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

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.


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), in a Hilbert-space V , where the noise input ω is a Hölder continuous function with Hölder exponent in the interval (1/3, 1/2), A is the infinitesimal generator of an analytic semigroup S(·) on V and G is 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 B H with 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)u 0 + t 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 equations (SEEs) driven by Brownian motion B 1/2 , 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 u → 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 B H . 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-Hölder-continuous paths (β ∈ (1/3, 1/2]) with a special quadratic nonlinearity. A different technique called Fractional Calculus was developed by Zähle [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 B H with 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 ⊗ S ω, depending on the noise path ω as well as on the semigroup S, satisfying useful properties as the Chen-equality. Under general hypothesis on the nonlinearity G we derive the existence and uniqueness of a local pathwise mild solution u to (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 B H with 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 S in the area equation will be given in terms of a particular tensor object ω ⊗ S ω. We also present an example to show a nonlinearity G that 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⊗ S ω by 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 B H with H ∈ (1/3, 1/2], while, in a less restrictive setting, the second one considers a Hilbert-valued trace-class Brownian motion B 1/2 . Furthermore, we refer to [10] for a short and recent announcement of our results.

Preliminaries
Let V = (V, (·, ·), | · |) be a separable Hilbert-space. On V we define A to be the negative and symmetric generator of an analytic semigroup S. We suppose that −A has a point spectrum 0 < λ 1 ≤ λ 2 ≤ · · · tending to infinite where the associated eigenelements (e i ) i∈N form a complete orthonormal system on V . D((−A) κ ) = V κ denotes the domain of (−A) κ for κ ∈ R, and as usual, L(V κ , V ζ ) denotes the space of continuous linear operators from V κ into V ζ , for κ, ζ ∈ R . We then have the following estimates for the semigroup S: From these properties we can derive easily the following result: Lemma 1. For any ν, η, µ ∈ [0, 1], κ, γ, ρ ∈ R such that κ ≤ γ+µ, there exists a constant c > 0 such that for 0 < q < r < s < t we have that 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 × V and V ⊗ V be the cartesian product and the tensor product of V , see [17]. The norm of V ⊗ V is denoted by · . For x, y ∈ V we denote by x ⊗ V y the rank-one tensor of V ⊗ V . Then (e i ⊗ V e j ) i,j∈N is a complete orthonormal system of V ⊗ V where (e i ) i∈N can be any complete orthonormal system for V , although for the following we choose the orthonormal system given at the beginning of this section. Let (V , | · |V , (·, ·)V ) be another separable Hilbert-space. By We note that G ∈ L 2 (V × V,V ) can be extended to a linear operatorĜ defined on V ⊗ V such thatĜ ∈ L 2 (V ⊗ V,V ), see [17] Chapter 2.6. More precisely, we can construct a weak Hilbert-Schmidt mapping p : ThenĜ on V ⊗ V is determined by factorization such that G =Ĝp. In addition, we have In the following we will write forĜ also the symbol G.
Let us now describe the coefficient of the evolution equation that we have in mind.
LetV be a subspace of V . Assume that the mapping G : is three times continuously Fréchet-differentiable with bounded first, second and third derivatives DG(u), D 2 G(u) and D 3 G(u), for u ∈ V . Let us denote, respectively, by c DG , c D 2 G and c D 3 G the bounds for DG, D 2 G and D 3 G, and let These estimates follow by the mean value theorem; for a proof of the last one see [20]. Notice that, in particular, DG : V → L 2 (V, L 2 (V,V )) (or equivalently, DG : V → L 2 (V × V,V )) is a bilinear map, that can be extended to DG : V → L 2 (V ⊗ V,V ), and D 2 G(u) is a trilinear map.
The proof of this result can be found in Chen et al. [4].
These functions may not be defined for s = 0 and can have a singularity for (s, t), s = 0.
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 We often use the following integral formula: for every s < t, µ, ν > −1 where c only depends on µ, ν. This property follows by the definition of the Beta function simply by performing a suitable change of variable.

