β′,∼(∆0,T , V ⊗V) of continuous functions vdefined on ∆0,T , which are zero for 0 < s =t, such that kvkβ+β′,∼= sup 0<s<t≤T sβkv(s, t)k (t−s)β+β′<∞. These functions may not be defined for s= 0 and can have a singularity for (s, t), s = 0. Lemma 4. The space Cβ+β′,∼(∆0,T ;V⊗V)is a Banach–space. The proof is similar to the proof of Lemma 3 and therefore we omit here. Let us define ¯ ∆0,T ={(s, t) : 0 ≤s≤t≤T}and consider the Banach–space Cβ+β′(¯ ∆0,T ;V⊗V) of continuous functions vdefined on ¯ ∆0,T , which are zero for s=t, equipped with the norm kvkβ+β′= sup 0≤s<t≤T kv(s, t)k (t−s)β+β′<∞. We often use the following integral formula: for every s < t,µ, ν > −1 (5) Zt s (r−s)µ(t−r)νdr =c(t−s)µ+ν+1 where conly depends on µ, ν. This property follows by the definition of the Beta function simply by performing a suitable change of variable. 3. Fractional Calculus In this paper the main instrument to treat (2) is fractional calculus. In this section we present the main features of this theory. We are going to assume that for some T > 0 we have that ω∈Cβ′([0, T ]; V), u∈Cβ,∼([0, T ]; V) and v∈Cβ+β′,∼(∆0,T ;V⊗V) for 1/3< β < β′<1/2, and that this triple of elements satisfies the Chen–equality given by (6) v(s, r) + v(r, t) + (u(r)−u(s)) ⊗V(ω(t)−ω(r)) = v(s, t) for 0 < s ≤r≤t≤T. We would like to emphasize that when ωis smooth an example for vis given by (7) (u⊗ω)(r, t) = Zt r (u(q)−u(r)) ⊗Vdω(q). This tensor area is clearly well defined and satisfies, for 0 < r < t, k(u⊗ω)(r, t)k ≤ c rβkukβ,∼kωkC1(t−r)1+β≤c rβ(t−r)β+β′, (8) and therefore (u⊗ω)∈Cβ+β′,∼(∆0,T ;V⊗V). Moreover, the Chen–equality easily follows in this case. 6 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS Let α∈(0,1). We define the right hand side fractional derivative of order αof u and the left hand side fractional derivative of order 1 −αof ωt−(·) := ω(·)−ω(t), given for 0 < s ≤r≤tby the expressions Dα s+u[r] = 1 Γ(1 −α)u(r) (r−s)α+αZr s u(r)−u(q) (r−q)1+αdq D1−α t−ωt−[r] =(−1)1−α Γ(α)ω(r)−ω(t) (t−r)1−α+ (1 −α)Zt r ω(r)−ω(q) (q−r)2−αdq, where Γ(·) denotes the Gamma function. For tensor valued elements vand for 0< r < t we define D1−α t−v[r] =(−1)1−α Γ(α)v(r, t) (t−r)1−α+ (1 −α)Zt r v(r, q) (q−r)2−αdq. Lemma 5. Suppose that β > α. Then there exists a constant c > 0such that for 0≤s < r ≤t≤T |Dα s+u(·)[r]| ≤ ckukβ,∼ (r−s)α,|D1−α t−ωt−[r]| ≤ c|||ω|||β′(t−r)α+β′−1, and for 0< r < q < t kD1−α t−v[q]−D1−α t−v[r]k ≤ c rβ(kukβ,∼|||ω|||β′+kvkβ+β′,∼)(q−r)α+β+β′−1. (9) The proof of the two first inequalities follows straightforwardly, and the proof of (9) is similar to the one of Lemma 6.3 in [16] with the difference that in that paper the authors work in different function spaces. We refer the reader to Lemma 21 and Corollary 29 in the Appendix section. As an extension of the fractional derivative of order α, for the mapping G:V7→ L2(V, ˆ V) and s < r we introduce the so–called compensated fractional derivative of order αgiven by ˆ Dα s+G(u(·))[r] = 1 Γ(1 −α)G(u(r)) (r−s)α +αZr s G(u(r)) −G(u(q)) −DG(u(q))(u(r)−u(q)) (r−q)1+αdq. It is immediate to prove the following result: Lemma 6. Suppose that α < 2βand Gsatisfies the assumptions of Lemma 2. Then there exists a positive constant csuch that for every 0≤s < r ≤T |ˆ Dα s+G(u(·))[r]| ≤ c(1 + kuk2 β,∼) (r−s)α. Let us assume for a while that V, ˆ V=R. We first recall the following useful property which is an integration by parts formula (−1)αZt s Dα s+u[r]ω(r)dr =Zt s u(r)Dα t−ω[r]dr,(10) see Z¨ahle [24], formula (21). For β > α and α+β′>1 the fractional integral is given by Zt s udω := (−1)αZt s Dα s+u[r]D1−α t−ωt−[r]dr, LOCAL PATHWISE SOLUTIONS TO SEES 7 see again [24], which is a version of the Young integral. By Lemma 5 and the property (5), for the above integral it is easy to derive that Zt s udω≤ckukβ,∼|||ω|||β′(t−s)β′. For H¨older–continuous uand ωthis kind of integral was defined by Young [23]. However, our function uis not H¨older–continuous in the strong sense but u∈ Cβ,∼([0, T ]; R), in which case that integral is also well defined in the above sense since, according to [24], what we need is that u∈Iα s+(Lp((s, t); R)), u(s+) bounded and ωt−∈I1−α t−(Lq((s, t); R)), with αp < 1, p−1+q−1≤1 (for the definition of these spaces we refer to Samko et al. [22]). In particular, under our conditions on α, β and β′, we know that ωt−∈I1−α t−(Lq((s, t); R)) for any q > 1 and u∈Iα s+(Lp((s, t); R)) when αp < 1, see Theorem 13.2 of [22]. Next we introduce integrals of fractional type with values in a separable Hilbert– space. To do that, we need a new separable Hilbert–space ( ˜ V , |·|˜ V,(·,·)˜ V). Lemma 7. Assume β > α and α+β′>1. Let ˆ V , ˜ Vbe two separable Hilbert– spaces, being (˜ei)i∈Nand (fj)j∈Ncomplete orthonormal basis of ˜ Vand ˆ Vresp., and let [s, t]∋r7→ F(r)∈L2(˜ V , ˆ V),[s, t]∋r7→ ξ(r)∈˜ V be measurable functions such that F∈Cβ,∼([0, T ]; L2(˜ V , ˆ V)),ξ∈Cβ′([0, T ]; ˜ V) and r7→ kDα s+F[r]kL2(˜ V , ˆ V)|D1−α t−ξ[r]|˜ Vis Lebesgue-integrable. Then for 0≤s≤ r≤t≤Twe can define Zt s F(r)dξ(r) := (−1)αX jX iZt s Dα s+(fj, F(·)˜ei)ˆ V[r]D1−α t−(˜ei, ξ(·))˜ V[r]drfj. That this expression is well defined follows by Zt s F(r)dξ(r)ˆ V =X jX iZt s Dα s+(fj, F(·)˜ei)ˆ V[r]D1−α t−(˜ei, ξ(·))˜ V[r]dr21 2 ≤X jZt sX i (Dα s+(fj, F(·)˜ei)ˆ V[r])2X i (D1−α t−(˜ei, ξ(·))˜ V[r])21 2 dr21 2 ≤Zt sX j,i (Dα s+(fj, F(·)˜ei)ˆ V[r])2X i (D1−α t−(˜ei, ξ(·))˜ V[r])21 2 dr ≤Zt sX j,i (Dα s+(fj, F(·)˜ei)ˆ V[r])21 2X i (D1−α t−(˜ei, ξ(·))˜ V[r])21 2 dr =Zt skDα s+F[r]kL2(˜ V , ˆ V)|D1−α t−ξ[r]|˜ Vdr < ∞.(11) 8 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS Observe that this last integral is finite since kDα s+F[r]kL2(˜ V , ˆ V)=X j,i (Dα s+(fj, F(·)˜ei)ˆ V[r])21 2 =X j,i 1 Γ(1 −α)(fj, F(r)˜ei)ˆ V (r−s)α+αZr s (fj, F(r)˜ei)ˆ V−(fj, F(q)˜ei)ˆ V (r−q)1+αdq21 2 ≤√2c(Pj,i(fj, F(r)˜ei)2 ˆ V)1 2 (r−s)α+X j,i Zr s (fj, F(r)˜ei)ˆ V−(fj, F(q)˜ei)ˆ V (r−q)1+αdq21 2 ≤√2ckF(r)kL2(˜ V , ˆ V) (r−s)α+Zr s kF(r)−F(q)kL2(˜ V , ˆ V) (r−q)1+αdq ≤qc(r−s)−αkFkCβ,∼([0,T ];L2(˜ V , ˆ V)), and |D1−α t−ξ[r]|˜ V≤c|||ξ|||β′(t−r)α+β′−1(see Lemma 5), so it suffices to apply (5). Now we can apply Lemma 7 to define integrals of fractional type with values in a separable Hilbert–space, as well as to consider integrators with values in Vor V⊗V. For example, consider an integrand of the type G(u(r)) where u∈Cβ,∼([0, T ]; V), and β < α < 2β,α+β′>1, β + 1 >2α. Note that if DG is bounded then G(u)∈Cβ,∼([0, T ], V ), but since we do not assume that β > α, then Dα s+G(u(·)) is not well-defined. However, we can apply Lemma 7 in the following way Zt s G(u(·))dω = (−1)αZt s Dα s+DG(u(·))(u(·)−u(s),·)[r]D1−α t−ωt−[r]dr +(−1)αZt s Dα s+(G(u(·)) −DG(u(·))(u(·)−u(s),·))[r]D1−α t−ωt−[r]dr. (12) Since u, ω, v are coupled by the Chen–equality (6), we get D1−α t−v(s, ·)t−[r] = (−1)1−α Γ(α)v(s, r)−v(s, t) (t−r)α+ (1 −α)Zt r v(s, r)−v(s, q) (q−r)2−αdq =(−1)1−α Γ(α)−v(r, t)−(u(r)−u(s)) ⊗V(ω(t)−ω(r)) (t−r)α + (1 −α)Zt r −v(r, q)−(u(r)−u(s)) ⊗V(ω(q)−ω(r)) (q−r)2−αdq =−D1−α t−v[r] + (u(r)−u(s)) ⊗VD1−α t−ωt−[r]. (13) Similarly, it is easy to derive that Dα s+(G(u(·)) −DG(u(·))(u(·)−u(s),·))[r] =ˆ Dα s+G(u(·))[r]−Dα