text:
\section{Introduction}
        We investigate the semilinear parabolic evolution equation with nonlinear rough boundary noise given by
        \begin{align}\label{eq:intro}
            \begin{cases}
                \frac{\partial}{\partial t} y = \mathcal{A} y +  f(y) & \text{ in } \mathcal{O},\\
                \mathcal{C} y = F(y)~\frac{{\textnormal{d}}}{{\textnormal{d}} t} \mathbf{X} & \text{ on } \partial \mathcal{O},\\
                y(0)=y_0.
            \end{cases}
        \end{align}
        where $\mathcal{O}\subset \mathbb{R}^d$ is a bounded domain with $C^\infty$-boundary, $\mathbf{X}$ is a $\gamma$-H\"older rough path with $\gamma\in(\frac{1}{3},\frac{1}{2}]$, $\mathcal{A}$ is a second order differential operator with a Neumann boundary operator $\mathcal{C}$, $y_0$ is the initial data and $f$ and $F$ are non-linear terms. Consequently, the theory developed in this article is applicable to the fractional Brownian motion with Hurst parameter $H\in(\frac{1}{3},\frac{1}{2}]$. For $H=\frac{1}{2}$ we recover the results for the Brownian motion in \cite{VeraarSchnaubelt}.
        Setting for the beginning the drift term $f=0$, we note that the solution of~\eqref{eq:intro} should be given by
        \begin{align}\label{eq:RoughConv}
            y_t = S_t y_0+ A\int_0^t S_{t-r} N(F(y_r))~{\textnormal{d}} \mathbf{X}_r,
        \end{align}
        where $A$ will be the $L^p$-realization of $\mathcal{A}$ which generates an analytic semigroup $(S_t)_{t\geq 0}$ on a suitable scale of Banach spaces. Furthermore, $N$ is the Neumann operator, which maps the boundary values inside the domain.
        The derivation of~\eqref{eq:RoughConv} was established in~\cite{daprato,m} for additive noise.
        One of the main goals of this work is to give a meaning to the convolution~\eqref{eq:RoughConv} using a controlled rough path approach and construct pathwise solutions for~\eqref{eq:intro}.
        Our approach is applicable only to Neumann boundary noise, the Dirichlet case is not included. This fact is not  surprising, since it is well-known that mild solutions for stochastic evolution equations with Brownian Dirichlet boundary noise fail to exist~\cite{daprato,VeraarSchnaubelt}.
        However, one can construct solutions for Dirichlet boundary noise, if the random input is given by a fractional Brownian motion where the Hurst index $H>\frac{3}{4}$~\cite{DuncanMaslowski}. Furthermore, one can show well-posedness of~\eqref{eq:intro} driven by an additive fractional Brownian motion if  $H\in(\frac{1}{4},1)$ as obtained in~\cite[Section 5]{DuncanMaslowski} and~\cite{DuncanMaslowski2}.
        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~\cite{DuncanMaslowski, DuncanMaslowski2,m,debussche}. Results on stochastic evolution equations of the type~\eqref{eq:intro} with multiplicative noise can be looked up in~\cite{Maslowski,VeraarSchnaubelt}. Here, the random input $\mathbf{X}$ is an infinite-dimensional Brownian motion. To our best knowledge, for fractional Brownian motion, only the additive case was considered in~\cite{DuncanMaslowski, DuncanMaslowski2}. Results for SPDEs with boundary L\'evy noise have been obtained in~\cite{BP,HP}.\\
        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~\cite{Gubinelli,GubinelliTindel} and the more recent approaches~\cite{GHairer,FrizHairer, GHN, HN19}. However, most of the results are stated on a torus and are 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~\eqref{eq:intro} with Neumann boundary noise. Our approach relies on controlled rough paths on a monotone scale of interpolation spaces introduced in~\cite{GHN}. The main difficulty in our setting is that we have to deal with controlled rough paths belonging to two different scales of Banach spaces.

        Another challenge is to obtain spatial regularity results in order to guarantee that~\eqref{eq:RoughConv} belongs to the domain of $A$ and to investigate its Gubinelli derivative. We can deal with these issues by introducing a suitable extrapolation operator in Section~\ref{main}. 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~\cite{DuncanMaslowski, DuncanMaslowski2} 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~\cite{VeraarSchnaubelt}. Similar to~\cite{VeraarSchnaubelt}, the main idea is to rewrite~\eqref{eq:intro} as a semilinear evolution equation without boundary noise using extrapolation operators and to analyze the interaction between these kinds of operators with a controlled rough path. \\
        Finally, we mention that the motivation of incorporating boundary noise arises in the study of transport models under random sources~\cite{Wang, BDS}.
        Furthermore, beyond the well-posedness theory mentioned above, optimal control results for SPDEs with boundary noise have been obtained in~\cite{debussche,bt}. Averaging principles and fluctuations around the averaged equation for such equations have been derived in~\cite{Cerrai}.\\

        This work is structured as follows. In section~\ref{preliminaries} we collect fundamental results from the controlled rough path approach according to a monotone interpolation family as introduced in~\cite{GHN}.
        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. We specify these function spaces in Section~\ref{preliminaries} and further provide a background on extrapolation operators. Section~\ref{main} contains the main results of this work. We analyze the interplay between the extrapolation operators and controlled rough paths. Based on this, we construct pathwise (local- and global-in-time) solutions for~\eqref{eq:intro} in Theorem~\ref{local} and \ref{thm:global}.
        Section~\ref{sec:rds} contains a direct application of our global well-posedness result. Namely, we establish in Theorem~\ref{rds} the existence of random dynamical systems associated to~\eqref{eq:intro}. 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~\cite{Stefanie, Munteanu}. However, such results are not available for fractional noise in the context of rough path approach.
        Finally, we conclude with some applications of our theory in Section~\ref{examples}. \\

        {\bf Acknowledgements.} We thank Robert Denk for very helpful discussions.

    \section{Preliminaries}\label{preliminaries}
        In this section we collect  basic results on rough paths and rough convolutions for semilinear parabolic problems~\cite{GHN} and
        provide the assumptions on the coefficients of~\eqref{eq:intro}, furthermore we will explain the concept of extrapolation spaces and operators.

        We first specify the type of noise we consider. The random input is a $d$-dimensional $\gamma$-H\"older rough path   $\textbf{X}:=(X,\mathbb{X})$, for $\gamma\in(1/3,1/2]$. Here we assume without loss of generality that $d=1$, since the generalization to $d>1$ can be made componentwise.
        More precisely, we have
        \begin{align*}
            X\in C^{\gamma}([0,T];\mathbb{R}) ~~\mbox{ and } ~~ \mathbb{X}\in C^{2\gamma}([0,T]^2;\mathbb{R}\otimes\mathbb{R})
        \end{align*}
        and the connection between $X$ and $\mathbb{X}$ is given by Chen's relation
        \begin{align}\label{chen}
            \mathbb{X}_{t,s}- \mathbb{X}_{u,s}-\mathbb{X}_{t,u}=X_{u,s}\otimes X_{t,u},
        \end{align}
        where $X_{u,s}:=X_u-X_s$.
        The term $\mathbb{X}$ is referred to as second-order process or L\`evy-area. We further introduce an appropriate distance between two $\gamma$-H\"older rough paths.
        \begin{definition}
            Let $J\subset\mathbb{R}$ be a compact interval, $\Delta_J:=\{ (s,t)\in J \times J : s\leq t \}$ and let $\mathbf{X}=(X,\mathbb{X})$ and $\mathbf{\tilde{X}}=(\tilde{X},\tilde{\mathbb{X}})$ be two $\gamma$-H\"older rough paths. We introduce the $\gamma$-H\"older rough path (inhomogeneous) metric
            \begin{align}\label{rp:metric}
                d_{\gamma,J}(\mathbf{X},\mathbf{\tilde{X}} )
                := \sup\limits_{(s,t)\in \Delta_J} \frac{|X_{t,s}-\tilde{X}_{t,s}|}{|t-s|^{\gamma}}
                + \sup\limits_{(s,t) \in \Delta_{J}}
                \frac{|\mathbb{X}_{t,s}-\tilde{\mathbb{X}}_{t,s}|} {|t-s|^{2\gamma}}.
            \end{align}
            We set $\rho_\gamma(\mathbf{X}):=d_{\gamma,[0,T]}(\mathbf{X},0)$.
        \end{definition}
        We specify the necessary assumptions on the linear part of~\eqref{eq:intro}. For concrete examples of operators satisfying such properties, see Section~\ref{examples}. Let $A$ be the $L^p(\mathcal{O})$-realization of $\mathcal{A}$ with respect to the Neumann boundary conditions. Therefore, its domain is given by $D(A):=\{u\in H^{2,p}(\mathcal{O}): \mathcal{C} u=0\}$.
        \begin{assumptions}\label{ass}
            \begin{itemize}
                \item [1)] The boundary value problem $(\mathcal{A},\mathcal{C})$ is normally elliptic.
                \item [2)] The operator $A$ generates an analytic semigroup $(S(t))_{t\in [0,\infty)}$. Furthermore, assume that $0\in \rho(A)$ and $A$ has bounded imaginary powers, that means there is a constant $C > 0$ such that
                \begin{align*}
                    \norm{A^{is}}_{\mathcal{L}(L^p(\mathcal{O}))} \leq C
                \end{align*}
            for all $s\in \mathbb{R}$ with $\abs{s}\leq 1$.
            \end{itemize}
        \end{assumptions}