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 ∈ C β ′ ([0, T ]; V ), u ∈ C β,∼ ([0, T ]; V ) and v ∈ C β+β ′ ,∼ (∆ 0,T ; V ⊗ V ) for 1/3 < β < β ′ < 1/2, and that this triple of elements satisfies the Chen-equality given by for 0 < s ≤ r ≤ t ≤ T . We would like to emphasize that when ω is smooth an example for v is given by This tensor area is clearly well defined and satisfies, for 0 < r < t, and therefore (u ⊗ ω) ∈ C β+β ′ ,∼ (∆ 0,T ; V ⊗ V ). Moreover, the Chen-equality easily follows in this case.
Let α ∈ (0, 1). We define the right hand side fractional derivative of order α of u and the left hand side fractional derivative of order 1 − α of ω t− (·) := ω(·) − ω(t), given for 0 < s ≤ r ≤ t by the expressions where Γ(·) denotes the Gamma function. For tensor valued elements v and for 0 < r < t we define Lemma 5. Suppose that β > α. Then there exists a constant c > 0 such that for and for 0 < r < q < t 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 : V → L 2 (V,V ) and s < r we introduce the so-called compensated fractional derivative of order α given bŷ It is immediate to prove the following result: Lemma 6. Suppose that α < 2β and G satisfies the assumptions of Lemma 2. Then there exists a positive constant c such that for every 0 ≤ s < r ≤ T Let us assume for a while that V,V = R. We first recall the following useful property which is an integration by parts formula see Zähle [24], formula (21). For β > α and α + β ′ > 1 the fractional integral is given by 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 For Hölder-continuous u and ω this kind of integral was defined by Young [23]. However, our function u is not Hölder-continuous in the strong sense but u ∈ C β,∼ ([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 ∈ I α s+ (L p ((s, t); R)), u(s+) bounded and ω t− ∈ I 1−α t− (L q ((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 ω t− ∈ I 1−α t− (L q ((s, t); R)) for any q > 1 and u ∈ I α s+ (L p ((s, t); R)) when αp < 1, see Theorem 13.2 of [22].
Lemma 7. Assume β > α and α + β ′ > 1. LetV ,Ṽ be two separable Hilbertspaces, being (ẽ i ) i∈N and (f j ) j∈N complete orthonormal basis ofṼ andV resp., and let That this expression is well defined follows by Observe that this last integral is finite since 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 V or V ⊗V . For example, consider an integrand of the type , but since we do not assume that β > α, then D α s+ G(u(·)) is not well-defined. However, we can apply Lemma 7 in the following way Since u, ω, v are coupled by the Chen-equality (6), we get Similarly, it is easy to derive that and thus, coming back to (12) we obtain that 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 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 Hölder-continuous at zero. The reason is because the f j -modes of this expression are Hölder continuous. Furthermore, the integrability condition (11) is satisfied by the regularity properties of the analytic semigroup S and the regularity of G and 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 G satisfies the assumptions of Lemma 2. Let β < α < 2β,

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 Hölder case in which we are interested in. Note that we could also add a nonlinear diffusion term F on 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 A has the properties of Section 2 and DG is bounded, we remind that for any u 0 ∈ V there 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 C γ,∼ ([0, T ]; V ) for any γ ∈ (0, 1). However, we do not obtain that the solution is in C γ ([0, T ]; V ), due to the fact that t → S(t)u 0 is not Hölder-continuous in general but in C γ,∼ ([0, T ]; V ).