s+DG(u(·))[r](u(r)−u(s),·), and thus, coming back to (12) we obtain that LOCAL PATHWISE SOLUTIONS TO SEES 9 Zt s G(u(·))dω =(−1)αZt s ˆ Dα s+G(u(·))[r]D1−α t−ωt−[r]dr −(−1)αZt s Dα s+DG(u(·))[r]D1−α t−v(·, t)[r]dr =(−1)αZt s ˆ Dα s+G(u(·))[r]D1−α t−ωt−[r]dr −(−1)2α−1Zt s D2α−1 s+DG(u(·))[r]D1−α t−D1−α t−v(·, t)[r]dr, (14) where the last equality is true thanks to the fractional integration by parts formula (10). The last integral on the right hand side of (14) is well-defined due to, for 0≤s≤r≤t≤T, (15) kD1−α t−D1−α t−v[r]k ≤ c(kvkβ+β′,∼+kukβ,∼|||ω|||β′)(r−s)−β(t−r)β+β′+2α−2 (see Lemma 21 above) and hence this integral can be defined by using Lemma 7. Remark 8. It may be checked that the integral given in Lemma 7 can be also defined for an integrator like F(r) = S(t−r)G(u(r)), which is not H¨older–continuous at zero. The reason is because the fj-modes of this expression are H¨older continuous. Furthermore, the integrability condition (11) is satisfied by the regularity properties of the analytic semigroup Sand the regularity of Gand u. Finally, on account of (14) and the estimates of the different fractional derivatives that take part in that expression, we can establish the following result: Lemma 9. Suppose u, v, ω satisfy the assumptions from the beginning of this section and Gsatisfies the assumptions of Lemma 2. Let β < α < 2β,α+β′> 1, β + 1 >2α. Then for 0≤s≤t≤Twe have Zt s G(u)dω≤c((1 + kuk2 β,∼)|||ω|||β′+kukβ,∼kvkβ+β′,∼)(t−s)β′. 4. Path-Area-solutions of stochastic evolution equations In this section we give the definition of a mild solution to (2) and establish a result about the local existence and uniqueness of such solution. In order to understand the notion of mild solution to (2), in a first step we consider the case in which the driving noise is regular, to later on consider the H¨older case in which we are interested in. Note that we could also add a nonlinear diffusion term Fon the right-hand side of the equation in (1). Nevertheless, to simplify the whole presentation we have not considered it since the dt-nonlinearity is not the interesting problem to be treated in the paper. 4.1. System (2) driven by smooth paths ω.In this situation, since Ahas the properties of Section 2 and DG is bounded, we remind that for any u0∈Vthere exists a unique solution u∈C([0, T ]; V) of (2) for any T > 0, see [21]. In addition, applying the properties (3), (4) we obtain that this solution is in Cγ,∼([0, T ]; V) for any γ∈(0,1). However, we do not obtain that the solution is in Cγ([0, T ]; V), due to the fact that t7→ S(t)u0is not H¨older-continuous in general but in Cγ,∼([0, T ]; V). 16 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS Lemma 17. Suppose that the conditions of Lemma 15 hold. Then there exists a c > 0depending on ωsuch that for U1, U2∈ˆ W0,T ||T1(U1)−T1(U2)||β,∼≤cT β′(1 + ||U1||2 W+||U2||2 W)||U1−U2||W+c|u1 0−u2 0|. Proof. Let us denote ∆u=u1−u2,∆U= (u1−u2, v1−v2). Then for r∈[0, T ] by Lemma 2 we have kG(u1(r)) −G(u2(r))kL2(V, ˆ V)≤cDGk∆ukβ,∼ and kG(u1(r)) −G(u1(q)) −DG(u1(q))(u1(r)−u1(q)) −(G(u2(r)) −G(u2(q)) −DG(u2(q))(u2(r)−u2(q)))kL2(V, ˆ V) ≤cD2G(ku1kβ,∼+ku2kβ,∼)k∆ukβ,∼ (r−q)2β q2β +cD3Gku2kβ,∼ (r−q)β qβsup r∈[0,T ]|∆u(r)|(2ku1kβ,∼+ku2kβ,∼)(r−q)β qβ ≤c(ku1kβ,∼+ku2kβ,∼)k∆ukβ,∼ (r−q)2β q2β(1 + ku2kβ,∼). Now we can follow the proof of Lemma 15.  