        In order to deal with the boundary data of~\eqref{eq:intro} we further need to introduce the solution operator for an abstract boundary value problem. To this aim, we consider for $p\in [2,3]$ and $\alpha>1+\frac{1}{p}$ the normally elliptic boundary value problem
        \begin{align}\label{eq:bvp}
            \begin{cases}
            \mathcal{A} u = 0 & \text{ in } \mathcal{O},\\
            \mathcal{C} u = g & \text{ on } \partial \mathcal{O}.
            \end{cases}
        \end{align}
        It is well-known, see for example \cite[Theorem 9.2]{Amann}, that the problem \eqref{eq:bvp} has for $g\in B^{\alpha-1-\frac{1}{p}}_{p,p}(\partial \mathcal{O})$, a unique strong solution in $H^{\alpha,p}(\mathcal{O})$. The corresponding solution operator $$N:B^{\alpha-1-\frac{1}{p}}_{p,p}(\partial\mathcal{O})\to H^{\alpha,p}(\mathcal{O})$$ is also called the Neumann-operator.
        In particular, the Neumann-operator is bounded between these two function spaces. Hereby, we immediately see that we need two scales of Banach spaces (as specified in Definition~\ref{raum}) for the solution and boundary data in order to investigate~\eqref{eq:intro}. These are given by the Besov spaces $\mathcal{B}^{\alpha-1-\frac{1}{p}}_{p,p}(\partial\mathcal{O})$  and a Banach scale generated by the operator $A$, see Definition \ref{def:banachScale}.
        \begin{definition}\label{raum}
            A family of separable Banach spaces $(\mathcal{B}_\alpha,|\cdot|_\alpha)_{\alpha\in\mathbb{R}}$ is called a monotone family of interpolation spaces if for $\alpha_1\leq \alpha_2$, there is a dense and continuous embedding $\mathcal{B}_{\alpha_2}\hookrightarrow \mathcal{B}_{\alpha_1}$  and the following interpolation inequality holds for $\alpha_1\leq \alpha_2\leq \alpha_3$ and $x\in  \mathcal{B}_{\alpha_3}$:
            \begin{align}\label{interpolation:ineq}
                |x|^{\alpha_3-\alpha_1}_{\mathcal{B}_{\alpha_2}} \lesssim |x|^{\alpha_3-\alpha_2}_{\mathcal{B}_{\alpha_1}} |x|^{\alpha_2-\alpha_1}_{\mathcal{B}_{\alpha_3}}.
            \end{align}
        \end{definition}
        \begin{remark}
            For example, the scale $(B^\alpha_{p,p}(\partial \mathcal{O}))_{\alpha\in \mathbb{R}}$ satisfies Definition \ref{raum}. The interpolation inequality follows from the fact that
            \begin{align*}
                \left[B^{\alpha_1}_{p,p}(\partial \mathcal{O}),B^{\alpha_2}_{p,p}(\partial \mathcal{O})\right]_{\vartheta}=B^\alpha_{p,p}(\partial \mathcal{O})
            \end{align*}
            for $\alpha_1< \alpha_2$ and $\alpha=(1-\vartheta)\alpha_1+\vartheta \alpha_2$ for $\vartheta\in[0,1]$. Since complex interpolation is exact, the inequality \eqref{interpolation:ineq} is satisfied. A further example is given by the Bessel potential scale, which is introduced in Section~\ref{main}.
        \end{remark}
        The main advantage of this approach is that we can view the semigroup $(S_t)_{t\in [0,\infty)}$ generated by $A$ as a linear mapping between these interpolation spaces and obtain the following standard bounds for the corresponding operator norms. If $S:[0,T]\to \mathcal{L}(\mathcal{B}_{\alpha},\mathcal{B}_{ \alpha+1})$ is such that for every $x\in \mathcal{B}_{\alpha+1}$ and $t\in(0,T]$ we have that $|(S_t-\text{id})x|_{\mathcal{B}_{\alpha}} \lesssim t|x|_{\mathcal{B}_{\alpha+1}}$ and $|S_tx|_{\mathcal{B}_{\alpha+1}}\lesssim t^{-1} |x|_{\mathcal{B}_{\alpha}}$, then for every $\sigma\in[0,1]$ we have that $S_t\in\mathcal{L}(\mathcal{B}_{\alpha+\sigma})$ and
        \begin{align}
            |(S_t-\text{id}) x|_{\mathcal{B}_{\alpha}}&\lesssim t^\sigma |x|_{\mathcal{B}_{\alpha+\sigma}},\label{hg:1}\\
            |S_tx|_{\mathcal{B}_{\alpha+\sigma}}&\lesssim t^{-\sigma}|x|_{\mathcal{B}_{\alpha}}\label{hg:2}.
        \end{align}
        Now we want to introduce the notion of Banach scales and within extrapolated spaces and operators.
        \begin{definition}{\em (\cite[Section V.1.1]{Amann2})}\label{def:banachScale}
            Let $J$ be an index set, for example $J\in\{\mathbb{N},\mathbb{Z},\mathbb{R},[0,\infty) \}$. We call the couple $(E_\alpha,A_\alpha)_{\alpha\in J} $ a Banach scale, if
            \begin{itemize}
                \item $(E_\alpha,\abs{\cdot}_{E_\alpha})$ is a Banach space for every $\alpha\in J$ and $E_{\alpha_1}\hookrightarrow E_{\alpha_2}$ for $\alpha_1>\alpha_2$. We denote the embedding with $j_{\alpha_1}^{\alpha_2}$.
                \item $A_\alpha\in \mathcal{L} (E_{\alpha+1},E_\alpha)$ is an isomorphism for every $\alpha\in J$.
                \item For every $\alpha_1>\alpha_2$ we have the equality
                \begin{align*}
                j_{\alpha_1}^{\alpha_2}A_{\alpha_1}=A_{\alpha_2}j_{\alpha_1+1}^{\alpha_2+1}.
            \end{align*}
        \end{itemize}
        If these embeddings are dense, we call the Banach scale densely injected.
        \end{definition}
        \begin{remark}\label{rem:Extrapolation}
            \begin{itemize}
                \item[i)] As a direct consequence of this definition, we know that for $\alpha_1>\alpha_2$, $A_{\alpha_1}$ is the $E_{\alpha_1}$-realization of $A_{\alpha_2}$. 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, \cite[Remark 1.1.2]{Amann2}.
                \item[ii)] One of the main advantages we get from this fact is that $A_{\alpha_1}\subset A_{\alpha_2}$ holds for all $\alpha_2<\alpha_1$. Therefore, for $x\in \mathcal{B}_{1+\alpha_1}$ we have the equality $A_{\alpha_1}x = A_{\alpha_2} x$.
            \end{itemize}
        \end{remark}
        In our case, we consider an operator $A$ satisfying Assumption \ref{ass}, set $\mathcal{B}_0:=L^p(\mathcal{O}), \mathcal{B}_1:=D(A)$. From this starting point, we want to build a Banach scale. For $\alpha >0$, we use the fractional powers of $A$ as follows. Define for $\alpha\in [0,\infty)$ the spaces
        \begin{align*}
            \mathcal{B}_\alpha:=D(A^\alpha),
        \end{align*}
        equipped with the norm $\abs{\cdot}_{\mathcal{B}_\alpha}:=\norm{A^\alpha\cdot}_{L^p}$ and let $A_\alpha$ be the $\mathcal{B}_\alpha$-realization of $A_0$. It can be shown that $(\mathcal{B}_\alpha, A_\alpha)_{\alpha\in [0,\infty)}$ forms a (one-sided) densely injected Banach scale, see \cite[Theorem 1.2.4]{Amann2}. \\
        To extend this to negative values, we make use of the theory of extrapolation spaces. The idea behind this concept comes from the fact, that $\mathcal{B}_0$ can be reconstructed from $\mathcal{B}_1$. To see this, note that for $x\in \mathcal{B}_1$ we have $\abs{A_1^{-1} x }_{\mathcal{B}_1}=\abs{x}_{\mathcal{B}_0}$, then, $\mathcal{B}_0$ is the completion of $\mathcal{B}_1$ equipped with $\abs{A_1^{-1}\cdot}_{\mathcal{B}_1}$. This now motivates the definition of extrapolation spaces as a super space of $\mathcal{B}_0$ constructed similarly. Define $\mathcal{B}_{-1}$ as the completion of $\mathcal{B}_0$ with respect to the norm $\abs{A^{-1}\cdot}_{\mathcal{B}_0}$ and let $A_{-1}$ be the closure of $A$ in $\mathcal{B}_{-1}$. For $\alpha \in (0,1)$ define $\mathcal{B}_{-\alpha}:=\left[\mathcal{B}_{-1},\mathcal{B}_0\right]_{1-\alpha}$ and let $A_{-\alpha}$ be the $\mathcal{B}_{-\alpha}$-realization of $A_{-1}$. The operators with negative indices are  also called {\em extrapolated operators}.
        \begin{remark}
            The same procedure can be done iteratively to extend the scale up to the index set $[-m,\infty)$ for an arbitrary $m\in \mathbb{N}$. Note that in every new step, we have to replace the previously constructed extrapolated spaces $E_{-m+1}$ by the isomorphic image $j_{-m+1}^{-m+2}(E_{-m+2})$ in order to ensure the validity of the dense embeddings of the form $E_{-m+2}\hookrightarrow E_{-m+1}\hookrightarrow E_{-m}$, see \cite[Section V.1.3]{Amann2}.
        \end{remark}
        \begin{theorem}{\em(\cite[Theorem V.1.2.1, Theorem V.1.5.4]{Amann2})}\label{thm:EpolOperator}
            The scale $(A_\alpha,\mathcal{B}_\alpha)_{\alpha\in [-2,\infty)}$ constructed above forms a densely injected Banach scale in the sense of Definition \ref{def:banachScale}. Furthermore, for $-2\leq \alpha_1<\alpha_2<\infty$ and $\vartheta \in [0,1]$ we have the reiteration property
            \begin{align*}
                \left[\mathcal{B}_{\alpha_1},\mathcal{B}_{\alpha_2}\right]_\vartheta=\mathcal{B}_{(1-\vartheta)\alpha_1+\vartheta \alpha_2},
            \end{align*}
            where $[\cdot,\cdot]_\vartheta$ stands for  complex interpolation.
            Moreover, $(\mathcal{B}_\alpha)_{\alpha\in[-2,\infty)}$ satisfies the interpolation inequality \eqref{interpolation:ineq} and $(A_\alpha)^{\alpha-\beta}:\mathcal{B}_\alpha\to \mathcal{B}_\beta$ is an isomorphism for $-2\leq \alpha<\beta<\infty$.
        \end{theorem}
        Based on this result, we conclude that the scale introduced above forms a family of monotone interpolation families as specified in Definition \ref{raum}.
        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 \ref{rem:Extrapolation} i) holds true.
        \begin{theorem}{\em(\cite[Proposition V.1.5.5]{Amann2},~\cite[Theorem V.2.1.3]{Amann2})} \label{thm:EpolSemigroup}
            Let $A$ satisfies the Assumption \eqref{ass} and consider the interpolation-extrapolation scale generated by $A$. Then for every $\alpha\in[-2,\infty)$ the operator $A_\alpha$ has bounded imaginary powers and generates an analytic semigroup. Further, for $\alpha>\beta$ we have that the semigroup generated by $A_\alpha$ is the same as the semigroup generated by $A_\beta$ restricted to $E_\alpha$.
        \end{theorem}
        Further details on the theory of Banach scales or extrapolation can be found in \cite{Amann,Amann2}.\\

        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~\cite{GHN}.

        \begin{definition} {\em (Controlled rough path according to a monotone family $(\mathcal{B}_\alpha)_{\alpha\in \mathbb{R}}$).}\label{def:crp}
            We call a pair $(y,y')$ a controlled rough path for a fixed $\alpha\in \mathbb{R}$ if
            \begin{itemize}
                \item $(y,y')\in C([0,T];\mathcal{B}_\alpha) \times ((C[0,T];\mathcal{B}_{\alpha-\gamma} ) \cap C^{\gamma}([0,T];\mathcal{B}_{\alpha-2\gamma} ))$. The component $y'$ is referred to as Gubinelli derivative of $y$.
                \item the remainder  \begin{align}\label{remainder}
                R^y_{t,s}= y_{t,s} -y'_s X_{t,s}
            \end{align}
            belongs to $ C^{\gamma}([0,T];\mathcal{B}_{\alpha-\gamma})\cap C^{2\gamma}([0,T];\mathcal{B}_{\alpha-2\gamma})$.
            \end{itemize}
        \end{definition}

        The space of controlled rough paths is denoted by $\mathcal{D}^{2\gamma}_{X,\alpha}$ and endowed with the norm $\|\cdot\|_{X,2\gamma,\alpha}$ given by
        \begin{align}\label{g:norm}
            \gubnorm{y}{y'}:= \left\|y \right\|_{\infty,\mathcal{B}_\alpha}
            + \|y' \|_{\infty,\mathcal{B}_{\alpha-\gamma}}
            + \left[y'\right]_{\gamma,\mathcal{B}_{\alpha-2\gamma}}

            + \left[R^y \right]_{2\gamma,\mathcal{B}_{\alpha-2\gamma}},
        \end{align}
        where $\left[\cdot\right]_{\gamma,\alpha}$ denotes the H\"older seminorm.
        The first index in the notation above always indicates the time regularity and the second one stands for the space regularity. For simplicity, we write if a confusion is excluded $|y|_{\alpha}:=|y|_{\mathcal{B}_\alpha}$,  $\|y\|_{\infty,\mathcal{B}_\alpha}:=\|y\|_{\infty,\alpha}$ and $[y']_{\gamma,\mathcal{B}_{\alpha-2\gamma}}:=[y']_{\gamma,\alpha-2\gamma}$ and analogously for the remainder. If we deal with different scales, we still will write out the full subscript.
        In order to emphasize the time horizon, we write $\mathcal{D}^{2\gamma}_{X,\alpha}([0,T])$ instead of $\mathcal{D}^{2\gamma}_{X,\alpha}$.
        \begin{remark}
            Note that we do not make the H\"older continuity of $y$ part of the definition of a controlled rough path, since
            using~\eqref{remainder} one immediately obtains for $\theta \in \left\{\gamma,2\gamma \right\}$ that
            \begin{align}\label{est:hoelder:y}
            \left[y \right]_{\gamma,\alpha-\theta}
            \leq \left\|y' \right\|_{\infty,\alpha-\theta} \left[X\right]_{\gamma} + \left[R^y \right]_{\gamma,\alpha-\theta}.
            \end{align}
        \end{remark}

        Given a controlled rough path, one can introduce the rough integral as follows~\cite[Theorem 4.5]{GHN}.

