Stochastic evolution equations with rough boundary noise

We investigate the pathwise well-posedness of stochastic partial differential equations perturbed by multiplicative Neumann boundary noise, such as fractional Brownian motion for H∈(1/3,1/2].\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H\in (1/3,1/2].$$\end{document} Combining functional analytic tools with the controlled rough path approach, we establish global existence of solutions and flows for such equations. For Dirichlet boundary noise we obtain similar results for smoother noise, i.e. in the Young regime.

where O ⊂ R d is a bounded domain with C ∞ -boundary, X is a γ-Hölder rough path with γ ∈ ( 1 3 , 1 2 ], A is a second order differential operator in divergence form with Neumann boundary conditions C as specified in (5.1), y 0 is the initial data and f and F are nonlinear terms.Consequently, the theory developed in this article is applicable to the fractional Brownian motion with Hurst parameter H ∈ ( 1 3 , 1  2 ].For H = 1 2 we recover the results for the Brownian motion in [35].Setting for now the drift term f = 0, we note that the solution of (1.1) should be given by where A will be the L p -realization of A which generates an analytic semigroup (S t ) t∈[0,∞) on a suitable scale of Banach spaces.Furthermore, N is the Neumann operator, which maps the boundary data into the interior of the domain.The derivation of (1.2) was established in [14,32] for additive noise.One of the main goals of this work is to give a meaning to the convolution (1.2) using a controlled rough path approach and construct pathwise solutions for (1.1).Our approach is applicable only to Neumann boundary noise for γ ∈ ( 1 3 , 1  2 ].The Dirichlet case can be included in the Young regime, provided that the Hölder regularity of the noise satisfies an additional condition.This fact is not surprising, since it is well-known that mild solutions for stochastic evolution equations with Brownian Dirichlet boundary noise generally fail to exist [14,35].Using different techniques and tools from Malliavin calculus, we mention that Dirichlet boundary noise can be treated for a Brownian motion, see [3].However, one can construct solutions for Dirichlet boundary noise, if the random input is given by a fractional Brownian motion where the Hurst index H > 3  4 [17].Furthermore, one can show well-posedness of (1.1) driven by an additive fractional Brownian motion if H ∈ ( 1 4 , 1) as obtained in [17,Section 5] and [18].We recover these thresholds for the Hurst parameters in the more general framework of rough path theory, which allows us to consider multiplicative noise in contrast to [13,15,17,18,32].Results on stochastic evolution equations of the type (1.1) with multiplicative noise can be looked up in [31,35].Here, the random input X is an infinite-dimensional Brownian motion.To our best knowledge, for fractional Brownian motion, only the additive case was considered in [13,17,18].Results for SPDEs with boundary Lévy noise have been obtained in [9,26].On the other hand, there has been a growing interest in developing a solution theory based on a semigroup approach for rough evolution equations starting with [24,25] and the more recent approaches [20,22,23,27].However, most of the results are stated on a torus and seem to be only applicable to rough PDEs with zero Dirichlet or Neumann boundary conditions.To our best knowledge, there are no works that deal with more complicated boundary conditions (nonlinear or dynamic) or with boundary noise using rough paths techniques.Here we contribute to this aspect and provide a solution theory for (1.1) with Neumann boundary noise.Our approach relies on controlled rough paths on a monotone scale of interpolation spaces introduced in [23].One major difficulty in our setting is that we have to deal with controlled rough paths belonging to two different scales of Banach spaces: one for the solution and one for the boundary data.However, the main technical challenge is to obtain enough spatial regularity in order to guarantee that (1.2) belongs to the domain of A and to investigate its Gubinelli derivative.It is well-known that [22,23,25,27] there is a trade-off between space and time regularity required in order to define the standard convolution t 0 S t−r F (y r ) dX r using pathwise arguments.In particular the nonlinear term F has to improve the spatial regularity as in [27,21] or is allowed to lose spatial regularity which is strictly less than the time regularity of the noise [22,23].In particular for the Brownian motion, which is 1/2− Hölder regular, this means that it is not yet possible to deal with transport-type noise in the mild formulation [22,23] as opposed to the Itô calculus.However, there are numerous advantages to consider a rough path formulation for (1.1) such as a continuous dependence of the solution with respect to the random input and the existence of random dynamical systems, which we discuss in Section 4. In our setting, we have to incorporate the Neumann operator in the rough convolution, compute a Gubinelli derivative and make sure that (1.2) belongs to the domain of A. We can deal with these issues by introducing a suitable extrapolation operator which entails a convenient representation of the mild solution.