Up to now we have obtained appropriate estimates for the first component T1of the operator Tgiven by (26). Now we aim at getting the corresponding estimates for T2, the second component of T. To this end, we need the two following Lemmata, dealing with estimates of some of the terms appearing in the expression of T2. Lemma 18. Under the Hypothesis (H)the following statements hold: (i) For the mapping e∈V7→ ωS(s, t)e= (−1)−αZt s (S(ξ−s)e)⊗Vdω(ξ) the following properties hold true: for 0≤s≤r≤t≤T, e ∈Vand 1/3< β′< β′′ < H, kωS(r, t)e−ωS(s, t)ek ≤ c(r−s)β′(|||ω|||β′+|||ω|||β′′ )|e|, kωS(s, t)ek ≤ c(t−s)β′|||ω|||β′|e|, kωS(s, t)(−A)β′ek ≤ c|||ω|||β′′ |e|. (ii) The mapping E∈L2(V, ˆ V)7→ Sω(s, t)E=Zt s S(t−r)Edω(r) is in L2(L2(V, ˆ V), V ), with norm bounded by c|||ω|||β′(t−s)β′kEkL2(V, ˆ V). LOCAL PATHWISE SOLUTIONS TO SEES 17 Proof. Let β′< β′′ < H such that for α < α′<1 we have β′+α′<1< β′′ +α. Then ωS(r, t)e−ωS(s, t)e=(−1)αZt r (S(ξ−r)e−S(ξ−s)e)⊗Vdω(ξ) −(−1)αZr s S(ξ−s)e⊗Vdω(ξ). (29) Interpreting these integrals in a fractional sense and using Lemma 1 we obtain |Dα r+(S(·−r)e−S(·−s)e)[ξ]| ≤ c(r−s)β′ (ξ−r)α+β′+αZξ r (r−s)β′(ξ−q)α′ (ξ−q)1+α(q−r)α′+β′dq|e| ≤c(r−s)β′(ξ−r)−α−β′|e|. Moreover, since β′′ < H, due to (H3) ω∈Cβ′′ ([0, T ]; V) (and in particular ω∈ Cβ′([0, T ], V )), then by Lemma 5 |D1−α t−ωt−[ξ]| ≤ c|||ω|||β′′ (t−ξ)α+β′′−1. Hence, by applying (5), the first integral on the right-hand side of (29) is bounded in particular by c(r−s)β′|||ω|||β′′ |e|. Furthermore, for the last term in (29), thanks to (H1) and the regularity properties of the semigroup we obtain Zr skDα s+S(·−s)e[ξ]⊗VD1−α r−ωr−[ξ]kdξ ≤c|||ω|||β′(r−s)β′|e|. The second statement of (i) follows directly from the first one taking r=t. For the last conclusion of (i), for the parameters chosen at the beginning of the proof, |Dα s+(S(·−s)(−A)β′e)[ξ]| ≤ c|e| (ξ−s)α+β′ +Zξ s (ξ−q)α′|e| (ξ−q)1+α(q−s)α′+β′dq≤c|e|(ξ−s)−α−β′, and therefore, since in particular β′+α < 1, Zt skDα s+(S(·−s)(−A)β′e)[ξ]⊗VD1−α t−ωt−[ξ]kdξ ≤c|||ω|||β′′ |e|Zt s (ξ−s)−α−β′(t−ξ)α+β′′−1dξ ≤c|||ω|||β′′ (t−s)β′′−β′|e| ≤ c|||ω|||β′′ |e|. Now we prove (ii). First of all, note that Sωis well-defined since, thanks to Remark 8, it suffices the H¨older–continuity of r7→ S(t−r)Eon any interval [ǫ, t]. In addition by the embedding ˆ V⊂Vthe mapping L2(V, ˆ V)∋E7→ ED1−α t−ωt−[r]∈V is in L2(L2(V, ˆ V), V ), and the norm can be estimated cV, ˆ V|D1−α t−ωt−[r]|, see Lemma 28 (ii). Furthermore, the integrand Dα s+S(t−·)[r]·D1−α t−ωt−[r] is weakly measurable with respect to L2(L2(V, ˆ V), V ) such that by Pettis’ theorem the integrand is 18 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS measurable. Because of D1−α t−Eωt−[r] = ED1−α t−ωt−[r], we get kSω(s, t)·kL2(L2(V, ˆ V),V )=   Zt s S(t−r)·dω(r)   L2(L2(V, ˆ V),V ) ≤Zt skDα s+S(t−·)[r]kL(V)k·D1−α t−ωt−[r]kL2(L2(V, ˆ V),V )dr ≤c|||ω|||β′(t−s)β′.  Corollary 19. Let (ωn)n∈Nbe a sequence converging to ωin Cβ′([0, T ]; V). Then lim n→∞ sup 0≤s < t ≤T |e|= 1 k(ω−ωn)S(s, t)ek (t−s)β′= 0, lim n→∞ sup 0≤s<t≤T kSω−ωn(s, t)·kL2(L2(V, ˆ V),V ) (t−s)β′= 0. From the previous result we also obtain that Dα s+(S(·−s)(Sω(q, s)·))[ξ]⊗VD1−α r−ωr−[ξ] is weakly L2(L2(V, ˆ V), V ⊗V)–measurable. Lemma 20. Suppose that the hypothesis (H) holds. For 0< s ≤q≤t≤T, ˜ E∈L2(V⊗V, ˆ V)and U= (u, v)∈ˆ W0,T consider the mapping w(t, s, q) : L2(V⊗V, ˆ V)∋˜ E7→ ˜ Ew(t, s, q)∈V⊗V given by (27) and (28). Then it is well-defined and satisfies, for β′< β′′ the estimate kw(t, s, q)kL2(L2(V⊗V, ˆ V),V ⊗V)≤c||U||Ws−β(q−s)β+β′(t−s)β′, where the constant cdepends on |||ω|||β′′ and k(ω⊗Sω)k2β′. In particular c∼(1 + |||ω|||2 β′′ )k(ω⊗Sω)k2β′. For the proof, see the Appendix. We would like to point out that wsatisfies the generalized Chen–equality ˜ Ew(t, s, r) + ˜ Ew(t, r, q)−˜ E(u(r)−u(s),·)(ω⊗Sω)(r, q) =˜ Ew(t, s, q) + ωS(q, t)Sω(q, r)˜ E(u(r)−u(s),·), (30) for 0 < s ≤r≤q≤t≤T, and the Chen–equality ˜ Ew(t, s, r) + ˜ Ew(t, r, t)−˜ E(u(r)−u(s),·)(ω⊗Sω)(r, t) = ˜ Ew(t, s, t)(31) where the latter one is obtained from (30) taking q=t. In order to prove these two properties we only need to follow an approximation argument, considering (ωn,(ωn⊗Sωn)) satisfying (H3), and therefore converging to (ω, (ω⊗Sω)) in Cβ′([0, T ], V )×C2β′(¯ ∆0,T ;L2(L2(V, ˆ V), V ⊗V)) and take into account that the approximating elements wn:= (u⊗(ωn⊗Sωn)) given in Lemma 10 satisfy the Chen–equalities (23) and (24). In particular, the terms (ωn S⊗ωn) converge to the LOCAL PATHWISE SOLUTIONS TO SEES 19 corresponding term (ωS⊗ω), see the proof of Lemma 20 in the Appendix section. The following result gives estimates for the fractional derivatives of vand w. Lemma 21. Let U= (u, v)∈ˆ W0,T . Then for 0< r < t ≤Twe have kD1−α t−D1−α t−v[r]kβ+β′,∼≤c||U||Wr−β(t−r)2α+β+β′−2, kD1−α t−D1−α t−w(t, ·,·)[r]k ≤ c||U||Wr−β(t−r)2α+β+2β′−2, where the first constant cdepends on |||ω|||β′′ , and the second one on |||ω|||β′′ and k(ω⊗Sω)k2β′. We have also shifted the proof of this result to the Appendix section. Lemma 22. Assume that u0∈V. There exists positive constants ˜cand csuch that for U∈ˆ W0,T kT2(U, ω,(ω⊗Sω), u0)kβ+β′,∼≤˜c|u0|+cT β′(1 + kUk2 W) and, in addition, for two elements U1, U2∈ˆ W0,T : kT2(U1, ω, (ω⊗Sω), u0)−T2(U2, ω, (ω⊗Sω), u0)kβ+β′,∼ ≤˜c|u1 0−u2 0|+cT β′(1 + kU1k2 W+kU2k2 W)kU1−U2kW. The constant cdepends on |||ω|||β′′ ,|||ω|||β′and k(ω⊗Sω)k2β′, while ˜con |||ω|||β′. Proof. Let us denote T2(U)(s, t) =: B1(s, t) + B2(s, t) + B3(s, t), corresponding to the three different addends of T2. For B1we can consider the following splitting: B1(s, t) = Zt s (S(ξ−s)−id)u(s)⊗Vdω(ξ) =Zt s (S(ξ)−S(s))u0⊗Vdω(ξ) +Zt s (S(ξ−s)−id)Zs 0 S(s−r)G(u(r))dω(r)⊗Vdω(ξ) =:B11(s, t) + B12(s, t). B1can be interpreted in the fractional sense thanks to the regularity of its integrand, which means that B11(s, t) = (−1)αZt s Dα s+((S(·)−S(s))u0)[ξ]⊗VD1−α t−ωt−[ξ]dξ. For α < α′where α′is sufficiently close to αand s > 0, applying (3) and (4), |Dα s+((S(·)−S(s))u0)[ξ]| ≤ c|(S(ξ)−S(s))u0| (ξ−s)α+Zξ s |(S(ξ)−S(q))u0| (ξ−q)1+αdq ≤c(ξ−s)β−α sβ+Zξ s (ξ−q)α′q−β (ξ−q)1+αqα′−βdq|u0| ≤c(ξ−s)β−α sβ+1 sβZξ s (ξ−q)α′ (ξ−q)1+α(q−s)α′−βdq|u0| ≤ c(ξ−s)β−α sβ|u0|, 20 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS hence, for s > 0, |B11(s, t)| ≤ c|||ω|||β′|u0| sβZt s (ξ−s)β−α(t−ξ)β′+α−1dξ ≤c|||ω|||β′|u0|s−β(t−s)β′+β, which implies kB11kβ+β′,∼≤˜c|u0|. Moreover, note that  Dα s+(S(·−s)−id) Zs 0 S(s−r)G(u(r))dω(r)[ξ] ≤c|S(ξ−s)−id) Rs 0S(s−r)G(u(r))dω(r)| (ξ−s)α +Zξ s |Rs 0S(ξ−r)−S(q−r))G(u(r))dω(r)| (ξ−q)1+αdq. To deal with B12 on account of Corollary 16 and thanks to the fact that β′> β we have Zξ s |Rs 0(S(ξ−q)−id)S(q−r)G(u(r))dω(r)| (ξ−q)1+αdq ≤Zξ s (ξ−q)α′ (q−s)α′−β(ξ−q)1+α (−A)βZs 0 S(s−r)G(u(r))dω(r) dq ≤c(1 + |||ω|||β′)(1 + kUk2 W)(ξ−s)β−αsβ′−β and by (5) kB12kβ+β′,∼≤cT β′|||ω|||2 β′(1 + kUk2 W). Finally, a similar estimate follows for B2and B3. In order to see this, note that B2and B3can be considered in a similar way to C11 and C12 in the proof of Lemma 15, with the difference that now we have to estimate D1−α t−(ω⊗Sω)(·, t)t−[r] and D1−α t−D1−α t−w(t, ·,·)[r], for which we use the 2β′-H¨older continuity of (ω⊗Sω) together with Lemma 21, arriving at kB2kβ+β′,∼≤cT β′k(ω⊗Sω)k2β′(1 + kUk2 W), kB3kβ+β′,∼≤cT β′(1 + |||ω|||2 β′′ )k(ω⊗Sω)k2β′(1 + kUk2 W). The second part of this lemma can be proven similarly and thus we omit it here.  