        \begin{theorem}
            \label{integral} Let $(y,y')\in \mathcal{D}^{2\gamma}_{X,\alpha}$. Then
            \begin{align}\label{Gintegral}
                \int\limits_{s}^{t} S_{t-r}y_{r}~{\textnormal{d}} \textbf{X}_{r} :=\lim\limits_{|\mathcal{P}|\to 0} \sum\limits_{[u,v]\in\mathcal{P}} S_{t-u}y_{u}X_{v,u} + S_{t-u}y'_{u}\mathbb{X}_{v,u},
            \end{align}
            where $\mathcal{P}$ denotes a partition of $[s,t]$ and the limit exists as an element in $\mathcal{B}_{\alpha-2\gamma}$. For $0\leq \beta< 3\gamma$ the following estimate
            \begin{align}\label{estimate:integral}
                \abs{ \int\limits_{s}^{t} S_{t-r} y_{r}~{\textnormal{d}}\textbf{X}_{r} - S_{t-s}y_{s}X_{t,s} -S_{t-s}y'_{s}\mathbb{X}_{t,s}}_{\alpha-2\gamma+\beta} \lesssim \rho_\gamma(\mathbf{X}) \gubnorm{y}{y'} (t-s)^{3\gamma-\beta}
            \end{align}
            holds true.
        \end{theorem}
        We emphasize that the stochastic convolution increases the spatial regularity of the controlled rough path,  see~\cite[Corollary 4.6]{GHN} and Lemma 3.5 in~\cite{HN21}. We recall this result, which will be used later on.
        \begin{corollary}\label{cor:higherreg}
            Let $(y,y')\in\mathcal{D}^{2\gamma}_{X,\alpha}([0,T])$, where $T\in[0,1]$ and $0\leq \sigma<\gamma$. Then the integral map
            \begin{align*}
                (y,y')\mapsto (z,z') := \Big( \int_0^\cdot S_{\cdot-r}y_r~{\textnormal{d}} \mathbf{X}_r, y_\cdot \Big)
            \end{align*}
            maps $\mathcal{D}^{2\gamma}_{X,\alpha}([0,T])$ into itself. Moreover, we have
            \begin{align*}
                \gubnormpar{z}{z'}{\alpha+\sigma} \leq |y_0|_{\alpha} + |y'_0|_{\alpha-\gamma} + C T^{\gamma-\sigma} (1+\rho_\gamma(\mathbf{X})) \gubnorm{y}{y'}.
            \end{align*}
        \end{corollary}
        If the random input $X$ is more regular,~i.e. $\gamma\in(\frac{1}2,1)$, then~\eqref{Gintegral}
reduces to the {\em Young integral} which can be defined for a path $y\in\mathcal{C}:= C([0,T];\mathcal{B}_\alpha)\cap C^\gamma([0,T];\mathcal{B}_{\alpha-\gamma})$ as
\begin{align}\label{young}
\int_s^t S_{t-r} y_r~{\textnormal{d}} X_r =\lim\limits_{|\mathcal{P}|\to 0} \sum\limits_{[u,v]\in\mathcal{P}} S_{t-u} y_u X_{v,u},
\end{align}
whereas~\eqref{estimate:integral} reads as
\begin{align}\label{est:young}
\abs{ \int\limits_{s}^{t} S_{t-r} y_{r}~{\textnormal{d}} X_{r} - S_{t-s}y_{s}X_{t,s}} _{\alpha-\gamma+\beta} \lesssim \|y\|_{\mathcal{C}} [X]_\gamma  (t-s)^{2\gamma-\beta} .
\end{align}

    \section{Main results}\label{main}
        We recall that $\gamma\in (\frac{1}{3},\frac{1}{2}]$ indicates the time regularity of the $\gamma$-H\"older rough path $\mathbf{X}:=(X,\mathbb{X})$.
        The main goal of this section is to prove that~\eqref{eq:RoughConv} is well-defined in the space of controlled rough paths.
        Since
        the boundary data of~\eqref{eq:intro} will belong to some Besov space, whereas the solution is expected to belong to the Bessel potential scale, we first fix two abstract scales of Banach spaces $(\mathcal{B}_\alpha)_{\alpha\in\mathbb{R}}$ and $(\widetilde\mathcal{B}_\alpha)_{\alpha\in\mathbb{R}}$. Furthermore, we denote the corresponding space of controlled rough paths by $\mathcal{D}^{2\gamma}_{X,\alpha}:=\mathcal{D}^{2\gamma}_{X}(\mathcal{B}_\alpha)$ respectively $\widetilde\mathcal{D}^{2\gamma}_{X,\alpha}:=\mathcal{D}^{2\gamma}_X(\widetilde\mathcal{B}_\alpha)$. For simplicity, we fix the time horizon $T\leq 1$ throughout this section. \\

        The first step is to define~\eqref{eq:RoughConv}
        using Theorem~\ref{integral}.
        Therefore, we begin this section by examining how a controlled rough path changes under the influence of the Neumann operator.

        \begin{lemma}\label{lem:CRP}
            Let $(\mathcal{B}_\alpha)_{\alpha\in \mathbb{R}}$ and $(\widetilde{\mathcal{B}}_\alpha)_{\alpha\in \mathbb{R}}$ monotone scales of interpolation spaces and $N\in \bigcap_{i=0}^2 \mathcal{L}(\widetilde{\mathcal{B}}_{\alpha-i\gamma},\mathcal{B}_{\alpha-i\gamma})$ for an $\alpha\in \mathbb{R}$. Then we obtain for every $(y,y^\prime)\in \widetilde{\mathcal{D}}^{2\gamma}_{X,\alpha}$ that $(Ny,Ny^\prime)\in \mathcal{D}^{2\gamma}_{X,\alpha}$.
        \end{lemma}
        \begin{proof}
            As a direct consequence of the assumption on $N$ we see $(Ny,Ny^\prime)\in C([0,T];\mathcal{B}_\alpha)\times C([0,T];\mathcal{B}_{\alpha-\gamma})$. Furthermore, define the remainder by $R^{Ny}:=NR^y$, then we get for $\vartheta\in \{\gamma,2\gamma\}$
            \begin{align*}
                [Ny^\prime]_{\gamma,\mathcal{B}_{\alpha-2\gamma}}&\leq \norm{N}_{\mathcal{L}(\widetilde{\mathcal{B}}_{\alpha-2\gamma},\mathcal{B}_{\alpha-2\gamma})}[y^\prime]_{\gamma,\widetilde{\mathcal{B}}_{\alpha-2\gamma}},\\
                [R^{Ny}]_{\theta,\mathcal{B}_{\alpha-\vartheta}}&\leq \norm{N}_{\mathcal{L}(\widetilde{\mathcal{B}}_{\alpha-\vartheta},\mathcal{B}_{\alpha-\vartheta})}[R^y]_{\theta,\widetilde{\mathcal{B}}_{\alpha-\vartheta}}.
            \end{align*}
            Consequently $Ny^\prime\in C^\gamma([0,T];\mathcal{B}_{\alpha-2\gamma})$ and $R^{Ny}\in C^\gamma([0,T];\mathcal{B}_{\alpha-\gamma})\cap C^{2\gamma}([0,T];\mathcal{B}_{\alpha-2\gamma})$.~Since $Ny_{t,s}=Ny^\prime_{s}X_{t,s}+R^{Ny}_{t,s}$, this concludes the proof. \hfill $\Box$\smallskip
        \end{proof}\\
        \\
        Regarding this, we can define the rough convolution based on Theorem~\ref{integral} as follows.
        \begin{corollary}\label{cor:RoughConv}
            Let $(y,y^\prime)\in \widetilde{\mathcal{D}}^{2\gamma}_{X,\alpha}$ and $(\mathcal{B}_\alpha)_{\alpha\in \mathbb{R}}$, $(\widetilde{\mathcal{B}}_\alpha)_{\alpha\in \mathbb{R}}$, $N$ as in Lemma~\ref{lem:CRP}. Then the rough convolution
            \begin{align}\label{def:RoughConv}
                \mathcal{I}_t:=\int_0^t S_{t-r}Ny_r~{\textnormal{d}} \mathbf{X}_r:=\lim\limits_{|\mathcal{P}|\to 0} \sum\limits_{[u,v]\in\mathcal{P}} S_{t-u}N\left(y_{u}X_{v,u} + y'_{u}\mathbb{X}_{v,u}\right),
            \end{align}
            exists as an element of $\mathcal{B}_{\alpha-2\gamma}$ for every $t\in [0,T]$, where $\mathcal{P}$ denotes a partition of $[s,t]$. Furthermore, we have for $0\leq s < t \leq T$ and $0\leq \beta<3\gamma$ the estimate
            \begin{align}\label{estimate:NewIntegral}
                \abs{\mathcal{R}_{t,s}^{Ny}}_{\mathcal{B}_{\alpha-2\gamma+\beta}} \lesssim \rho_\gamma(\mathbf{X}) \gubnorm{Ny}{Ny'} (t-s)^{3\gamma-\beta},
            \end{align}
            where $\mathcal{R}^y_{t,s}:=\int_{s}^{t} S_{t-r} y_{r}~{\textnormal{d}}\textbf{X}_{r} - S_{t-s}\left(y_{s}X_{t,s} -y'_{s}\mathbb{X}_{t,s}\right)$. Consequently, $(\mathcal{I},Ny)\in \mathcal{D}^{2\gamma}_{X,\alpha+ \theta}$ for every $\theta \in [0,\gamma)$.
        \end{corollary}
        \begin{proof}
            The existence of the rough convolution follows directly from Theorem~\ref{integral} and Lemma~\ref{lem:CRP}. Moreover, we obtain that $(\mathcal{I},Ny)\in \mathcal{D}^{2\gamma}_{X,\alpha+ \theta}$ for every $\theta \in [0,\gamma)$ due to Corollary \ref{cor:higherreg}.
            \hfill $\Box$\smallskip
        \end{proof}\\

        In order to make sense of~\eqref{eq:RoughConv} in the space of controlled rough paths, we need to make sure that $\mathcal{I}_t\in D(A)$ for every $t\in[0,T]$ and to find a suitable Gubinelli derivative for $A \mathcal{I}$. Therefore, we first specify the scales of Banach spaces which are required in our framework.\\
       We let from now on $p\in [2,3]$ and $\alpha>1+\frac{1}{p}$. We set $\widetilde\mathcal{B}_\beta:=B^{\beta-1-\frac{1}{p}}_{p,p}(\partial \mathcal{O})$ for $\beta \in \mathbb{R}$ and let $(A_\beta,\mathcal{B}_\beta)_{\beta \in [-2,\infty)}$ be the interpolation-extrapolation scale generated by $A:D(A)\subset L^p(\mathcal{O})\to L^p(\mathcal{O})$. When $A$ satisfies Assumption \ref{ass}, we know that the scale of Banach spaces can be expressed by
        \begin{align*}
            \mathcal{B}_{\frac{\beta}{2}}= H^{\beta,p}_\mathcal{C}(\mathcal{O}) :=
            \begin{cases}
                \{ u\in H^{\beta,p}(\mathcal{O}) : \mathcal{C} u =0 \}, & \beta>1+\frac{1}{p}\\
                H^{\beta,p}(\mathcal{O}), & -1+\frac{1}{p}<\beta<1+\frac{1}{p}
            \end{cases},
        \end{align*}
        for $\beta\leq 2$, see for example \cite[Theorem 7.1]{Amann}. Recalling now that the Neumann operator $N$ is the solution operator of \eqref{eq:bvp}, we obtain for all $0<\varepsilon<\frac{1}{2}+\frac{1}{2p}$ that $N\in \mathcal{L}(\widetilde\mathcal{B}_{\alpha},\mathcal{B}_\varepsilon)$, since $H^{\alpha,p}(\mathcal{O}) \hookrightarrow  H^{2\varepsilon,p}(\mathcal{O}) $. If we further replace $\alpha$ by $\alpha- \vartheta$ with $\vartheta\in \{\gamma, 2\gamma \}$, then $N$ is still bounded, but $Ng$ is only a weak solution of the boundary value problem, see \cite[Section 9]{Amann}. This leads to $N\in \bigcap_{i=0}^2 \mathcal{L}(\widetilde{\mathcal{B}}_{\alpha-i\gamma},\mathcal{B}_{\varepsilon-i\gamma})$, which means that Corollary~\ref{cor:RoughConv} is applicable.
        \begin{corollary}\label{cor:RoughConvInDomain}
            Let $0<\varepsilon<\frac{1}{2}+\frac{1}{2p}$ and $(y,y^\prime)\in \widetilde{\mathcal{D}}^{2\gamma}_{X,\alpha}$. Then the rough integral $\mathcal{I}_t$ defined by \eqref{def:RoughConv} belongs to $D(A)$ for every $t\in[0,T]$.
        \end{corollary}
        \begin{proof}
            According to Corollary \ref{cor:RoughConv} we get for every $\theta\in [0,\gamma)$ that $(\mathcal{I},Ny)\in \mathcal{D}^{2\gamma}_{X,\varepsilon+\theta}$. We take $\theta:=\frac{1}{3}+\delta<\gamma$ with $\delta>0$ small enough. Then we can choose $\varepsilon:=\frac{1}{2}+\frac{1}{2p}-\delta$, so that we obtain $\varepsilon+\theta\geq 1$. Therefore, we conclude that $\mathcal{I}_t\in D(A)$ for every $t\in[0,T]$.
            \hfill $\Box$\smallskip
        \end{proof}
        \begin{remark}\label{rem:dirichlet}
        \begin{itemize}
           \item [1)] The essential step in the proof of Corollary \ref{cor:RoughConvInDomain} is that we can choose an $\varepsilon$ such that $\varepsilon>1-\gamma$. Here we recall that $\gamma$ 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 $\varepsilon$ only up to $\frac{1}{2p}$. This comes from the fact that the Dirichlet-operator $D$ is  bounded from $B^{\beta-\frac{1}{p}}_{p,p}(\partial \mathcal{O})$ to $H^{\beta,p}(\mathcal{O})$ for a $\beta>\frac{1}{p}$. Since $p\geq 2$ and $\gamma < \frac{1}{2}$, it is not possible to find an $\varepsilon$ such that $\varepsilon>1-\gamma$ and $D\in \mathcal{L}(B^{\beta-\frac{1}{p}}_{p,p}(\partial \mathcal{O}), \mathcal{B}_\varepsilon)$.
       \item   [2)]  The results obtained in this manuscript remain valid in the Young regime, i.e.~for $\gamma\in(\frac{1}{2},1)$, recall~\eqref{young}. Moreover, in this case, we can incorporate Dirichlet boundary noise since the conditions $\varepsilon>1-\gamma$ and $\varepsilon<\frac{1}{2p}$ can simultaneously be fulfilled. This result was also established for additive fractional noise in~\cite{DuncanMaslowski}.
       \end{itemize}
            \end{remark}