The idea of working with extrapolation operators in the context of rough path theory is new, and the results obtained are of independent interest.In the works [17,18] due to the presence of an additive fractional noise on the boundary, the application of extrapolation operators is not required.In order to deal with multiplicative Brownian noise on the boundary, extrapolation operators have been introduced in [35].Similar to [35], the main idea is to rewrite (1.1) as a semilinear evolution equation without boundary noise using extrapolation operators.The novel aspect of this work is to analyze the interaction between extrapolation operators, controlled rough paths and rough integrals which is necessary in order to investigate the well-poosedness of (1.1).Finally, we mention that the motivation of incorporating boundary noise arises in the study of transport models under random sources [10,36].In this framework, the recent work [6] considers a variant of the 3D Navier-Stokes equations subject to stochastic wind driven boundary conditions which are modelled by an additive cylindrical Brownian motion.Furthermore, beyond the well-posedness theory mentioned above, optimal control results for SPDEs with boundary noise have been obtained in [7,15], whereas numerical aspects have been discussed in [8].Averaging principles and fluctuations around the averaged equation for such equations have been derived in [12].This work is structured as follows.In Section 2 we collect fundamental results from the controlled rough path approach according to a monotone interpolation family as introduced in [23].In our case, we have to work with two different scales of interpolation spaces in order to deal with the boundary data.More precisely, the boundary data belongs to the Besov scale, whereas the solution is expected to belong to the Bessel potential scale according to the boundary conditions Cu = 0. We specify these function spaces in Section 2 and further provide a background on extrapolation operators.Section 3 contains the main results of this work.We analyse the interplay between the extrapolation operators and controlled rough paths.Based on this, we construct pathwise (local-and global-in-time) solutions for (1.1) in Theorem 3.16 and Theorem 3.20.Moreover, we also establish in Theorem 3.24 a well-posedness result for Dirichlet boundary noise if H ∈ ( 3 4 , 1) using Young's integral.Section 4 contains a direct application of our global well-posedness result.Namely, we establish in Theorem 4.2 the existence of random dynamical systems associated to (1.1).Its long-time behavior will be addressed in a future work.For example, it is known that white noise on the boundary can have a stabilization effect [19,33].However, such results are not available for fractional noise in the context of the rough path approach.Finally, we conclude with some applications of our theory in Section 5.
We first specify the type of noise we consider.The random input is a d-dimensional γ-Hölder rough path X := (X, X), for γ ∈ (1/3, 1/2] and X 0 = 0.Here we assume without loss of generality that d = 1, since the generalization to d > 1 can be made componentwise.More precisely, we have and the connection between X and X is given by Chen's relation where we write X u,s := X u − X s for any path.The term X is sometimes referred to in the literature as a second order process.We further introduce an appropriate distance between two γ-Hölder rough paths. Definition 2.1 Let J ⊂ R be a compact interval, ∆ J := {(s, t) ∈ J × J : s ≤ t} and X = (X, X) and X = ( X, X) be two γ-Hölder rough paths.We introduce the γ-Hölder rough path (inhomogeneous) metric We set ρ γ (X) := d γ,[0,T ] (X, 0).
We specify the necessary assumptions on the linear part of (1.1).For concrete examples of operators satisfying such properties, see Section 5. Let A be the L p (O)-realization of A with respect to the Neumann boundary conditions.Therefore, its domain is given by D(A) := {u ∈ H 2,p (O) : Cu = 0}.