Since we want to find the appropriate definition of mild solution to (2) when the driving noise is only Hölder continuous, in what follows we will use the fractional integration techniques of the previous section to rewrite (2) in a way that allows us to shed light on that issue. Assuming that G satisfies the conditions of Lemma 2, similarly to the expression (14) of last section, we can rewrite (2) as where the tensor (u ⊗ ω) is given by (7). Looking at (16) we realize that we also need an equation to determine the corresponding counterpart of (u ⊗ ω). In order to get such an equation, let us introduce the mapping (ω ⊗ S ω)(s, t) given by , for which it suffices to execute the corresponding integrations. In addition, we set It is an easy exercise to check that (ω ⊗ S ω) satisfies the Chen-equality which follows by joining the integrals given in (17) over∆ s,r and∆ r,t and the double integral over the rectangle [s, r] × [r, t] given by the composition of the expressions in (18).
Since ω is regular, the tensor (u ⊗ ω) given by (7) can be expressed as and, exchanging the order of integration, the last integral of (19) can be written as Interpreting (20) in the sense of (14) and coming back to (19), it gives us where the element w = (u ⊗ (ω ⊗ S ω)) is defined bỹ forẼ ∈ L 2 (V ⊗ V,V ). Therefore, to evaluate the last integral of (21) we have to give a meaning to (22).
where, for s ≤ τ ≤ t and E ∈ L 2 (V,V ), (ω S (t) ⊗ ω) satisfies The proof of this result is in the Appendix section. Note that joining the integrals on the right hand side of (22) with respect their domain of integration we obtaiñ that for q = t becomes Ew(t, s, r) +Ẽw(t, r, t) −Ẽ(u(r) − u(s), ·)(ω ⊗ S ω)(r, t) =Ẽw(t, s, t) (24) since ω S (t, t) = 0, and this last formula corresponds to the Chen-equality for w.
We have finally arrived at the following definition: Definition 11. Under the described conditions on A and G, for any u 0 ∈ V , a mild solution of (2) is given by the pair (u, u ⊗ ω) satisfying, respectively, the equations (16) and (21).
We stress once more that in order to reach the previous definition we have considered a regular driving path ω, although in that situation the solution is simply given by the path component, as stated at the beginning of this subsection. Definition 11 has to be seen as the guide to establish the definition of a mild solution in the general case of dealing with driving functions that are only Hölder continuous. (H1) Let H ∈ (1/3, 1/2] and let 1/3 < β ≤ β ′ < H. Suppose that there is an α such that 1 − β < α < 2β, α < β+1 2 . (H2) Let A be the generator of an analytic semigroup S given at the beginning of Section 2 and let G : V → L 2 (V,V ) be a non-linear mapping satisfying the assumptions of Lemma 2, withV ⊂ V such that the embedding operator is Hilbert-Schmidt with norm c V,V . In particular, we could chooseV = V κ assuming that

System
(H3) Let (ω n ) n∈N be a sequence of piecewise linear functions with values in V such that ((ω n ⊗ S ω n )) n∈N is defined by (17). Assume then that for any β ′ < H the sequence ((ω n , (ω n ⊗ S ω n ))) n∈N converges to (ω, ) on a set of full measure.
We would like to mention that in Garrido-Atienza et al. [12] we propose two different settings where assumption (H3) is satisfied. In particular, in that paper we focus on the case of considering as driving noise a fractional Brownian-motion B H with H ∈ (1/3, 1/2] and values in a Hilbert-space, and, by another less restrictive method, on the infinite-dimensional trace-class Brownian-motion B 1/2 . Both constructions are based on some results by Deya et al. [5] . Take a fixed ω ∈ C β ′ ([0, T ], V ) and consider γ such that α < γ < 1. We denote by (W 0,T , · W ) the subspace of elements U = (u, v) of the Banach-space such that the Chen-equality holds. Let us also consider a subsetŴ 0,T of this space given by the limit points in this space of the set (ω n , ω n ⊗ S ω n ) satisfies (H3)}.