        Now we have to find a  suitable Gubinelli derivative for $A\mathcal{I}$.
        \begin{remark}
        Since $Ny$ is a Gubinelli derivative for $\mathcal{I}$, it would make sense to consider $ANy$. However, regarding the definition of $N$, $Ny$ 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 \ref{preliminaries}.
        \end{remark}
        From now on we assume that $\varepsilon>1-\gamma$ so that $\mathcal{I}_t\in D(A)$, as proved in Corollary~\ref{cor:RoughConv}. \\

       We set $\eta:=1-\varepsilon$, where $\varepsilon=\frac{1}{2} +\frac{1}{2p}-\delta$ for a small $\delta>0$, as determined in the proof of Corollary~\ref{cor:RoughConvInDomain}.
        \begin{theorem}\label{thm:RoughConv}
            For every $(y,y^\prime)\in \widetilde{\mathcal{D}}^{2\gamma}_{X,\alpha}$ we have $(A\mathcal{I},A_{-\eta}Ny)\in \mathcal{D}^{2\gamma}_{X,-\eta}$.
        \end{theorem}
        \begin{proof}
            We set $ z_t:=A\mathcal{I}_t$ and $z_t^\prime:=A_{-\eta}Ny$ for $t\in[0,T]$. By construction $z_t\in \mathcal{B}_1\hookrightarrow \mathcal{B}_{-\eta}$ and $Ny_t\in \mathcal{B}_\varepsilon= \mathcal{B}_{1-\eta}$. Therefore, $z_t^\prime$ is well-defined with values in $\mathcal{B}_{-\eta}\hookrightarrow\mathcal{B}_{-\eta-\gamma}$. Consequently, we get $(z,z^\prime)\in C([0,T];\mathcal{B}_{-\eta})\times C([0,T];\mathcal{B}_{-\eta-\gamma})$. \\
            We recall that $A$ can be viewed as the $\mathcal{B}_{0}$-realization of $A_{-\eta-\gamma}$, see Remark~\ref{rem:Extrapolation}.~Then for $x\in D(A)$, we have the equality $A_{-\eta-\gamma}x=Ax$. Now let $0\leq s<t\leq T$ and obtain, using the fact that $A_{-\eta}\in \mathcal{L}(\mathcal{B}_{1-\eta},\mathcal{B}_{-\eta})$
            \begin{align*}
                \abs{z^\prime_{t,s}}_{-\eta-2\gamma}\lesssim \abs{Ny_{t,s}}_{1-\eta-2\gamma}=\abs{Ny_{t,s}}_{\varepsilon-2\gamma}\lesssim \abs{Ny_{t,s}}_{\varepsilon-\gamma}\leq  \left[Ny\right]_{\gamma,\varepsilon-\gamma}\abs{t-s}^\gamma.
            \end{align*}
            Therefore $z^\prime \in C^\gamma([0,T]; \mathcal{B}_{-\eta-2\gamma})$. The tricky part is  to estimate the remainder $R^z_{t,s}:= z_{t,s}-z^\prime_s X_{t,s}.$
            For this reason, we rewrite the reminder $R^z$, so that we can use the previous estimates derived for $\mathcal{R}^{Ny}$:
            \begin{align}
                \begin{split}\label{eq:remainder}
                    R^z_{t,s}&=A\int_0^t S_{t-r}Ny_r~{\textnormal{d}} \mathbf{X}_r - A\int_0^s S_{s-r}Ny_r~{\textnormal{d}} \mathbf{X}_r-A_{-\eta}Ny_sX_{t,s}\\
                    & = A\left(\int_s^t S_{t-r}Ny_r~{\textnormal{d}} \mathbf{X}_r + (S_{t-s}-\text{id} )\int_0^s S_{s-r}Ny_r~{\textnormal{d}} \mathbf{X}_r  \right)+AS_{t-s}Ny^\prime_s\mathbb{X}_{t,s}\\
                    & - AS_{t-s}Ny^\prime_s\mathbb{X}_{t,s}+AS_{t-s}Ny_sX_{t,s}-AS_{t-s}Ny_sX_{t,s}-A_{-\eta}Ny_sX_{t,s}\\
                    &=A\mathcal{R}^{Ny}_{t,s}+AS_{t-s}Ny^\prime_s \mathbb{X}_{t,s}+(AS_{t-s}-A_{-\eta})Ny_sX_{t,s}+A(S_{t-s}-\text{id})\mathcal{I}_s.
                \end{split}
            \end{align}
            Therefore we get for $\vartheta\in \{\gamma, 2\gamma  \}$
            \begin{align*}
                \abs{R^z_{t,s}}_{-\eta-\vartheta}&\leq \underbrace{\abs{A\mathcal{R}^{Ny}_{t,s}}_{-\eta-\vartheta}}_{I_1}+\underbrace{\abs{AS_{t-s}Ny^\prime_s \mathbb{X}_{t,s}}_{-\eta-\vartheta}}_{I_2}+\underbrace{\abs{(AS_{t-s}-A_{-\eta})Ny_sX_{t,s}}_{-\eta-\vartheta}}_{I_3}\\
                &+\underbrace{\abs{A(S_{t-s}-\text{id})\mathcal{I}_s}_{-\eta-\vartheta}}_{I_4},
            \end{align*}
            so we can estimate the individual terms separately. Applying Corollary \ref{estimate:NewIntegral} with $\beta:=2\gamma-\vartheta$ entails
            \begin{align*}
                I_1\lesssim \abs{\mathcal{R}^{Ny}_{t,s}}_{1-\eta -\vartheta}\lesssim \rho_\gamma(\mathbf{X}) \norm{Ny,Ny^\prime}_{X,2\gamma,\varepsilon}\abs{t-s}^{\gamma+ \vartheta}\leq \rho_\gamma(\mathbf{X}) \norm{Ny,Ny^\prime}_{X,2\gamma,\varepsilon}\abs{t-s}^{\vartheta}T^\gamma.
            \end{align*}
            To deal with the second term, we use that $S_{t-s}\in\mathcal{L}(\mathcal{B}_{\varepsilon-\gamma})$ and $1-\eta-\vartheta=\varepsilon-\vartheta$ to further infer that
            \begin{align*}
                I_2&\lesssim \rho_\gamma(\mathbf{X}) \abs{S_{t-s}Ny_s^\prime}_{1-\eta-\vartheta}\abs{t-s}^{2\gamma}\lesssim \rho_\gamma(\mathbf{X})  \abs{Ny_s^\prime}_{\varepsilon-\vartheta}\abs{t-s}^{2\gamma}\\
                &\lesssim \rho_\gamma(\mathbf{X}) \abs{Ny_s^\prime}_{\varepsilon-\gamma}\abs{t-s}^{2\gamma}\leq \rho_\gamma(\mathbf{X}) \norm{Ny,Ny^\prime}_{X,2\gamma,\varepsilon}\abs{t-s}^{\vartheta}T^{2\gamma-\vartheta}.
            \end{align*}
            Combining Remark \ref{rem:Extrapolation} with the smoothing property \eqref{hg:1} and use that $|\cdot|_{-\eta-\vartheta}\lesssim |\cdot|_{-\eta}$ since $\eta+\vartheta\geq \eta$,
            we get the estimate
            \begin{align*}
                I_3&\lesssim \rho_\gamma (\mathbf{X})  \abs{(AS_{t-s}-A_{-\eta})Ny_s}_{-\eta-\vartheta}\abs{t-s}^\gamma\\
                &=\rho_\gamma (\mathbf{X})  \abs{A_{-\eta}(S_{t-s}-\text{id})Ny_s}_{-\eta-\vartheta}\abs{t-s}^\gamma\\
                &\lesssim \rho_\gamma (\mathbf{X}) \abs{(S_{t-s}-\text{id})Ny_s}_{1-\eta-\vartheta}\abs{t-s}^\gamma \\
                &= \rho_\gamma (\mathbf{X}) \abs{(S_{t-s}-\text{id})Ny_s}_{\varepsilon-\vartheta}\abs{t-s}^\gamma \\
                & \lesssim \rho_\gamma (\mathbf{X}) \|S_{t-s}-\text{id}\|_{\mathcal{L}(\mathcal{B}_\varepsilon,\mathcal{B}_{\varepsilon-\vartheta})} |Ny_s|_{\varepsilon} |t-s|^\gamma \\
                &\lesssim \rho_\gamma(\mathbf{X}) \|Ny\|_{\infty,\varepsilon}\abs{t-s}^{\gamma+\vartheta}\\
                &\leq \rho_\gamma(\mathbf{X}) \norm{Ny,Ny^\prime}_{X,2\gamma,\varepsilon} \abs{t-s}^{\vartheta}T^{\gamma}.
            \end{align*}
            In order to estimate $I_4$ we first apply the smoothing property~\eqref{hg:1} and recall that $\eta=1-\varepsilon$, to further obtain
            \begin{align*}
                I_4&=\abs{A_{-\eta-\vartheta}(S_{t-s}-\text{id})\mathcal{I}_s}_{-\eta-\vartheta}\lesssim\abs{(S_{t-s}-\text{id})\mathcal{I}_s}_{1-\eta-\vartheta}\\
                &= \abs{(S_{t-s}-\text{id})\mathcal{I}_s}_{\varepsilon-\vartheta}\lesssim \abs{\mathcal{I}_s}_{\varepsilon} \abs{t-s}^\vartheta.
            \end{align*}
            Now we can use the same decomposition as in \eqref{eq:remainder} for $\mathcal{I}_s$. This  leads together with \eqref{hg:2}, \eqref{estimate:NewIntegral} and $S_t\in \mathcal{L}(\mathcal{B}_\varepsilon)$ to
            \begin{align*}
                I_4&\lesssim \left(\abs{\mathcal{R}^{Ny}_{s,0}}_{\varepsilon}+\abs{S_sNy_0X_{s,0}}_{\varepsilon}+\abs{S_sNy_0^\prime \mathbb{X}_{s,0}}_{\varepsilon} \right)\abs{t-s}^{\vartheta}\\
                &\lesssim \rho_\gamma(\mathbf{X})\left(\norm{Ny,Ny^\prime}_{X,2\gamma,\varepsilon}s^\gamma+\abs{S_sNy_0}_\varepsilon s^\gamma+\abs{S_sNy_0^\prime}_\varepsilon s^{2\gamma}\right)\abs{t-s}^{\vartheta}\\
                &\lesssim \rho_\gamma(\mathbf{X})\left(\norm{Ny,Ny^\prime}_{X,2\gamma,\varepsilon}s^\gamma+\norm{Ny,Ny^\prime}_{X,2\gamma,\varepsilon} s^\gamma+\abs{Ny_0^\prime}_{\varepsilon-\gamma} s^{\gamma}\right)\abs{t-s}^{\vartheta}\\
                &\leq 3 \rho_\gamma(\mathbf{X}) \norm{Ny,Ny^\prime}_{X,2\gamma,\varepsilon} \abs{t-s}^{\vartheta} T^\gamma.
            \end{align*}
            Putting all the previous estimates together, we conclude that
            \begin{align*}
                \left[R^z\right]_{\vartheta,-\eta-\vartheta}\lesssim \rho_\gamma (\mathbf{X}) \norm{Ny,Ny^\prime}_{X,2\gamma,\varepsilon} T^{\gamma },
            \end{align*}
            so $R^z\in C^\gamma([0,T];\mathcal{B}_{-\eta-\gamma})\cap C^{2\gamma}([0,T];\mathcal{B}_{-\eta-2\gamma})$.
            \hfill $\Box$\smallskip\\
        \end{proof}