Throughout this manuscript we make the following assumptions.
Assumptions 2.2 1) The boundary value problem (A, C) is normally elliptic.
The main advantage of this approach is that we can view the semigroup (S t ) t∈[0,∞) generated by A as a linear mapping between these interpolation spaces and obtain the following standard bounds for the corresponding operator norms.If S : [0, T ] → L(B α , B α+1 ) is such that for every x ∈ B α+1 and t ∈ (0, T ] we have that Now we introduce the notion of Banach scales and within extrapolated spaces and operators. Definition 2.5 ([2, Section V.1.1])Let J be an index set such that for any α ∈ J, α + 1 ∈ J.
• For every α 1 > α 2 we have the equality If these embeddings are dense, we call the Banach scale densely injected.
Remark 2.6 As a direct consequence of this definition, we know that for Furthermore, if the scale is densely injected, all operators are completely determined by A 0 .Based on this, it is possible to construct the scale out of a single operator, [2, Remark 1.1.2].One of the main advantages we get from this fact is that A α 1 ⊂ A α 2 holds for all α 2 < α 1 .Therefore, for every x ∈ B 1+α 1 we have the equality We frequently use this property throughout this manuscript.
In our case, we consider an operator A =: A 0 satisfying Assumption 2.2 and set B 0 := L p (O), B 1 := D(A).From this starting point, we build a Banach scale, which is uniquely determined by A. For α > 0, we use the fractional powers of A as follows.We define for α > 0 and λ in the resolvent of A, the spaces and let A α be the B α -realization of A 0 .Furthermore λ = 0 can be assumed without loss of generality by shifting the operator A.
It can be shown that To extend this to negative indices, we use the theory of extrapolation spaces.The idea behind this concept comes from the fact, that B 0 can be reconstructed from B 1 .To see this, note that for . This now motivates the definition of extrapolation spaces as a super space of B 0 similarly constructed.We define B −1 as the completion of B 0 with respect to the norm The operators with negative indices are also called extrapolated operators.

Remark 2.7
The same procedure can be done iteratively to extend the scale up to the index set [−m, ∞) for an arbitrary m ∈ N. Note that in every new step, we have to replace the previously constructed extrapolated spaces B −m+1 by the isomorphic image j −m+2 −m+1 (B −m+2 ) in order to ensure the validity of the dense embeddings of the form above forms a densely injected Banach scale in the sense of Definition 2.5.Furthermore, for −m ≤ α 1 < α 2 < ∞ and ϑ ∈ [0, 1] we have the reiteration property Based on this result, we conclude that the space part of the scale introduced above forms a family of monotone interpolation families as specified in Definition 2.3.We will refer to it as the interpolation-extrapolation scale generated by A. Moreover, we are interested in semigroups generated by the operators in a Banach scale.In this case, a similar statement to Remark 2.6 i) holds true.Keeping this in mind, we now introduce the following definition of a controlled rough path tailored to the parabolic structure of the PDE we consider, in the spirit of [23].This is convenient for our aims, since the semigroup will not be incorporated in the definition of the controlled rough path as in [22] or alternative approaches [25,27] which iterate the stochastic convolution into itself.
Definition 2.10 (Controlled rough path according to a monotone family (B α ) α∈R ).We call a pair (y, y ′ ) a controlled rough path for a fixed α ∈ R if to as Gubinelli derivative of y.
• the remainder ) The space of controlled rough paths is denoted by D 2γ X,α and endowed with the norm • X,2γ,α given by [23] Note that for paths h : [0, T ] → B and second order processes g : If we deal with different scales, we write the full subscript.In order to emphasize the time horizon, we write D 2γ X,α ([0, T ]) instead of D 2γ X,α .Furthermore, when the time interval is clear from the context, we use the abbreviation C γ ′ (B α ′ ) for suitable γ ′ and α ′ to point out the interplay between space and time regularity.