Here (u n ⊗ ω n ) is well defined by the integral (7). Note that this set of limit points is a subspace of C β,∼ ([0, T ]; V )×C β+β ′ ,∼ (∆ 0,T ; V ⊗V ) which is closed. HenceŴ 0,T itself is a complete metric space depending on ω with a metric generated by the In addition, elements U ∈Ŵ 0,T satisfy the Chen-equality (6) with respect to ω. Furthermore, for the elements u n , (u n ⊗ ω n ) we can reformulate the right hand side of (16) as in (2). Indeed, all expressions that appear in this procedure (even those which cancel) are well defined. Furthermore, the additivity of integrals follows, and in particular: where these integrals are defined in the sense of (16).
Proof. Replacing (u, v) by an approximating sequence (u n , (u n ⊗ ω n )) then the above formula holds which follows because for F (·) = G(u n (·)) the integral can be defined in the sense of Lemma 7 and (u n ⊗ ω n ) can be defined by (7). Then the additivity of that integral follows by the method in Zähle [24]. Now it suffices to rewrite these integrals according to (16) and consider their V -limit for n → ∞.
Remark 13. Let us stress that for smooth elements ω both integrals of (14) are additive for elements (6). But for the sake of conciseness we did not to include the proof in the paper. We also would like to emphasize that some of the results that we are going to establish below still remain true when considering insteadŴ 0,T the bigger space W 0,T . Nevertheless, for uniqueness of presentation we always keep the spaceŴ 0,T .
In order to study the existence of solutions to (2) we consider the operator (26) T (U, ω, (ω ⊗ S ω), u 0 ) = (T 1 (U, ω, u 0 ), T 2 (U, ω, (ω ⊗ S ω), u 0 )) defined for U = (u, v) ∈Ŵ 0,T by the expressions being (ω S (t) ⊗ ω) defined, for s ≤ τ ≤ t and E ∈ L 2 (V,V ), by where the last integral is defined by fractional derivatives due to the regularity of the semigroup S. Having in mind the Definition 11, the corresponding definition of a mild solution of (2) is given as follows: In what follows we want to study the existence of a fixed point for the operator T .
Lemma 15. Assume that u 0 ∈ V . There exists a c > 0 independent of s < t, u 0 and ω such that for U ∈Ŵ 0,T Proof. By (3) and (4), for 0 < s < t ≤ T we have that |(S(t) − S(s))u 0 | ≤ c (t−s) β s β |u 0 |, and then S(·)u 0 β,∼ ≤ c|u 0 |. We now use the following abbreviations We are going to estimate the V -norm of the following expression:  where this equality holds by Lemma 12. For the first expression, for an α ′ with α ′ − α > 0 sufficiently small we obtain Now evaluating these integrals by (5) we see that Now let us consider C 22 (0, s, t). By using Lemma 1 and (15) we have that and by (H1) and (5) we obtain that The remaining terms can be estimated in a similar manner. Setting s = 0 and considering C 11 (0, t)+C 12 (0, t) we obtain the same estimate for the norm of C([0, T ]; V ), hence the proof is complete.
The following conclusion with respect to the regularity of the above integrals holds true: Corollary 16. Let β < β ′ < H, then there exists a c > 0 independent of s < t and ω such that for U ∈Ŵ 0,T We omit the proof of this result since we only would need to estimate |(−A) β C 11 (0, t)+ (−A) β C 12 (0, t)|, where C 11 and C 12 were defined in the proof of Lemma 15. Hence, it suffices to follow the estimates of that result together with the properties of the semigroup S.
Lemma 17. Suppose that the conditions of Lemma 15 hold. Then there exists a c > 0 depending on ω such that for U 1 , U 2 ∈Ŵ 0,T . Then for r ∈ [0, T ] by Lemma 2 we have Now we can follow the proof of Lemma 15.