Now we approximate T(U, ω, (ω⊗Sω), u0) by piecewise linear noise. Lemma 23. Let U∈ˆ W0,T and assume that (ωn,(ωn⊗Sωn)) satisfies (H3). Then lim n→∞ kT(U, ωn,(ωn⊗Sωn), u0)−T(U, ω, (ω⊗Sω), u0)kW= 0. In addition, T(U, ω, (ω⊗Sω), u0)∈ˆ W0,T . Proof. Note that if a term of Tcontains ωor (ω⊗Sω) then this term depends on these expressions linearly or bi-linearly (see the definition of T1,T2together with (27)) which just gives the convergence conclusion. To see the second part of the statement we take U= (u, v)∈ˆ W0,T . For this element we choose an approximating sequence (un,(un⊗ωn)) from (25). We note that T1((un,(un⊗ ωn)), ωn, un 0)∈Cγ([0, T ], V ) and T1((un,(un⊗ωn)), ωn, un 0)(0) ∈D((−A)) for any γ∈(0,1), see Pazy [21] Theorem 4.3.1 and (3)-(4). Therefore T2((un,(un⊗ ωn)), ωn,(ωn⊗Sωn), un 0) can be defined as (T1((un,(un, ωn)), ωn, un 0)⊗ωn) given LOCAL PATHWISE SOLUTIONS TO SEES 21 by (7). By definition of the space ˆ W0,T , by Lemmas 17, 22, and the first part of this lemma we have that that T(U, ω, (ω⊗Sω), u0)∈ˆ W0,T . Let us now prove the uniqueness of the path-area-solution in ˆ W0,T if such a solution exists. Theorem 24. Suppose that U1= (u1, v1), U2= (u2, v2)∈ˆ W0,T are two path-area solutions related to the initial condition u0∈V. Then we have U1=U2. Proof. Let T1∈[0, T ) such that [0, T1] represents the maximal interval of uniqueness. Here T1= 0 means that two different solutions are just branching from 0. Note that the restrictions ˜ Ui= (ui [T1,˜ T], vi ∆[T1,˜ T]) are path-area solutions too with respect to the interval [T1,˜ T],˜ T≤Twith initial condition u(T1). This follows because for T1we can apply Lemma 12. On the other hand, T2restricted to T1< s < t ≤˜ Tkeeps the same structure as the original T2. Then, if the constant C is an estimate of k˜ Uik2 WT1,˜ T, similar to the estimates that we obtained in Lemmas 17 and 22, we obtain 06=k˜ U1−˜ U2kWT1,˜ T≤c(˜ T−T1)β′(1 + 2C)k˜ U1−˜ U2kWT1,˜ T or equivalently 1≤c(˜ T−T1)β′(1 + 2C) for any ˜ T > T1, which is a contradiction if ˜ T−T1is sufficiently small.  Now we present the main theorem of the paper. Theorem 25. Suppose that the standing conditions (H) are satisfied and suppose that T > 0is chosen sufficiently small depending on the parameters of the problem. Then Thas a fixed point in ˆ W0,T ⊂W0,T that defines a mild path-area solution to (2) which is unique. Proof. Lemmas 15, 22 and 23 prove that Tmaps a closed centered ball from ˆ W0,T into itself if T > 0 is sufficiently small and by Lemma 17 and Lemma 22 we obtain that this mapping is a contraction if T > 0 is chosen sufficiently small.  Corollary 26. Let U∈ˆ W0,T be a mild path-area solution given by Theorem 25 and let Unbe the path-area solution corresponding to the equations (2) and (21) and driven by a smooth noise ωn. Then we have lim n→∞ kUn−UkW= 0. In fact, for large enough nwe can find a ball in ˆ W0,T which is mapped into itself by T(·, ωn,(ωn⊗Sωn), u0) and T(·, ω, (ω⊗Sω), u0), and where in addition these mappings are contractions with uniform contraction condition. Then the conclusion follows by the parameter version of the Banach-fixed point Theorem together with Lemma 23. Remark 27. An application of the main results of this article is to prove that the (pathwise) mild path–area solutions are global, see the forthcoming paper [11]. In addition, the results presented in this paper are the basement to prove that SEEs with non-trivial diffusion coefficients Gand driven by fractional Brownian–motions BHwith H∈(1/3,1/2] generate a random dynamical system, see also [11]. That property has been established recently by the same authors when dealing with a more regular fBm, namely BHwith H∈(1/2,1), see [9]. 