Remark 2.11 Note that we do not make the Hölder continuity of y part of the definition of a controlled rough path, since using (2.7) one immediately obtains for θ ∈ {γ, 2γ} that (2.9) Given a controlled rough path, one can introduce the rough integral as follows [23,Theorem 4.5].
where P denotes a partition of [s, t] and the limit exists as an element in B α−2γ .For 0 ≤ β < 3γ the following estimate holds true.
We emphasize that the stochastic convolution increases the spatial regularity of the controlled rough path, see [23,Corollary 4.6] and [29,Lemma 3.5].We recall this result, which will be used later on.

Main results
We recall that γ ∈ ( 1 3 , 1 2 ] indicates the time regularity of the γ-Hölder rough path X := (X, X).The main goal of this section is to prove that (1.2) is well-defined in the space of controlled rough paths.Since the boundary data of (1.1) will belong to some Besov space, whereas the solution is expected to belong to a Bessel potential scale, see (3.3), we first fix two abstract scales of Banach spaces (B α ) α∈R and ( B α ) α∈R .Furthermore, we denote the corresponding space of controlled rough paths by For simplicity, we fix the time horizon T ≤ 1 throughout this section.The first step is to define (1.2) using Theorem 2.12.Therefore, we begin this section by examining how a controlled rough path changes under the influence of the Neumann operator.
As a direct consequence of the assumption on N we see . Furthermore, we define the remainder by R N y := N R y and get for ϑ ∈ {γ, 2γ} Regarding this, we can define the rough convolution based on Theorem 2.12 as follows.
Corollary 3.2 Let (B α ) α∈R , ( B α ) α∈R and N as in Lemma 3.1 and (y, y ′ ) ∈ D 2γ X,α 1 .Then the rough convolution exists as an element of B α 2 −2γ for every t ∈ [0, T ], where P denotes a partition of [s, t].Furthermore, we have for 0 ≤ s < t ≤ T and 0 ≤ β < 3γ the estimate where R N y t,s := t s S t−r N y r dX r −S t−s N (y s X t,s − y ′ s X t,s ) is the integral remainder.Consequently, (I, N y) ∈ D 2γ X,α 2 +θ holds for every θ ∈ [0, γ).Proof.
In order to make sense of (1.2) in the space of controlled rough paths, we need to make sure that I t ∈ D(A) for every t ∈ [0, T ] and to find a suitable Gubinelli derivative for AI.Therefore, we first specify the scales of Banach spaces which are required in our framework.We let from now on p ∈ [2, 3] and 2 > α > 1 + 1 p .In this case we define the spaces for −1 + 1 p < β ≤ 2, see for example [1,Theorem 7.1].Recalling now that the Neumann operator N is the solution operator of (2.3), we obtain for all 0 < ε < 1  2 Proof.According to Lemma 3.1 and Corollary 3.2 we get for every θ ∈ [0, γ) that (I, N y) ∈ D 2γ X,ε+θ .We take θ := 1 3 + δ < γ with δ > 0 small enough.Then we can choose ε := 1 2 + 1 2p − δ, so that we obtain ε + θ = 5  6 + 1 2p ≥ 1 since p ≤ 3. Therefore, we conclude that 1) The essential step in the proof of Corollary 3.3 is that we can choose an ε such that ε > 1 − γ.Here we recall that γ stands for the regularity of the noise.In the case of Neumann conditions, we have seen that this is possible.But for Dirichlet boundary conditions, we can choose ε only up to 1 2p .This comes from the fact that the Dirichletoperator D is bounded from B β− 1 p p,p (∂O) to H β,p (O) and provides a strong solution for β > 1 p .Since p ≥ 2 and γ < 1 2 , it is not possible to find an ε such that ε > 1 − γ.