Up to now we have obtained appropriate estimates for the first component T 1 of the operator T given by (26). Now we aim at getting the corresponding estimates for T 2 , 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 T 2 . the following properties hold true: for 0 ≤ s ≤ r ≤ t ≤ T, e ∈ V and 1/3 < β ′ < β ′′ < H, (ii) The mapping Proof. Let β ′ < β ′′ < H such that for α < α ′ < 1 we have β ′ + α ′ < 1 < β ′′ + α. Then Interpreting these integrals in a fractional sense and using Lemma 1 we obtain Moreover, since β ′′ < H, due to (H3) ω ∈ C β ′′ ([0, T ]; V ) (and in particular ω ∈ C β ′ ([0, T ], V )), then by Lemma 5 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 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, and therefore, since in particular β ′ + α < 1, Now we prove (ii). First of all, note that S ω is well-defined since, thanks to Remark 8, it suffices the Hölder-continuity of r → S(t−r)E on any interval [ǫ, t]. In addition by the embeddingV ⊂ V the mapping is in L 2 (L 2 (V,V ), V ), and the norm can be estimated c V,V |D 1−α t− ω t− [r]|, see Lemma 28 (ii). Furthermore, the integrand D α s+ S(t − ·)[r] · D 1−α t− ω t− [r] is weakly measurable with respect to L 2 (L 2 (V,V ), V ) such that by Pettis' theorem the integrand is From the previous result we also obtain that Lemma 20. Suppose that the hypothesis (H) holds. For 0 < s ≤ q ≤ t ≤ T , E ∈ L 2 (V ⊗ V,V ) and U = (u, v) ∈Ŵ 0,T consider the mapping w(t, s, q) : L 2 (V ⊗ V,V ) ∋Ẽ →Ẽw(t, s, q) ∈ V ⊗ V given by (27) and (28). Then it is well-defined and satisfies, for β ′ < β ′′ the estimate where the constant c depends on |||ω||| β ′′ and (ω ⊗ S ω) 2β ′ . In particular For the proof, see the Appendix.
The following result gives estimates for the fractional derivatives of v and w.
We have also shifted the proof of this result to the Appendix section.
Lemma 22. Assume that u 0 ∈ V . There exists positive constantsc and c such that for U ∈Ŵ 0,T and, in addition, for two elements U 1 , U 2 ∈Ŵ 0,T : Proof. Let us denote T 2 (U )(s, t) =: B 1 (s, t) + B 2 (s, t) + B 3 (s, t), corresponding to the three different addends of T 2 . For B 1 we can consider the following splitting: B 1 can be interpreted in the fractional sense thanks to the regularity of its integrand, which means that For α < α ′ where α ′ is sufficiently close to α and s > 0, applying (3) and (4), hence, for s > 0, To deal with B 12 on account of Corollary 16 and thanks to the fact that β ′ > β we have . Finally, a similar estimate follows for B 2 and B 3 . In order to see this, note that B 2 and B 3 can be considered in a similar way to C 11 and C 12 in the proof of Lemma 15, with the difference that now we have to estimate D 1−α t− (ω ⊗ S ω)(·, t) t− [r] and D 1−α t− D 1−α t− w(t, ·, ·)[r], for which we use the 2β ′ -Hölder continuity of (ω ⊗ S ω) together with Lemma 21, arriving at . The second part of this lemma can be proven similarly and thus we omit it here. Now we approximate T (U, ω, (ω ⊗ S ω), u 0 ) by piecewise linear noise.
Let us now prove the uniqueness of the path-area-solution inŴ 0,T if such a solution exists.
Proof. Let T 1 ∈ [0, T ) such that [0, T 1 ] represents the maximal interval of uniqueness. Here T 1 = 0 means that two different solutions are just branching from 0. Note that the ) are path-area solutions too with respect to the interval [T 1 ,T ],T ≤ T with initial condition u(T 1 ). This follows because for T 1 we can apply Lemma 12. On the other hand, T 2 restricted to T 1 < s < t ≤T keeps the same structure as the original T 2 . Then, if the constant C is an estimate of Ũ i 2 , similar to the estimates that we obtained in Lemmas 17 and 22, we obtain Now we present the main theorem of the paper.
Theorem 25. Suppose that the standing conditions (H) are satisfied and suppose that T > 0 is chosen sufficiently small depending on the parameters of the problem. Then T has a fixed point inŴ 0,T ⊂ W 0,T that defines a mild path-area solution to (2) which is unique.