22 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS 5. Example Let V=l2be the space of square additive sequences with values in R. In addition, let Abe a negative symmetric operator defined on D(−A)⊂l2with compact inverse. In particular, we can assume that −Ahas a discrete spectrum 0 < λ1≤ λ2≤ ··· ≤ λi≤ ··· → ∞ where the associated eigenelements (ei)i∈Nform a complete orthonormal system in l2. The spaces D((−A)ν) = Vνare then defined by {u= (ui)i∈N∈l2:X i λ2ν iu2 i=: |u|2 Vν<∞}. Assume that there exists κ > 0 such that the Hilbert-Schmidt embedding Vκ⊂V holds true, and take ˆ V=Vκ. Consider a sequence of functions (gij )i,j∈N, with gij :V→R, and define G(u) for u∈Vby G(u)v=X j gij(u)vji∈N for all v∈V. We assume that kG(u)k2 L2(V,Vκ)=X j|G(u)ej|2 Vκ=X jX i λ2κ i(G(u)ej)2 i=X i,j λ2κ ig2 ij(u)≤c uniformly with respect to u∈V. In addition, assume that gij are four times differentiable and their derivatives are uniformly bounded in the following way |Dgij(u)(ek)| ≤ cijk g,1,|D2gij(u)(ek, h1)| ≤ cijk g,2|h1|,|D3gij(u)(ek, h1, h2)| ≤ cijk g,3|h1||h2|, |D4gij(u)(ek, h1, h2, h3)| ≤ cijk g,4|h1||h2||h3|for any u∈V, and these bounds satisfy X ijk λ2κ i(cijk g,1)2<∞,X ijk λ2κ i(cijk g,2)2<∞,X ijk λ2κ i(cijk g,3)2<∞,X ijk λ2κ i(cijk g,4)2<∞. To see for instance that DG exists, note that by Taylor expansion |gij(u+h)−gij(u)−Dgij(u)(h)|2≤1 2|D2gij(u+ηh)(h, h)|2≤(cijk g,2)2|h|4 where u, h ∈Vand η∈[0,1]. In particular, we also note that X jk  DG(u)(ek, ej) 2 Vκ =X ijk λ2κ i|Dgi,j(u)(ek)|2≤X ijk λ2κ i(cijk g,1)2=: c2 DG <∞. This condition ensures the Lipschitz continuity of Gas well as the Hilbert-Schmidt property of DG. Similarly, we obtain that DG is also Lipschitz with respect to the Hilbert-Schmidt norm. We also obtain the existence of the second and third derivative. Hence the conditions on Gin Hypothesis (H) hold. The developed theory can be also applied to other examples of G, like kernel integrals, see [11]. LOCAL PATHWISE SOLUTIONS TO SEES 23 6. Appendix In the appendix we prove some technical estimates related to w= (u⊗(ω⊗Sω)). Proof of Lemma 10 Proof. For ˜ E∈L2(V⊗V, ˆ V) we define f˜ E:L2(V, V ⊗V)×V→L2(V, V ⊗V) given by f˜ E(Q, u) = Q(˜ E(u, ·)). From (17) for smooth ωwe have that ˜ E(u⊗(ω⊗Sω)(t))(s, q) = −(−1)αZq s ωS(r, t)˜ E(u(r)−u(s), ω′(r))dr =−(−1)αZq s f˜ E(ωS(r, t), u(r)−u(s))ω′(r)dr. Following Theorem 3.3 in [16], we have Zq s f˜ E(ωS(r, t), u(r)−u(s))ω′(r)dr =(−1)αZq s ˆ Dα s+f˜ E(ωS(·, t), u(·)−u(s))[r]D1−α q−ωq−[r]dr −(−1)αZq s (ωS(r, t)−ωS(s, t))Dα s+˜ E(u(·)−u(s),·)[r]D1−α q−ωq−[r]dr −(−1)αZq s Dα s+ωS(·, t)[r]˜ E(u(r)−u(s),·)D1−α q−ωq−[r]dr +Zq s Df ˜ E(ωS(r, t), u(r)−u(s))(ωS(r, t)−ωS(s, t), u(r)−u(s))ω′(r)dr. (32) Now we calculate the derivative of f˜ E: Df ˜ E(ωS(r, t), u(r)−u(s))(ωS(r, t)−ωS(s, t), u(r)−u(s))ω′(r) =(ωS(r, t)−ωS(s, t)) ˜ E(u(r)−u(s), ω′(r)) + ωS(r, t)˜ E(u(r)−u(s), ω′(r)) =˜ E(u(r)−u(s),·)D2(ωS(t)⊗ω)(s, r) + ωS(r, t)˜ ED2(u⊗ω)(s, r). Substituting the above expression in (32), after applying fractional integration (10) to the last two terms, we have to calculate ED1−α q−(ωS(t)⊗ω)q−(s, ·)[r], for E∈ L2(V, ˆ V), and ˜ ED1−α q−(u⊗ω)(t)(s, ·)q−[r]. First, we have ED1−α q−(ωS(t)⊗ω)q−(s, ·)[r] = D1−α q−E(ωS(t)⊗ω)q−(s, ·)[r] =(−1)1−α Γ(α)E(ωS(t)⊗ω)(s, r)−E(ωS(t)⊗ω)(s, q) (q−r)1−α + (1 −α)Zq r E(ωS(t)⊗ω)(s, r)−E(ωS(t)⊗ω)(s, θ) (θ−r)2−αdθ =−(−1)1−α Γ(α)E(ωS(t)⊗ω)(r, q) + (ωS(r, t)−ωS(s, t))E(ω(q)−ω(r)) (q−r)1−α + (1 −α)Zq r E(ωS(t)⊗ω)(r, θ) + (ωS(r, t)−ωS(s, t))E(ω(θ)−ω(r)) (θ−r)2−αdθ =−ED1−α q−(ωS(t)⊗ω))[r] + (ωS(r, t)−ωS(s, t))ED1−α q−ωq−(r). 