2) In the Young regime, i.e. γ ∈ ( 1 2 , 1), we can incorporate Dirichlet boundary noise since the conditions ε > 1 − γ and ε < 1 2p can simultaneously be fulfilled.For additive fractional noise, it is known that Dirichlet boundary conditions can be incorporated provided that H ∈ ( 3 4 , 1) as established in [17].We provide further details on the well-posedness of (1.1) with multiplicative Dirichlet boundary noise in Theorem 3.24 and an example in Section 5.
Remark 3.5 Since N y is a Gubinelli derivative for I, it would make sense to consider AN y as one for AI.However, regarding the definition of N , N y does not belong to D(A).Due to this reason, we need an extension of A, which is given by the extrapolated operator introduced in Section 2. In fact, A −η is the weakest possible extrapolation operator such that A −η N y is well-defined.
Theorem 3.6 For every (y, y ′ ) ∈ D 2γ X,α we have (AI, We recall that A −η can be viewed as the B −η -realization of A −η−2γ , see Remark 2.6.Then for x ∈ B 1−η , we have the equality A −η−2γ x = A −η x.Now we let 0 ≤ s < t ≤ T and obtain, using additionally the fact that where we used Remark 2.11 in the last inequality.This means that z ′ ∈ C γ (B −η−2γ ).The tricky part is to estimate the remainder R z t,s := z t,s − z ′ s X t,s .For this reason, we rewrite R z , so that we can use the estimate (3.2) derived for the integral remainder R N y : With this representation we get for ϑ ∈ {γ, 2γ} , so we can estimate the individual terms separately.Applying Corollary 3.2 with β := 2γ − ϑ entails where we note that due to Corollary 3.3 it holds that R N y t,s ∈ D(A) = B 1 .To deal with the second term, we use Combining Remark 2.6 with the smoothing property (2.5) to get the estimate In order to estimate I 4 we first apply (2.5) to obtain where we used again Remark 2.6 for A(S t−s − Id)I s = A −η−ϑ (S t−s − Id)I s .Now we can use a similar decomposition as in (3.4) for I s .This leads, together with (2.6), (3.2) and the fact that S t ∈ L(B ε ) to Putting all the previous estimates together, we conclude that ) which completes the proof.
Even if Theorem 3.6 is an interesting result on its own, the statement is not enough for our purposes due to the presence of the operator A in front of the rough integral.In particular, it is not possible to show that (1.1) has a global solution working with the controlled rough path (AI, A −η N y) ∈ D 2γ X,−η and using the techniques in [29], even though we could establish a local solution using a fixed-point argument.Therefore, we further show that we can plug the operator A in the rough integral, see [32] for an analogous result for additive fractional noise.As a consequence of Corollary 3.2, the limit on the right-hand side of (3.1), exists in B ε−2γ .So the equality holds for a bounded, and therefore continuous, operator A with domain B ε−2γ .However, in our case, we only have A ∈ L(B 1 , B 0 ) and ε − 2γ < 1.Nevertheless, we can show the following statement.
Lemma 3.7 Under the assumptions of Corollary 3.2, the limit Proof.