        Even if Theorem~\ref{thm:RoughConv} 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~\eqref{eq:intro} has a global solution working with the controlled rough path $(A\mathcal{I}, A_{-\eta}Ny)\in \mathcal{D}^{2\gamma}_{X,-\eta}$ and using the techniques in~\cite{HN21}, 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~\cite{m} for an analogous result for additive fractional noise. As a consequence of Corollary \ref{cor:RoughConv}, the limit on the right-hand side of \eqref{def:RoughConv}, exists in $\mathcal{B}_{\varepsilon-2\gamma}$. So
        the equality
        \begin{align}\label{interchangeIntegral}
            \widetilde{A}\int_0^{t} S_{t-r} N(y_r)~{\textnormal{d}}\mathbf{X}_r=\int_0^{t} \widetilde{A}S_{t-r} N(y_r)~{\textnormal{d}}\mathbf{X}_r,
        \end{align}
        holds for a bounded, and therefore continuous, operator $\widetilde{A}$ with domain $\mathcal{B}_{\varepsilon-2\gamma}$.
        However, in our case, we only have $A\in \mathcal{L}(\mathcal{B}_1,\mathcal{B}_0)$ and $\varepsilon-2\gamma<1$. Nevertheless we can show the following statement.
        \begin{lemma}
            Under the assumptions of Corollary \ref{cor:RoughConv}, the limit
            \begin{align*}
                \int_s^t S_{t-r}Ny_r~{\textnormal{d}} \mathbf{X}_r= \lim\limits_{|\mathcal{P}|\to 0} \sum\limits_{[u,v]\in\mathcal{P}} S_{t-u}N\left(y_{u}X_{v,u} + y'_{u}\mathbb{X}_{v,u}\right),
            \end{align*}
            exists in the $\mathcal{B}_{\alpha-2\gamma+\beta}$ topology for every $\beta\in [0,3\gamma)$.
        \end{lemma}
        \begin{proof}
            For the sake of completeness, we indicate a straightforward proof of this statement based on~\cite[Theorem 4.1]{GHN}. Let $\mathcal{P}^n:=\{t_i=s+2^{-n} i (t-s)~:~i=0,\ldots, 2^n \}$ be the $n$-th dyadic partition of the interval $[s,t]$, and $\mathcal{I}^{\mathcal{P}^n}_{t,s}:=\sum\limits_{[u,v]\in\mathcal{P}^n} S_{t-u}N\left(y_{u}X_{v,u} + y'_{u}\mathbb{X}_{v,u}\right)$  the sum associated to the partition $\mathcal{P}^n$. Using Lemma~\ref{lem:CRP} and the estimates for analytic semigroups~\eqref{hg:1} and~\eqref{hg:2}, it follows that
            \begin{align*}
                \abs{\mathcal{I}^{\mathcal{P}^n}_{t,s}-\mathcal{I}^{\mathcal{P}^{n+1}}_{t,s}}_{\alpha-2\gamma+\beta}\lesssim 2^{-n(3\gamma-1-\delta)} \abs{t-s}^{3\gamma-\beta},
            \end{align*}
            for a $\delta\in (\beta-1,3\gamma-1)$, see the proof of \cite[Theorem 4.1]{GHN}. Since the right side is summable over $n$, the sequence $(\mathcal{I}_{t,s}^{\mathcal{P}^n})_{n\in \mathbb{N}}$ is Cauchy in $\mathcal{B}_{\alpha-2\gamma+\beta}$ and therefore, there is a limit $\widetilde{\mathcal{I}}_{t,s}\in \mathcal{B}_{\alpha-2\gamma+\beta}$. Since $\mathcal{B}_{\alpha-2\gamma+\beta}\hookrightarrow \mathcal{B}_{\alpha-2\gamma}$, and the limit in Corollary~\ref{cor:RoughConv} is unique, we get that $\widetilde{\mathcal{I}}_{t,s}=\mathcal{I}_{t,s}$. In conclusion, the limit exists in the $\mathcal{B}_{\alpha-2\gamma+\beta}$ topology.
            \hfill $\Box$\smallskip
        \end{proof}\\

        Now, going back to the situation in~\eqref{interchangeIntegral}, we choose $\beta:=2\gamma-\varepsilon+1<3\gamma$, due to our restriction on $\varepsilon$. Since $A$ satisfies Assumption \ref{ass}, Theorem \ref{thm:EpolSemigroup} ensures that every extrapolated operator generates again an analytic semigroup. Together with Theorem \ref{thm:EpolOperator}  and the fact that $Ny\in \mathcal{B}_{\varepsilon}=\mathcal{B}_{1-\eta}$, this leads to
        \begin{align}\label{newInterchangeIntegral}
            A\int_0^{t} S_{t-r} N(y_r)~{\textnormal{d}}\mathbf{X}_r=\int_0^{t} S_{t-r}A_{-\eta} N(y_r)~{\textnormal{d}}\mathbf{X}_r.
        \end{align}

      \begin{remark}\label{two:i}
          To make sure that the right-hand side is well-defined as a controlled rough integral, we need to find a Gubinelli derivative for $A_{-\eta}Ny$. A natural choice would be $A_{-\eta}Ny^\prime$, but since $y^\prime$ loses spatial regularity, this would not be well-defined. Therefore, in order to choose an appropriate Gubinelli derivative for $A_{-\eta}N y$, the extrapolated operator $A_{-\eta}$ has to be lifted. Due to this reason, one can show that $(A_{-\eta}Ny, A_{-\sigma}Ny')\in \mathcal{D}^{2\gamma}_{X,-\eta}$ with $\sigma:=\eta+\gamma$. 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 \ref{rem:Extrapolation}. Therefore, we have $A_{-\eta}Ny=A_{-\sigma}Ny$ since $-\eta>-\sigma$ and $Ny\in \mathcal{B}_\varepsilon=\mathcal{B}_{1-\eta}\hookrightarrow\mathcal{B}_{1-\sigma}$.
      \end{remark}
        This enables us to formulate the next result.

        \begin{lemma}\label{lem:GubDeriv}
            For every $(y,y^\prime)\in \widetilde{\mathcal{D}}^{2\gamma}_{X,\alpha}$ we have $(A_{-\sigma}Ny,A_{-\sigma}Ny^\prime)\in \mathcal{D}^{2\gamma}_{X,-\eta}$.
        \end{lemma}
        \begin{proof}
            Note that we cannot use Lemma \ref{lem:CRP}. We set $z_t:=A_{-\sigma}Ny_t, z^\prime_t:=A_{-\sigma}Ny^\prime_t$ and $R^z_{t,s}:=z_{t,s}-z^\prime_{s}X_{t,s}$. We know that $z_t\in \mathcal{B}_{-\eta}$, since $Ny_t\in \mathcal{B}_\varepsilon = \mathcal{B}_{1-\eta}$, and $z^\prime_t\in \mathcal{B}_{-\eta-\gamma}=\mathcal{B}_{-\sigma}$ since $Ny^\prime_t\in \mathcal{B}_{\varepsilon-\gamma}=\mathcal{B}_{1-\sigma}$. Furthermore, we have
            \begin{align*}
                \abs{z'_{t,s}}_{-\eta-2\gamma}=
            \abs{A_{-\sigma} N y^\prime_{t,s}}_{-\eta-2\gamma}&=\abs{A_{-\eta-2\gamma} N y^\prime_{t,s}}_{-\eta-2\gamma}\lesssim \abs{N y^\prime_{t,s}}_{1-\eta-2\gamma}\\
            &=\abs{N y^\prime_{t,s}}_{\varepsilon-2\gamma}\lesssim \abs{t-s}^{\gamma} \left[Ny^\prime\right]_{\gamma,\varepsilon-2\gamma}.
            \end{align*}
            To investigate the remainder, we let $\vartheta\in \{\gamma,2\gamma\}$ and establish using  Remark \ref{rem:Extrapolation}
            \begin{align*}
                \abs{R^z_{t,s}}_{-\eta-\vartheta}=\abs{A_{-\sigma}R^{Ny}_{t,s}}_{-\sigma -(\vartheta-\gamma)}\lesssim \abs{R^{Ny}_{t,s}}_{1-\sigma-(\vartheta-\gamma)}=\abs{R^{Ny}_{t,s}}_{\varepsilon-\vartheta}\lesssim \abs{t-s}^\vartheta \left[R^{Ny}\right]_{\vartheta, \varepsilon -\vartheta}.
            \end{align*}
            \hfill $\Box$\smallskip
        \end{proof}\\
       Based on the previous deliberations, we conclude.
       \begin{lemma}\label{sigma}
           The right-hand side of~\eqref{newInterchangeIntegral} is well-defined as a rough convolution with the controlled rough path $(A_{-\sigma} Ny, A_{-\sigma} N y')\in \mathcal{D}^{2\gamma}_{X,-\eta}$. Moreover, a Gubinelli derivative of
            $\int_0^t S_{t-r} A_{-\sigma} N(y_r)~{\textnormal{d}} \mathbf{X}_r$ is $A_{-\sigma} N y_t$.
       \end{lemma}

        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. The idea of the proof is  similar to the one in \cite[Lemma 4.7]{GHN}. In our setting, the nonlinearity is allowed to map between different scales of Banach spaces.
        \begin{lemma} (Composition of a controlled rough path with a smooth function)\label{lem:composition}
            Let $\beta,\delta\in \mathbb{R}$ and $F:\mathcal{B}_{\beta-\vartheta}\to \mathcal{B}_{\beta-\vartheta+\delta}$ two times continuously differentiable with bounded derivatives for any $\vartheta \in \{0,\gamma, 2\gamma \}$. For $(y,y^\prime)\in \mathcal{D}^{2\gamma}_{X,\beta}$ we define $(z_t,z_t^\prime):=(F(y_t),DF(y_t)\circ y_t^\prime)$ for $t\in [0,T]$.
            \begin{itemize}
            \itemsep -2pt
            \item[i)] We have $(z,z^\prime)\in \widetilde{\mathcal{D}}^{2\gamma}_{X,\beta+\delta}$ and the estimate
            \begin{align}\label{est:SmoothComposition}
                 \norm{z,z^\prime}_{X,2\gamma,\beta+\delta}\lesssim \norm{F}_{C^2}(1+\rho_\gamma(\mathbf{X}))^2 \norm{y,y^\prime}_{X,2\gamma,\beta}(1+\norm{y,y^\prime}_{X,2\gamma,\beta}),
            \end{align}
            holds.
            \item[ii)] Assume additionally that $F$ is three times differentiable with bounded third derivative, and let $(\widetilde{z},\widetilde{z}^\prime)$ be the composition of another controlled rough path $(\widetilde{y},\widetilde{y}^\prime)\in \mathcal{D}^{2\gamma}_{X,\beta}$ with $F$. Then
            \begin{align}\label{est:SmoothComposition2}
                \norm{z-\widetilde{z},z^\prime-\widetilde{z}^\prime}_{X,2\gamma,\beta+\delta} \lesssim \norm{F}_{C^3}&(1+\rho_\gamma(\mathbf{X}))^2(1+\norm{y,y^\prime}_{X,2\gamma,\beta}+\norm{\widetilde{y},\widetilde{y}^\prime}_{X,2\gamma,\beta})^2\nonumber\\ &\times \norm{y-\widetilde{y},y^\prime-\widetilde{y}^\prime}_{X,2\gamma,\beta}
            \end{align}
            is satisfied.
            \end{itemize}
            Where we set $\norm{F}_{C^k}:=\max\limits_{\vartheta\in \{\gamma,2\gamma\}}  \norm{F}_{C^k(\mathcal{B}_{\beta-\vartheta},\widetilde{\mathcal{B}}_{\beta-\vartheta+\delta})}$.
        \end{lemma}
        \begin{proof}
            The proof is similar to \cite[Lemma 4.7]{GHN}. We point out the main differences that occur in our case. We can view for $t\in [0,T]$ the derivative $D^kF(y_t)$ as an element of $\mathcal{L}(\mathcal{B}^{\otimes k}_{\beta-\vartheta},\widetilde{\mathcal{B}}_{\beta-\vartheta+\delta})$ for $k=1,2,3$ and $\vartheta\in \{\gamma,2\gamma\}$. Since $\mathcal{B}$ and $\widetilde{\mathcal{B}}$ are both Banach scales (recall Definition~\ref{raum}), all the necessary estimates remain valid. For instance, one can estimate the Gubinelli derivative $z'$ as
            \begin{align*}
                \abs{z^\prime_{t,s}}_{\widetilde{\mathcal{B}}_{\beta-2\gamma+\delta}}&\leq \abs{DF(y_s)\circ y_{t,s}^\prime}_{\widetilde{\mathcal{B}}_{\beta-2\gamma+\delta}} + \abs{(DF(y_t)-DF(y_s))\circ y_t^\prime}_{\widetilde{\mathcal{B}}_{\beta-2\gamma+\delta}}\\
                &\leq \norm{DF(y_s)}_{\mathcal{L}(\mathcal{B}_{\beta-2\gamma},\widetilde{\mathcal{B}}_{\beta-2\gamma+\delta})}\abs{y_{t,s}^\prime}_{\mathcal{B}_{\beta-2\gamma}}+\norm{DF(y_t)-DF(y_s)}_{\mathcal{L}(\mathcal{B}_{\beta-2\gamma},\widetilde{\mathcal{B}}_{\beta-2\gamma+\delta})}\abs{y_s^\prime}_{\mathcal{B}_{\beta-2\gamma}}\\
                &\leq \norm{F}_{C^1} \norm{y,y^\prime}_{X,2\gamma,\beta}\abs{t-s}^\gamma+ \norm{D^2F}_{C} \abs{y_{t,s}}_{\mathcal{B}_{\beta-2\gamma}}\abs{y^\prime_s}_{\mathcal{B}_{\beta-2\gamma}} \\
                &\lesssim \norm{F}_{C^2} \norm{y,y^\prime}_{X,2\gamma,\beta} \abs{t-s}^\gamma + \norm{F}_{C^2} \norm{y,y^\prime}_{X,2\gamma,\beta} \left[y \right]_{\gamma,\mathcal{B}_{\beta-2\gamma}} \abs{t-s}^\gamma\\
                &\leq \norm{F}_{C^2} \norm{y,y^\prime}_{X,2\gamma,\beta} \abs{t-s}^\gamma + \norm{F}_{C^2} \norm{y,y^\prime}_{X,2\gamma,\beta} (1+\rho_\gamma(\mathbf{X}))\norm{y,y^\prime}_{X,2\gamma,\beta} \abs{t-s}^\gamma,
            \end{align*}
            where we use \eqref{est:hoelder:y}. The estimates of the remainder follow by analogue computations.
            \hfill $\Box$\smallskip\\
        \end{proof}