Proof. Lemmas 15, 22 and 23 prove that T maps a closed centered ball fromŴ 0,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 ∈Ŵ 0,T be a mild path-area solution given by Theorem 25 and let U n be the path-area solution corresponding to the equations (2) and (21) and driven by a smooth noise ω n . Then we have In fact, for large enough n we can find a ball inŴ 0,T which is mapped into itself by T (·, ω n , (ω n ⊗ S ω n ), u 0 ) and T (·, ω, (ω ⊗ S ω), u 0 ), 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 G and driven by fractional Brownian-motions B H with 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 B H with H ∈ (1/2, 1), see [9].

Example
Let V = l 2 be the space of square additive sequences with values in R. In addition, let A be a negative symmetric operator defined on D(−A) ⊂ l 2 with compact inverse. In particular, we can assume that −A has a discrete spectrum 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ i ≤ · · · → ∞ where the associated eigenelements (e i ) i∈N form a complete orthonormal system in l 2 . The spaces D((−A) ν ) = V ν are then defined by Assume that there exists κ > 0 such that the Hilbert-Schmidt embedding V κ ⊂ V holds true, and takeV = V κ . Consider a sequence of functions (g ij ) i,j∈N , with g ij : V → R, and define G(u) for u ∈ V by We assume that In addition, assume that g ij are four times differentiable and their derivatives are uniformly bounded in the following way |Dg ij (u)(e k )| ≤ c ijk g,1 , |D 2 g ij (u)(e k , h 1 )| ≤ c ijk g,2 |h 1 |, |D 3 g ij (u)(e k , h 1 , h 2 )| ≤ c ijk g,3 |h 1 ||h 2 |, |D 4 g ij (u)(e k , h 1 , h 2 , h 3 )| ≤ c ijk g,4 |h 1 ||h 2 ||h 3 | for any u ∈ V, and these bounds satisfy To see for instance that DG exists, note that by Taylor expansion where u, h ∈ V and η ∈ [0, 1]. In particular, we also note that This condition ensures the Lipschitz continuity of G as 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 G in Hypothesis (H) hold.
The developed theory can be also applied to other examples of G, like kernel integrals, see [11].

Appendix
In the appendix we prove some technical estimates related to w = (u ⊗ (ω ⊗ S ω)).
Substituting the above expression in (32), after applying fractional integration (10) to the last two terms, we have to calculate Secondly, similar to (13) we obtain that , and substituting the last two expressions into (32) we obtain the conclusion.
In the following we want to prove that w is well-defined for U ∈Ŵ 0,T . To this end we will consider a mapping w(t, s, q) = w(U, ω, (ω ⊗ S ω))(t, s, q) which coincides for smooth ω with the expression introduced in Lemma 10. In order to prove the regularity of w stated in Lemma 20 we need several properties that we collect and prove in the following result. Proof. Consider the separable Hilbert-space L 2 (V ⊗ V,V ) equipped with the complete orthonormal basis (Ẽ ijk ) i,j,k∈N given bỹ E ijk (e l ⊗ V e m ) =Ẽ ijk (e l , e m ) = 0 : j = l or k = m f i : j = l and k = m.
We remind that (e i ) i∈N and (f i ) i∈N are, respectively, complete orthonormal basis of V andV . Then for v ∈ V ⊗ V , where v jk is the mode of v with respect to e j ⊗ V e k . This completes (i). The second statement can be proven similarly and therefore we omit its proof. In what follows we abbreviate the notation in the following way: let us denote Proof of Lemma 20.
Proof of Lemma 21.
Similar we obtain an estimate for the first expression on the right side of (34) setting θ = t. Finally, note that the appearing constant c depends on |||ω||| β ′′ and (ω ⊗ S ω) 2β ′ .
Corollary 29. The proof of (9) follows the same steps than the last proof, with the difference that we do not have to use (23) but the Chen-equality (6).