For the sake of completeness, we indicate a sketch of the proof of this statement based on a classical sewing lemma, see [23,Theorem 4.1] and [22,Theorem 2.4].Let P n := {t i = s + 2 −n i(t − s) : i = 0, . . ., 2 n } be the n-th dyadic partition of the interval [s, t], and the sum associated to the partition P n .To prove now that (I P n t,s ) n∈N is a Cauchy sequence, set m = v−u 2 n+1 the midpoint of an interval [u, v].Then we can write With this representation one can show that for a δ ∈ (β − 1, 3γ − 1), using similar ideas as in [23,Theorem 4.1,4.5].The only difference is the appearance of the operator N ∈ 2 i=0 L( B α 1 −iγ , B α 2 −iγ ), which is bounded and therefore only changes the space we end up with.For instance, we consider the first part of the sum.With the help of Chen's relation it can be shown that and therefore, with the regularity property (2.6) and the Hölder conditions of the controlled rough path (N y, N y ′ ) ∈ D 2γ X,α 2 , one gets The second term can be treated analogously which means that (3.6) holds for a constant C ξ which depends on the Hölder norms of ξ and on the semigroup.Furthermore, since the right-hand side of (3.6) is summable over n, the sequence (I P n t,s ) n∈N is Cauchy in B α 2 −2γ+β and therefore has a limit I t,s ∈ B α 2 −2γ+β .Since B α 2 −2γ+β ֒→ B α 2 −2γ , and the limit in Corollary 3.2 is unique, we get that I t,s = I t,s .In conclusion, the limit exists in the B α 2 −2γ+β topology.Now, going back to the situation in (3.5), we choose β := 2γ − ε + 1 < 3γ, due to our restriction on ε.Since A satisfies Assumption 2.2, Theorem 2.9 ensures that every extrapolated operator generates again an analytic semigroup.Together with Theorem 2.8, Remark 2.6 and the fact that N y ∈ B ε = B 1−η , this leads to (3.7) Remark 3.8 To make sure that the right-hand side is well-defined as a controlled rough integral, we need to find a Gubinelli derivative for A −η N y.A natural choice would be A −η N y ′ , but since y ′ loses spatial regularity, this is not well-defined.Therefore, in order to choose an appropriate Gubinelli derivative for A −η N y, the extrapolated operator A −η has to be lifted.Due to this reason, one can show that (A −η N y, A −σ N y ′ ) ∈ D 2γ X,−η holds with σ := η + γ.In order to avoid working with two different indices for the extrapolation operator in the path component and its Gubinelli derivative, we rely on Remark 2.6.Therefore we have This enables us to formulate the next result.Lemma 3.9 For every (y, y ′ ) ∈ D 2γ X,α we have Note that we cannot use Lemma 3.1, since A −σ is not defined for every element in B ε−2γ .So we have to take advantage of the fact that This leads to z To investigate the remainder R z t,s := A −σ R N y t,s ∈ B 1−σ , we let ϑ ∈ {γ, 2γ} and establish using again Remark 2.6.Regarding the previous deliberations, this computation concludes the proof.
Consequently this allows us to define the rough convolution.
Lemma 3.10 The right-hand side of (3.7) is well-defined as a rough convolution with the controlled rough path Further, since we want to solve equations with multiplicative noise, the next step is to consider the composition of a controlled rough path with a smooth function.In our setting, in contrast to [23], the nonlinearity is allowed to map between different scales of Banach spaces.
holds.ii) Assume additionally that F is three times Fréchet differentiable with bounded third derivative, and let ( z, z ′ ) be the composition of another controlled rough path ( y, y ′ ) ∈ D 2γ X,β with F .Then Where we set Proof.The proof is similar to [23,Lemma 4.7].We point out the main differences that occur in our case.We can view for t ∈ [0, T ] the derivative D k F (y t ) as an element of L(B ⊗k β−ϑ , B β−ϑ+δ ) for k = 1, 2, 3 and ϑ ∈ {γ, 2γ}.Since (B β ) β∈R and ( B β ) β∈R are both Banach scales (recall Definition 2.3), all the necessary estimates remain valid.For instance, one can estimate the Gubinelli derivative z ′ = DF (y) • y ′ as where we use (2.9).The estimates of the remainder follow by analogue computations.
Returning to (1.1) and regarding that according to (3.7) and Lemma 3.10 it holds and we can now rewrite (1.1) as a semilinear evolution equation without boundary noise We recall that A −σ is the extrapolation operator introduced in Section 2, Remark 3.12 The idea to rewrite (1.1) as a semilinear problem without boundary noise as in (3.10) using an extrapolation operator was also applied in [35].There the extrapolated operator A −1 was used.We note that the index of the extrapolation operator needed there is α 2 − 1 where α ∈ 1, 1 + 1 p .Therefore, our result is consistent with the one in [35] for Brownian noise, since in both cases the index satisfies We work here with the extrapolation operator A −η−γ , as pointed out in Remark 3.8 because this seems to fit well in the rough path framework.