Coming back to~\eqref{eq:intro} and regarding that according to~\eqref{newInterchangeIntegral} and Lemma~\ref{sigma} it holds  $$A\int_0^{t} S_{t-r} N(F(y_r))~{\textnormal{d}}\mathbf{X}_r=\int_0^{t} S_{t-r}A_{-\sigma} N(F(y_r))~{\textnormal{d}}\mathbf{X}_r,$$
        we can now rewrite \eqref{eq:intro} as a semilinear evolution equation without boundary noise
        \begin{align}\label{eq:introNew}
            \begin{cases}
                  {\textnormal{d}} y = \left(Ay + f(y)\right)~{\textnormal{d}} t + A_{-\sigma}N(F(y))~{\textnormal{d}} \mathbf{X}_t,\\
                  y(0)=y_0\in \mathcal{B}_{-\eta}.
            \end{cases}
        \end{align}
        We recall that $A_{-\sigma}$ is the extrapolation operator introduced in Section~\ref{preliminaries}, $\sigma=1-\varepsilon-\gamma$ with $\varepsilon=\frac{1}{2}+\frac{1}{2p}-\delta$ for a small $\delta>0$.

        \begin{remark}
            The idea to rewrite~\eqref{eq:intro} as a semilinear problem without boundary noise as in~\eqref{eq:introNew} using an extrapolation operator was also applied in~\cite{VeraarSchnaubelt}. There the extrapolated operator $A_{-1}$ was used. We note that the index of the extrapolation operator needed is $\frac{\widetilde{\alpha}}{2}-1$ where $\widetilde{\alpha} \in \left(1,1+\frac{1}{p}\right)$. Therefore, our result is consistent with the one in \cite{VeraarSchnaubelt} for Brownian noise, since in both cases the index satisfies $-\eta=\frac{\widetilde{\alpha}}{2}-1\in \left(-\frac{1}{2},-\frac{1}{2}+\frac{1}{2p}\right)$ due to the restriction $\varepsilon>1-\gamma$ and $\gamma\in \left(\frac{1}{3},\frac{1}{2}\right]$. We work here with the extrapolation operator $A_{-\eta-\gamma}$, as pointed out in Remark~\ref{two:i}.
        \end{remark}

        We give now the main assumptions on the nonlinear drift and diffusion coefficients of \eqref{eq:intro}, that will guarantee the local-~as well as global-in-time existence of solutions based on the results in~\cite{GHN,HN21}.
        \begin{assumptions}\label{ass2}
            \begin{itemize}
                \item[1)] There exists a $\delta_1\in[0,1)$ such that the drift term $f:\mathcal{B}_{-\eta}\to \mathcal{B}_{-\eta-\delta_1}$ is Lipschitz continuous.
                \item[2)] There exists a $\delta_2>\eta+1+\frac{1}{p}$ such that for any $\vartheta \in \{0,\gamma, 2\gamma \}$ the diffusion term $F:\mathcal{B}_{-\eta-\vartheta}\to \widetilde{\mathcal{B}}_{-\eta-\vartheta+\delta_2}$ is three times continuously differentiable with bounded derivatives.

                \item[3)] Let 1) and 2) be satisfied. Assume additionally that $f:\mathcal{B}_{-\eta}\to \mathcal{B}_{-\eta-\delta_1}$ for $\delta_1\in[2\gamma,1)$ satisfies a linear growth condition and that the derivative of
                \begin{align*}
                    DF(\cdot) \circ G(\cdot):\mathcal{B}_{-\eta-\gamma}\to \mathcal{B}_{-\eta-\gamma+\delta_2}
                \end{align*}

                is bounded, where $G:=A_{-\sigma} N F$.
            \end{itemize}
        \end{assumptions}
        \begin{remark}
            \begin{itemize}
                \item[i)] The assumption on the diffusion coefficient ensures that $-\eta+\delta_2>1+\frac{1}{p}$ such that $NF:\mathcal{B}_{-\eta}\to \mathcal{B}_{\varepsilon}=\mathcal{B}_{1-\eta}\hookrightarrow \mathcal{B}_{1-\sigma}$ due to the definition of the Neumann operator. Therefore $G:\mathcal{B}_{-\eta}\to\mathcal{B}_{-\eta}$ is well-defined.
                \item[i)] Due to the rough path techniques, the smoothness assumption on the diffusion coefficient $F$ is more restrictive than in \cite{VeraarSchnaubelt}. There $F$ maps into $\mathcal{B}_{-1}$, is Lipschitz continuous and has to satisfy a linear growth condition.

            \end{itemize}
        \end{remark}

        \begin{corollary}\label{new:crp}
            Let Assumption \ref{ass2} 2) be fulfilled. Then for every $(y,y^\prime)\in \mathcal{D}^{2\gamma}_{X,-\eta}$ we have $(A_{-\sigma}NF(y),A_{-\sigma}N(DF(y)\circ y^\prime))\in \mathcal{D}^{2\gamma}_{X,-\eta}$.
        \end{corollary}
        \begin{proof}
            Let be $(y,y^\prime)\in \mathcal{D}^{2\gamma}_{X,-\eta}$. Due to Lemma \ref{lem:composition} we know that $(F(y),DF(y)\circ y^\prime)\in \mathcal{D}^{2\gamma}_{X,-\eta+\delta_2}$. Since $-\eta+\delta_2>1+\frac{1}{p}$ the claim follows, applying Lemma \ref{lem:GubDeriv} to the controlled rough path $(F(y), DF(y)\circ y')$.
            \hfill $\Box$\smallskip
        \end{proof}
        \begin{theorem}\em{(Existence of a local-in-time solution for~\eqref{eq:introNew})}\label{local}
            Assume that $F$ and $f$ satisfy Assumption \ref{ass2} 1) and 2). Then there exists for every initial condition $y_0\in \mathcal{B}_{-\eta}$ a time $T^*\leq T$ and a unique solution $(y,A_{-\sigma}N F(y))\in \mathcal{D}^{2\gamma}_{X,-\eta}\left([0,T^*)\right)$ to \eqref{eq:intro} such that
            \begin{align}\label{eq:mildSolution}
                y_t=S_t y_0 + \int_0^t S_{t-r} f(y_r)~{\textnormal{d}} r+\int_0^t S_{t-r} A_{-\sigma}N(F(y_r))~{\textnormal{d}} \mathbf{X}_r, \text{ for all } t<T^*.
            \end{align}

        \end{theorem}
        \begin{proof} Note that $(F(y),DF(y)\circ y^\prime)\in \mathcal{D}^{2\gamma}_{X,-\eta+\delta_2}$ by Lemma \ref{lem:composition} and that $(A_{-\sigma}NF(y), A_{-\sigma} N(DF(y)\circ y'))\in \mathcal{D}^{2\gamma}_{X,-\eta} $ due to Corollary~\ref{new:crp}. Therefore one can apply  a fixed-point argument in $\mathcal{D}^{2\gamma}_{X,-\eta}$, similar to~\cite[Theorem 5.1]{GHN}.
            \hfill $\Box$\smallskip
    \end{proof}\\

        In order to prove global existence of the solution, we use Assumption~\ref{ass2} 3) in order to avoid the quadratic terms appearing by the composition of a controlled rough path with a smooth function, as in Lemma~\ref{lem:composition}. To this aim, we derive the following key result, recalling that $G=A_{-\sigma} NF$.
        \begin{lemma}\label{no:q}
            Let Assumption \ref{ass2} be satisfied and further assume that $(y,G(y))\in \mathcal{D}^{2\gamma}_{X,-\eta}$. Then $(G(y), DG(y)\circ G(y))\in\mathcal{D}^{2\gamma}_{X,-\eta}$ and the following bound is valid
            \begin{align*}
                \norm{G(y), DG(y)\circ G(y)}_{X,2\gamma,-\eta}\lesssim 1+\norm{y,G(y)}_{X,2\gamma,-\eta}.
            \end{align*}
        \end{lemma}
        \begin{proof}