24 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS Secondly, similar to (13) we obtain that ˜ ED1−α q−(u⊗ω)(t)(s, ·)q−[r] = −˜ ED1−α q−(u⊗ω)(t)[r] + ˜ E(u(r)−u(s), D1−α q−ωq−(r)), and substituting the last two expressions into (32) we obtain the conclusion.  In the following we want to prove that wis well–defined for U∈ˆ W0,T . To this end we will consider a mapping w(t, s, q) = w(U, ω, (ω⊗Sω))(t, s, q) which coincides for smooth ωwith the expression introduced in Lemma 10. In order to prove the regularity of wstated in Lemma 20 we need several properties that we collect and prove in the following result. Lemma 28. (i) Let ˜ E∈L2(V⊗V, ˆ V)and let v∈V⊗Vbe fixed. Then the mapping ˜ E7→ ˜ Ev is in L2(L2(V⊗V, ˆ V), V ). (ii) Let E∈L2(V, ˆ V)and let u∈Vbe fixed. Then the mapping E7→ Eu is in L2(L2(V, ˆ V), V ). (iii) Let ˜ E∈L2(V⊗V, ˆ V)and u∈V. Then k˜ E(u, ·)kL2(V, ˆ V)=k˜ E(u⊗V·)kL2(V, ˆ V)≤ |u|k˜ EkL2(V⊗V, ˆ V). Proof. Consider the separable Hilbert–space L2(V⊗V, ˆ V) equipped with the complete orthonormal basis ( ˜ Eijk)i,j,k∈Ngiven by ˜ Eijk(el⊗Vem) = ˜ Eijk(el, em) = 0 : j6=lor k6=m fi:j=land k=m. We remind that (ei)i∈Nand (fi)i∈Nare, respectively, complete orthonormal basis of Vand ˆ V. Then for v∈V⊗V, sX ijk (˜ Eijkv)2=sX i|fi|2X jk v2 jk =cV, ˆ Vkvk where vjk is the mode of vwith respect to ej⊗Vek. This completes (i). The second statement can be proven similarly and therefore we omit its proof. Finally, we have k˜ E(u, ·)k2 L2(V, ˆ V)=X i|˜ E(u, ei)|2 ˆ V =X i|X k uk˜ E(ek, ei)|2 ˆ V≤X iX k|uk||˜ E(ek, ei)|ˆ V2 ≤X iX k|uk|21 2X k|˜ E(ek, ei)|2 ˆ V1 22 =|u|2k˜ Ek2 L2(V⊗V, ˆ V).  LOCAL PATHWISE SOLUTIONS TO SEES 25 In what follows we abbreviate the notation in the following way: let us denote L2,⊗=L2(L2(V, ˆ V), V ⊗V), L2,⊗,⊗=L2(L2(V⊗V, ˆ V), V ⊗V). Proof of Lemma 20. Proof. Let us consider separately the three terms of w(t, s, q) given by (27). Precisely, we start estimating the third term I3(˜ E) := Zq s D2α−1 s+ωS(·, t)[r]D1−α q−D1−α q−˜ Ev[r]dr. As we have seen in Lemma 28 (i), for a fixed v∈V⊗Vthe mapping L2(V⊗V, ˆ V)∋ ˜ E7→ ˜ Ev is in L2(L2(V⊗V, ˆ V), V ) where an estimate of the norm of this operator is given by cV, ˆ Vkvk. Then, since Lemma 18 (i) in particular implies that D2α−1 s+ωS(r, t) is in L(V, V ⊗V), the mapping ˜ E7→ I3(˜ E) is in L2,⊗,⊗. We have kI3(·)kL2,⊗,⊗≤Zq skD2α−1 s+ωS(·, t)[r]·D1−α q−D1−α q−v[r]kL2,⊗,⊗dr ≤Zq skD2α−1 s+ωS(·, t)[r]kL(V,V ⊗V)k·D1−α q−D1−α q−v[r]kL2(L2(V⊗V, ˆ V),V )dr. In order to estimate the second factor in the integrand of I3, note that due to Lemma 21, for r∈(s, q) and U∈ˆ W0,T , we obtain kD1−α q−D1−α q−v[r]k ≤ ckUkWr−β(q−r)β+β′+2α−2 and as we have said at the beginning of this proof kD1−α q−D1−α q−·v[r]kL2(L2(V⊗V, ˆ V),V )≤ccV, ˆ VkUkWr−β(q−r)β+β′+2α−2. On the other hand, by Lemma 18 (i), kD2α−1 s+ωS(·, t)[r]kL(V,V ⊗V)≤c(t−r)β′ (r−s)2α−1+Zr s (r−ξ)β′ (r−ξ)2αdξ(|||ω|||β′+|||ω|||β′′ ). Combining the previous estimates we can conclude kI3(·)kL2,⊗,⊗≤ckUkWs−β(t−s)β′(q−s)β+β′, where cdepends on |||ω|||β′′ and |||ω|||β′. Next we deal with I2(˜ E) := Zq s ˆ Dα s+ωS(·, t)˜ E(u(·)−u(s),·)[r]D1−α q−ωq−[r]dr. Observe that kˆ Dα s+ωS(·, t)˜ E(u(·)−u(s),·)[r]D1−α q−ωq−[r]kL2,⊗,⊗ ≤kωS(r, t)˜ E(u(r)−u(s), D1−α q−ωq−[r])kL2,⊗,⊗ (r−s)α +cZr s k(ωS(r, t)−ωS(θ, t)) ˜ E(u(r)−u(θ), D1−α q−ωq−[r])kL2,⊗,⊗ (r−θ)1+αdθ ≤ccV, ˆ V(q−r)α+β′−1s−βk˜ EkL2(V⊗V, ˆ V)kukβ,∼|||ω|||β′ ×(|||ω|||β′+|||ω|||β′′ )((t−r)β′(r−s)β−α+ (r−s)β′+β−α),