We give now the main assumptions on the nonlinear drift and diffusion coefficients of (1.1), that will guarantee the local-as well as the global-in-time existence of solutions based on the results in [23,29].
3) Let 1) and 2) be satisfied.Assume additionally that f : satisfies a linear growth condition and that the derivative of The assumption on the diffusion coefficient ensures that is also well-defined.However, it is not true that G : B −η−2γ → B −η−2γ , since the extrapolated operator A −σ is no longer defined on this space.Nevertheless, we can deal with this technical issue, recall Lemma 3.9 for a similar situation.
i) Due to the rough path techniques, the assumptions on the diffusion coefficient F are more restrictive than in [35], where F maps into B −1 , is Lipschitz continuous and satisfies a linear growth condition.Moreover, due to the presence of the Neumann operator, F is supposed to improve the spatial regularity, in order to define the rough convolution as in (2.10).Such issues are common for rough convolutions and have been also encountered in [27,21].
For the second remainder term, note that due to the boundedness of D [DF (•) • G(•)], we get the Lipschitz type estimate Collecting all the estimates proves the statement.ξ t,s := y s X t,s + y ′ s X t,s and ξ t,s := y s X t,s + y ′ s X t,s are the approximations of the individual integrals z and z.This reads as Combining now (3.14) with the proofs of Lemma 3.7 and Corollary 3.2 we derive for i = 0, 1, 2. To get now (3.13) we have to estimate the individual terms of the distance (3.12) similar to Theorem 3.6, see also [22,Lemma 3.13].For example, we have The remaining terms of (3.12) can be handled analogously.
ii) The statement can be obtained following the steps of the proof of Lemma 3.11, see also [22,Lemma 3.14].
Based on this result we can establish the continuous dependence of the solution with respect to the noise and initial data.
The Young case.For the sake of completeness, we now consider Dirichlet boundary noise in the Young regime, i.e. if the random input X ∈ C γ (R) for γ ∈ ( 1 2 , 1).We denote, as in Remark Then, as justified in Remark 3. p .In this case the theory of the interpolation-extrapolation scale in [1] breaks down.To overcome this issue, we additionally assume that β Furthermore, just as in the Neumann case, the condition γ > 1 − ε D also needs to be satisfied.In the Neumann case, this condition automatically holds for rough noise, i.e. γ ∈ ( 1 3 , 1  2 ].For Dirichlet boundary noise in the Young regime, this leads to an additional restriction on the regularity of the noise, compare [17].Therefore we choose γ ∈ (1 − 1 2p , 1).Under these assumptions, we show that it is possible to incorporate Dirichlet boundary noise in (1.1).This SPDE can be now rewritten as where the integral is understood in the Young sense (2.13).obtain a random dynamical system from an SPDE, since its solution is defined almost surely, which contradicts the cocycle property.In particular, this means that there are exceptional sets which depend on the initial condition and it is not clear how to define a random dynamical system if more than countably many exceptional sets occur.This issue does not occur in a pathwise approach.Provided that global existence of solutions is ensured, rough path driven equations generate random dynamical systems if the driving rough path forms a rough path cocycle, as established in [5].
According to [5, Section 2] rough path lifts of various stochastic processes define cocycles.These include Gaussian processes with stationary increments under certain assumption on the covariance function [20,Chapter 10] and particularly apply to the fractional Brownian motion with Hurst index H > 1 4 .
Based on Theorem 3.20 we immediately derive the existence of a random dynamical system associated to (1.1).Using a classical flow transformation, such a statement together with the existence of a random attractor was obtained for a system of SPDEs with dynamical boundary conditions in [10].precisely, one considers a sequence of (classical) solutions (y n , (y n ) ′ ) n∈N of (1.1) corresponding to smooth approximations (X n , X n ) n∈N of (X, X).Obviously, the mapping (t, X, ξ) → y n t is (B([0, T ]) ⊗ F ⊗ B(B −η ), B(B −η ))-measurable for any T > 0. Since y continuously depends on the rough input X = (X, X), according to [22,Lemma 3.12], one concludes that lim n→∞ y n t = y t .This gives the measurability of y with respect to F ⊗ B(B −η ).Due to the time-continuity of y, we obtain by [11,Chapter 3] the (B([0, T ]) ⊗ F ⊗ B(B −η ), B(B −η ))-measurability of the mapping (t, ω, ξ) → y t for any t ≥ 0.