            By definition we have
            \begin{align*}
                 \norm{G(y), DG(y)\circ G(y)}_{X,2\gamma,-\eta}&= \norm{G(y)}_{\infty, -\eta}+ \norm{DG(y)\circ G(y)}_{\infty, -\eta-\gamma}+ \left[DG(y)\circ G(y)\right]_{\gamma,-\eta-2\gamma}\\
                 &+  \left[R^{G(y)}\right]_{\gamma,-\eta-\gamma}+\left[R^{G(y)}\right]_{2\gamma,-\eta-2\gamma},
            \end{align*}
            where $R^{G(y)}_{t,s}:=G(y_t)-G(y_s)-\left(DG(y_s)\circ G(y_s)\right)X_{t,s}$ is  the remainder. The first two terms can be estimated directly, since it is assumed that $F$ has bounded derivatives, thus also $G$. Therefore, we have
            \begin{align*}
                \norm{G(y)}_{\infty, -\eta}&\leq \abs{G(y_0)}_{-\eta}+T^\gamma \left[G(y)\right]_{\gamma, -\eta}\lesssim 1+\left[y\right]_{\gamma,-\eta}\lesssim 1+\norm{y,G(y)}_{X,2\gamma, -\eta},\\ \norm{DG(y)\circ G(y)}_{\infty, -\eta-\gamma}&\lesssim \norm{G(y)}_{\infty, -\eta-\gamma}\lesssim \norm{y,G(y)}_{X,2\gamma, -\eta}.
            \end{align*}
            For the third term we recall that for $\alpha:=\delta_2-\eta>1+\frac{1}{p}$ the Neumann operator maps $\widetilde{\mathcal{B}}_{\alpha-\gamma}$ to $\mathcal{B}_{\varepsilon-\gamma}$. Together with \ref{ass2} 3) and Remark \ref{rem:Extrapolation} we have
            \begin{align*}
                \left[DG(y)\circ G(y)\right]_{\gamma,-\eta-2\gamma}&=\left[A_{-\sigma}N(DF(y)\circ G(y))\right]_{\gamma,-\eta-2\gamma}\lesssim \left[N(DF(y)\circ G(y))\right]_{\gamma,\varepsilon-\gamma}\\
                &\lesssim \left[DF(y)\circ G(y)\right]_{\gamma,\alpha-\gamma}\lesssim \left[y\right]_{\gamma,-\eta-\gamma} \lesssim \norm{y,G(y)}_{X,2\gamma,-\eta}.
            \end{align*}
         Consequently we only have to estimate the remainder. For $0\leq s< t \leq T$ we write
            \begin{align*}
                R^{G(y)}_{t,s}=\int_0^1 (DG(y_s+ry_{t,s})-DG(y_s))~{\textnormal{d}} r~G(y_s) X_{t,s}+\int_0^1 DG(y_s+ry_{t,s})~{\textnormal{d}} r~ R^y_{t,s},
            \end{align*}
            and use again the boundedness of the derivatives to obtain
            \begin{align*}
                \left[R^{G(y)}\right]_{\gamma,-\eta-\gamma}&\lesssim\norm{DG(y)}_{\infty,\mathcal{L}(\mathcal{B}_{-\eta-\gamma})}(\norm{G(y)}_{\infty,-\eta-\gamma}\left[X\right]_\gamma+\left[R^y\right]_{\gamma,-\eta-\gamma})\\
                &\lesssim \norm{y,G(y)}_{X,2\gamma,-\eta}.
            \end{align*}
            For the second term, note that due to the boundedness of $D\left[DF(\cdot)\circ G(\cdot)\right]$, we get the Lipschitz type estimate
            \begin{align*}
                \abs{(DF(x_1)-DF(x_2))G(x_1)}_{-\eta-\gamma}&\leq \abs{DF(x_1)G(x_1)-DF(x_2)G(x_2)}_{-\eta-\gamma}\\&
                +\abs{DF(x_2)(G(x_1)-G(x_2))}_{-\eta-\gamma}\lesssim \abs{x_1-x_2}_{-\eta-\gamma},
            \end{align*}
            for $x_1,x_2\in \mathcal{B}_{-\eta-\gamma}$. With this property, we get similarly to the first remainder term
            \begin{align*}
                 \abs{R^{G(y)}_{t,s}}_{-\eta-2\gamma}&\lesssim\int_0^1 \abs{(DG(y_s+ry_{t,s})-DG(y_s))G(y_s)}_{-\eta-2\gamma}~{\textnormal{d}} r~ \left[X\right]_{\gamma}\abs{t-s}^\gamma\\ &+\norm{DG(y)}_{\infty,\mathcal{L}(\mathcal{B}_{-\eta-2\gamma})}\left[R^y\right]_{2\gamma,-\eta-2\gamma}\abs{t-s}^{2\gamma}\\
                 &\lesssim\int_0^1 \abs{A_{-\sigma}N((DF(y_s+ry_{t,s})-DF(y_s))G(y_s))}_{-\eta-2\gamma}~{\textnormal{d}} r \abs{t-s}^{\gamma}\\&+\norm{y,G(y)}_{X,2\gamma,-\eta} \abs{t-s}^{2\gamma}\\
                 &\lesssim\int_0^1 \abs{(DF(y_s+ry_{t,s})-DF(y_s))G(y_s)}_{\alpha-2\gamma}~{\textnormal{d}} r \abs{t-s}^\gamma \\ &+\norm{y,G(y)}_{X,2\gamma,-\eta}\abs{t-s}^{2\gamma}\\
                 &\lesssim\int_0^1 \abs{(DF(y_s+ry_{t,s})-DF(y_s))G(y_s)}_{\alpha-\gamma}~{\textnormal{d}} r \abs{t-s}^\gamma+\norm{y,G(y)}_{X,2\gamma,-\eta}\abs{t-s}^{2\gamma}\\
                 &\lesssim \abs{y_t-y_s}_{\eta-\gamma}\abs{t-s}^\gamma+\norm{y,G(y)}_{X,2\gamma,-\eta}\abs{t-s}^{2\gamma}\lesssim \norm{y,G(y)}_{X,2\gamma,-\eta}\abs{t-s}^{2\gamma},
            \end{align*}
            so $R^{G(y)}\in C^{2\gamma}([0,T];\mathcal{B}_{-\eta-2\gamma})$. Collecting all the previous estimates proves the statement.
            \hfill $\Box$\smallskip
        \end{proof}\\

     Based on Lemma~\ref{no:q} we can derive an estimate for the solution of~\eqref{eq:introNew} which does not contain quadratic terms.
    \begin{corollary}\label{bound:int}
        Let $F$ satisfy the Assumption \ref{ass2} and let $(y,G(y))\in \mathcal{D}^{2\gamma}_{X,-\eta}$.
        Then we have
        \begin{align*}
            (z,z^\prime):=\left(\int_0^\cdot S_{\cdot-r} G(y_r)~{\textnormal{d}} \mathbf{X}_r, G(y_r)\right)\in \mathcal{D}^{2\gamma}_{X,-\eta}
        \end{align*}
        and the following estimate holds true
        \begin{align*}
            \norm{z,z^\prime}_{X,2\gamma,-\eta}\lesssim \abs{G(y_0)}_{-\eta}+\abs{DG(y_0)\circ G(y_0)}_{-\eta-\gamma}+ T^\gamma \norm{G(y), DG(y)\circ G(y)}_{X,2\gamma,-\eta}.
        \end{align*}
    \end{corollary}
     \begin{proof} The claim follows by Corollary \ref{cor:higherreg} combined with the fact that $(G(y), DG(y)\circ G(y))\in \mathcal{D}^{2\gamma}_{X,-\eta}$ as established in Lemma \ref{no:q}.

        \hfill $\Box$\smallskip
    \end{proof}
    \begin{corollary}\label{bound:s}
        Let $f$ and $F$ satisfy Assumptions \ref{ass2} 3) and let $(y,G(y))\in \mathcal{D}^{2\gamma}_{X,-\eta}$ be the solution of~\eqref{eq:introNew} on a time interval $[0,T]$ with initial data $y_0\in\mathcal{B}_{-\eta}$. Then we have
        \begin{align*}
        \|y,G(y)\|_{X,2\gamma,-\eta} \lesssim 1 + |y_0|_{\mathcal{B}_{-\eta}} + T^\theta \|y,G(y)\|_{X,2\gamma,-\eta},
        \end{align*}
where $\theta:=\min\{1-\delta_1,\gamma\} . $
    \end{corollary}
    \begin{proof}
        Since $(y,G(y))$ solves~\eqref{eq:introNew} we have
        \begin{align*}
            \norm{y, G(y)}_{X,2\gamma,-\eta}&\leq \norm{S(\cdot)y_0,0}_{X,2\gamma,-\eta}+\norm{\int_0^\cdot S_{\cdot-r} f(y_r)~{\textnormal{d}} r,0}_{X,2\gamma,-\eta}\\
            &+\norm{\int_0^\cdot S_{\cdot-r} A_{-\sigma}NF(y_r)~{\textnormal{d}} \mathbf{X}_r, A_{-\sigma}NF(y_r)}_{X,2\gamma,-\eta}.
        \end{align*}
        Here it is easily to see that  $(S(\cdot)y_0,0)\in \mathcal{D}^{2\gamma}_{X,-\eta}$ and $(\int_0^\cdot S_{\cdot-r} f(y_r)~{\textnormal{d}} r,0)\in \mathcal{D}^{2\gamma}_{X,-\eta}$ satisfy the bounds
        \begin{align*}
         \norm{S(\cdot)y_0,0}_{X,2\gamma,-\eta}\lesssim \abs{y_0}_{-\eta}, \quad \norm{\int_0^\cdot S_{\cdot-r} f(y_r)~{\textnormal{d}} r,0}_{X,2\gamma,-\eta}\lesssim (1+\norm{y}_{\infty, -\eta}) T^{1-\delta_1},
        \end{align*}
       using Assumption~\ref{ass2} 1).

        If we now combine these estimates with the one from Lemma \ref{no:q} and Lemma \ref{bound:int}, we get
        \begin{align*}
            \norm{y, G(y)}_{X,2\gamma,-\eta}&\lesssim \abs{y_0}_{-\eta}+(1+\norm{y}_{\infty, -\eta}) T^{1-\delta_1}\\
            &+\abs{G(y_0)}_{-\eta}+\abs{DG(y_0)\circ G(y_0)}_{-\eta-\gamma}+ T^\gamma \norm{G(y), DG(y)\circ G(y)}_{X,2\gamma,-\eta}\\
            &\lesssim 1+ \abs{y_0}_{-\eta} +T^{1-\delta_1}\norm{y,G(y)}_{X,2\gamma,-\eta} +T^{\gamma}\norm{y,G(y)}_{X,2\gamma,-\eta},
        \end{align*}
        where also the boundedness of the derivative $DG$ was used.
        \hfill $\Box$\smallskip
    \end{proof}\\

    Putting the previous deliberations together, we finally obtain.

        \begin{theorem}\em{(Existence of a global-in-time solution for~\eqref{eq:introNew})}\label{thm:global}
            Assume that $f$ and $F$ satisfy the Assumptions \ref{ass2} 3). Then there exists for every initial condition $y_0\in \mathcal{B}_{-\eta}$ a unique solution $(y,A_{-\sigma}NF(y))=(y,G(y))\in \mathcal{D}^{2\gamma}_{X,-\eta}\left([0,T]\right)$ to \eqref{eq:introNew}) such that $y$ fulfils \eqref{eq:mildSolution} for all $t\in [0,T]$.
        \end{theorem}
        \begin{proof}
         Due to Corollary~\ref{bound:s} we obtain by a standard concatenation argument (compare~\cite[Lemma 3.8]{HN21}) the following a-priori bound for the path component of the solution on a time interval $[0,T]$ for an arbitrary $T>0$. Namely, there exist constants $M_1,M_2>0$ such that
         \[
         \|y\|_{\infty,\mathcal{B}_{-\eta},[0,T]}\leq M_1 r e^{M_2T},
         \]
    where $r:=1\vee|y_0|_{\mathcal{B}_{-\eta}}$. Therefore, the local solution obtained in Theorem~\ref{local} cannot exhibit a finite-time blow-up as justified in~\cite[Theorem 3.9]{HN21}.
            \hfill $\Box$\smallskip
        \end{proof}
        \begin{remark}
         We believe that the theory developed in this work can be extended to time-dependent operators $A(t)$ generating parabolic evolution families $(U(t,s))_{t\geq s}$ as in~\cite{VeraarSchnaubelt}. Analogously, such operators satisfy for $t>s$ similar estimates to~\eqref{hg:1} and~\eqref{hg:2}, i.e.
            \begin{align}
                |(U(t,s)-\text{Id}) x|_{\mathcal{B}_{\alpha}}&\lesssim |t-s|^\sigma |x|_{\mathcal{B}_{\alpha+\sigma}},\\
     |U(t,s)x|_{\mathcal{B}_{\alpha+\sigma}}&\lesssim |t-s|^{-\sigma}|x|_{\mathcal{B}_\alpha}.
            \end{align}
             However the extrapolation scale has to be extended to cover this setting as well.
        \end{remark}

    \section{Random dynamical systems}\label{sec:rds}

        Based on our global well-posedness result, under suitable assumptions on the driving
        rough path, we are able to construct a random dynamical system corresponding to~\eqref{eq:intro}. To this aim we introduce some concepts from the theory of random dynamical systems~\cite{Arnold}. \\

        The following concept describes a model of the driving noise.