where the coefficients a ij , b : O → R are smooth, (a ij ) d i,j=1 is symmetric and uniform elliptic, meaning that there exists some constant k > 0 such that for all ξ ∈ R d and x ∈ O we have

1 Introduction
We investigate the semilinear parabolic evolution equation with nonlinear rough boundary noise given by Ay + f (y) in O, Cy = F (y) d dt X on ∂O, y(0) = y 0 .(1.1)

Theorem 2 . 9 ([ 2 ,
Proposition V.1.5.5], [2, Theorem V.2.1.3])Let A satisfy the Assumption (2.2) and consider the interpolation-extrapolation scale generated by A. Then for every α ∈ [−m, ∞) the operator A α has bounded imaginary powers and generates an analytic semigroup.Further, for α > β we have that the semigroup generated by A α is the same as the semigroup generated by A β restricted to B α .Further details on the theory of Banach scales or extrapolation can be found in [1, Section 6-7] and [2, Chapter V].

Theorem 3 . 16 ( 0 S 0 S
Existence of a local-in-time solution for (3.10))Assume that F and f satisfy Assumption 3.13 1) and 2).Then there exists for every initial condition y 0 ∈ B −η a time T * ≤ T and a unique solution (y,A −σ N F (y)) ∈ D 2γ X,−η ([0, T * )) to (1.1) such that y t = S t y 0 + t t−r f (y r ) dr + t t−r A −σ N F (y r ) dX r , for all t < T * .(3.11)

3 . 4 ,
by D the solution operator of (2.3) with C = γ ∂ .In this case, the domain D(A) is different now, which means that the extrapolation-interpolation scale according to A changes.To point that out, we denote the extrapolation spaces by B D β and B D β := B β− 1 p p,p (∂O).In this case the spaces B D β are given by ) where η D := 1 − ε D and σ D := η D − γ.Assumptions 3.23 (Young case) There exists δ 2 > η D + 1 + 1 p such that for any ϑ ∈ {0, γ} the diffusion term F : B D −η D −ϑ → B D −η D −ϑ+δ 2 is two times continuously Fréchet differentiable with bounded derivatives.Based on the arguments of Theorem 3.20 we derive.Theorem 3.24 (Dirichlet boundary noise in the Young case) Let X ∈ C γ (R) with γ ∈ (1− 1 2p , 1).Assume that f and F satisfy Assumption 3.13 3) replacing 3.13 2) with 3.23.Then there exists for every initial condition y 0 ∈ B D −η D a unique mild solution y

Remark 4 . 4
Naturally, based on the statement of Theorem 3.24, we obtain a random dynamical system in the Young case for the SPDE (1.1) with multiplicative Dirichlet boundary noise.5 ExamplesHere we provide an application of our theory, specifying concrete examples for A and F .Since the condition on the drift term f is less restrictive, examples such as polynomial nonlinearities or Nemytskii type operators are possible.Therefore we focus here on examples for the diffusion coefficient F .In both examples, we consider p = 2 and the formal operators are augmented by either Neumann or Dirichlet boundary conditions Au := d i,j=1 ∂ i (a ij ∂ j ) u + bu, Cu := d i,j=1
4, the Dirichlet operator D is bounded from B D β to B D ε D with ε D < 1 2p .Furthermore, the boundary value problem (2.3) (with C replaced by C) has a strong solution for β > 1 p .However, for the definition of Young's integral (2.13) we need to consider paths which are continuous in B D β and γ-Hölder continuous with values in B D β− γ .This means that the index β − γ can become negative if we only assume that β > 1