        \begin{definition}\label{mds}
            Let $(\Omega,\mathcal{F},\mathbb{P})$ stand for a probability space and
            $\theta:\mathbb{R}\times\Omega\rightarrow\Omega$ be a family of
            $\mathbb{P}$-preserving transformations (i.e.~$\theta_{t}\mathbb{P}=
            \mathbb{P}$ for $t\in\mathbb{R}$) having the following properties:
            \begin{description}
                \item[(i)] The mapping $(t,\omega)\mapsto\theta_{t}\omega$ is
                $(\mathcal{B}(\mathbb{R})\otimes\mathcal{F},\mathcal{F})$-measurable, where
                $\mathcal{B}(\cdot)$ denotes the Borel $\sigma$-algebra;
                \item[(ii)] $\theta_{0}=\textnormal{Id}_{\Omega}$;
                \item[(iii)] $\theta_{t+s}=\theta_{t}\circ\theta_{s}$ for all
                $t,s,\in\mathbb{R}$.
            \end{description}
            Then the quadrupel $(\Omega,\mathcal{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})$
            is called a metric dynamical system.
        \end{definition}

        \begin{definition}
            \label{rds}
            A continuous random dynamical system on a separable Banach space $\cX$ over a metric dynamical
            system $(\Omega,\mathcal{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})$
            is a mapping $$\varphi:[0,\infty)\times\Omega\times \cX\to \cX,
            \mbox{  } (t,\omega,x)\mapsto \varphi(t,\omega,x), $$
            which is $(\mathcal{B}([0,\infty))\otimes\mathcal{F}\otimes
            \mathcal{B}(\cX),\mathcal{B}(\cX))$-measurable and satisfies:
            \begin{description}
                \item[(i)] $\varphi(0,\omega,\cdot{})=\textnormal{Id}_{\cX}$
                for all $\omega\in\Omega$;
                \item[(ii)]$ \varphi(t+\tau,\omega,x)=
                \varphi(t,\theta_{\tau}\omega,\varphi(\tau,\omega,x)),
                \mbox{ for all } x\in \cX, ~t,\tau\in[0,\infty),~\omega\in\Omega;$
                \item[(iii)] $\varphi(t,\omega,\cdot{}):\cX\to \cX$ is
                continuous for all $t\in[0,\infty)$ and all $\omega\in\Omega$.
            \end{description}
        \end{definition}

        The second property in Definition~\ref{rds} is referred to as the
        cocycle property. The generation of a random dynamical system from an It\^{o}-type stochastic partial differential equation (SPDE) has been a long-standing open problem, since Kolmogorov's theorem breaks down for random fields parametrized by infinite-dimensional Banach spaces.~As a consequence it is not known how to 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~\cite{BRiedelScheutzow}.\\

        The next concept describes a model of the driving noise. Let $(\Omega,\mathcal{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})$ be a metric dynamical system as in Definition~\ref{mds}. We say that
        \begin{align*}
            \mathbf{X}=(X,\mathbb{X}):\Omega\to C^{\gamma}_{\text{loc}}([0,\infty);\mathbb{R}^d) \times C^{2\gamma}_{\text{loc}}([0,\infty);\mathbb{R}^{d\times d})
        \end{align*}
        is a continuous ($\gamma$-H\"older) rough path cocycle~\cite{BRiedelScheutzow} if $\mathbf{X}|_{[0,T]}$ is a continuous $\gamma$-H\"older rough path for every $T>0$ and for every $\omega\in\Omega$ and the following cocycle property holds true for every $s,t\in[0,\infty)$ and $\omega\in\Omega$
        \begin{align*}
            &X_{s,s+t}(\omega)= X_t(\theta_s\omega),\\
            &\mathbb{X}_{s,s+t}(\omega)=\mathbb{X}_{0,t}(\theta_s\omega).
        \end{align*}
        According to~\cite[Section 2]{BRiedelScheutzow} rough path lifts of various stochastic processes define cocycles.
        These include Gaussian processes with stationary increments under certain
        assumption on the covariance function~\cite[Chapter 10]{FrizHairer} and particularly apply to the fractional Brownian motion with Hurst index $H>\frac{1}{4}$.\\

        Based on Theorem~\ref{thm:global} we immediately derive the existence of a random dynamical system associated to~\eqref{eq:intro}. 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~\cite{BDS}.

        \begin{theorem}\label{thm:rds}  Let $\mathbf{X}=(X,\mathbb{X})$ be a $\gamma$-H\"older rough path cocycle.
            Under the assumptions of Theorem~\ref{thm:global}, the solution operator of~\eqref{eq:intro} generates a random dynamical system on $\mathcal{B}_{-\eta}$.
        \end{theorem}
        \begin{proof}
            In order to verify the cocycle property, regarding Theorem~\ref{thm:global}, we let $t,\tau\in\mathbb{R}_{+}$, recall that $G=A_{-\sigma}NF$  and compute
            \begin{align*}
                \varphi({t+\tau},\omega,y_0):=y_{t+\tau} &=S_{t+\tau}y_0 + \int\limits_{0}^{t+\tau} S_{t+\tau-r}
                f(y_{r}) ~{\textnormal{d}} r + \int\limits_{0}^{t+\tau} S_{t+\tau-r}
                G(y_{r}) ~{\textnormal{d}} \mathbf{X}_{r}\\
                & = S_t S_\tau y_0 + \int\limits_{0}^{\tau}  S_{t+\tau- r}
                f(y_{r}) ~{\textnormal{d}} r + \int\limits_{\tau}^{t+\tau} S_{t+\tau-r}
                f(y_{r}) ~{\textnormal{d}} r\\
                & + \int\limits_{0}^{\tau}S_{t+\tau-r} G(y_{r}) ~{\textnormal{d}}
                \mathbf{X}_{r} + \int\limits_{\tau}^{t+\tau} S_{t+\tau-r} G(y_{r})
                ~{\textnormal{d}} \mathbf{X}_{r}\\
                &=S_t \left( S_{\tau}y_0 + \int\limits_{0}^{\tau} S_{\tau-r}
                f(y_{r}) ~{\textnormal{d}} r + \int\limits_{0}^{\tau} S_{\tau-r} ~{\textnormal{d}}
                \mathbf{X}_{r}  \right)\\
                & + \int\limits_{0}^{t}  S_{t-r} f(y_{r+\tau}) ~{\textnormal{d}} r +
                \int\limits_{0}^{t} S_{t-r} G(y_{r+\tau}) ~{\textnormal{d}} \theta_{\tau}
                \mathbf{X}_{r}\\
                & = S_{t} y_{\tau} + \int\limits_{0}^{t} S_{t-r} f(y_{r+\tau})
                ~{\textnormal{d}} r + \int\limits_{0}^{t}S_{t-r}G(y_{r+\tau}) ~{\textnormal{d}}
                \theta_{\tau}\mathbf{X}_{r}.
            \end{align*}
            Note that the shift property of the rough integral~\eqref{Gintegral} is immediate, see \cite[Corollary 4.5]{HN20}. The $(\mathcal{B}([0,\infty))\otimes\mathcal{F}\otimes\mathcal{B}(\mathcal{B}_{-\eta}), \mathcal{B}(\mathcal{B}_{-\eta}))$-measurability of $\varphi$ follows by standard arguments. More precisely, one considers a sequence of (classical) solutions $(y^{n},(y^{n})')_{n\in\mathbb{N}}$ of~\eqref{eq:intro} corresponding to smooth approximations $(X^{n},\mathbb{X}^{n})_{n\in\mathbb{N}}$ of $(X,\mathbb{X})$. Obviously, the mapping $(t,X,\xi)\mapsto y^{n}_{t}$ is $(\mathcal{B}([0,T])\otimes\mathcal{F}\otimes\mathcal{B}(\mathcal{B}_{-\eta}), \mathcal{B}(\mathcal{B}_{-\eta}))$-measurable for any $T>0$. Since $y$ continuously depends on the rough input $X=(X,\mathbb{X})$, according to \cite[Lemma 3.12]{GHairer}, one concludes that $\lim\limits_{n\to\infty}y^{n}_t=y_{t}$. This gives the measurability of $y$ with respect to $\mathcal{F}\otimes \mathcal{B}(\mathcal{B}_{-\eta})$. Due to the time-continuity of $y$, we obtain by~\cite[Chapter~3]{CastaingValadier} the $(\mathcal{B}([0,T])\otimes\mathcal{F}\otimes\mathcal{B}(\mathcal{B}_{-\eta}), \mathcal{B}(\mathcal{B}_{-\eta}))$-measurability of the mapping $(t,\omega,\xi)\mapsto y_{t}$ for any $t\geq 0$.
            \hfill $\Box$\smallskip
        \end{proof}

    \section{Examples}\label{examples}
        Here we provide an application of our theory, specifying concrete examples for A and F. We let $p=2$ and note that in this case $\varepsilon$ can be chosen up to $\frac{3}{4}$. We  consider the operators
        \begin{align}
            \mathcal{A} u:= \sum_{i,j=1}^d \partial_i a_{ij}\partial_j u+a_i\partial_i u+b\quad \mathcal{C} u:= \sum_{i,j=1}^d \gamma_\partial (\nu_ia_{ij}\partial _ju+  a_iu),
        \end{align}
        where the coefficients $a_{ij}, a_j, b:\overline{\mathcal{O}}\to \mathbb{R}$ are all smooth,  $(a_{ij})_{i,j=1}^d$ is symmetric and uniform elliptic, so there exists some constant $k>0$ such that for all $\xi \in \mathbb{R}^d$ and $x\in \overline{\mathcal{O}}$ we have
        \begin{align*}
            \sum_{i,j=1}^d a_{ij}(x)\xi_i \xi_j\geq k \abs{\xi}^2.
        \end{align*}
        In this case $(\mathcal{A},\mathcal{C})$ is a normally elliptic boundary value problem and the $L^2(\mathcal{O})$-realization $A$, after a possible shift, has bounded imaginary powers \cite[Theorem 2.3]{DDHPV}.\\
        Furthermore, the symmetry of the coefficients leads to a nice characterization of the interpolation-extrapolation scale even for negative values. More precisely, due to \cite[Section 7]{Amann2} we get for $\beta\geq-3$
        \begin{align*}
            \mathcal{B}_{\frac{\beta}{2}}:=H^\beta_\mathcal{C}(\mathcal{O}):=
            \begin{cases}
                \{ u\in H^{\beta}(\mathcal{O}) : \mathcal{C} u =0 \},& \beta>\frac{3}{2}\\
                H^{\beta}(\mathcal{O}) ,& -\frac{1}{2}<\beta<\frac{3}{2}\\
                \left(H^{-\beta}(\mathcal{O})\right)^\prime, & -\frac{3}{2}<\beta\leq-\frac{1}{2}\\
                \{ u\in H^{-\beta}(\mathcal{O}) : \mathcal{C} u =0 \}^\prime, & \beta<-\frac{3}{2}.
            \end{cases}
        \end{align*}
        We are interested in the case $\beta=2(-\eta-2\gamma)$. Now, since $-\eta >-\gamma$ we have $-\eta-2\gamma>-\frac{3}{2}$ and therefore
        \begin{align*}
            \mathcal{B}_{-\eta-2\gamma}\hookrightarrow \mathcal{B}_{-\frac{3}{2}}=\{ u\in H^{3}(\mathcal{O}) : \mathcal{C} u =0 \}^\prime\hookrightarrow H^{-3}(\mathcal{O}),
        \end{align*}
        where the last embedding holds due to the fact that $H^3_0(\mathcal{O})\hookrightarrow \{ u\in H^3(\mathcal{O}) : \mathcal{C} u =0 \}$ and $\left(H^3_0(\mathcal{O})\right)^\prime=H^{-3}(\mathcal{O})$. We now want to give an example for a suitable diffusion coefficient $F$ that fulfils the conditions in Theorem ~\ref{local}. Based on the above considerations, it is sufficient to find a linear continuous function from $H^{-3}(\mathcal{O})$ to $H^{\tilde{\delta}}(\partial \mathcal{O})=B^{\tilde{\delta}}_{2,2}(\partial \mathcal{O})$ for a $\tilde{\delta}>0$. An example for this, is a slightly adapted lifting operator. More precisely, set $\nu:=-\frac{7}{2}-\tilde{\delta}$ and consider
        \begin{align*}
            \Lambda^{\nu}:H^{-3}(\mathbb{R}^d)\to H^{\tilde{\delta}+\frac{1}{2}}(\mathbb{R}^d),f\mapsto\mathcal{F}^{-1}(1+\abs{\cdot}^2)^{\frac{\nu}{2}}\mathcal{F} f,
        \end{align*}
        where $\mathcal{F}$ is the Fourier transform, which is a classic result in the theory of pseudo-differential operators, see for example \cite[Theorem 18.1.13]{Hor}.
        There are also linear and continuous  operators $e_\mathcal{O}:H^{-3}(\mathcal{O})\to H^{-3}(\mathbb{R}^d)$ and  $r_\mathcal{O}:H^{-3}(\mathbb{R}^d)\to H^{-3}(\mathcal{O})$. These are called {\em retraction/co-retraction}, see \cite[Theorem 4.2.2]{Triebel78}. Since the trace operator $\gamma_\mathcal{O}$ is also linear and continuous, we conclude that the operator $F:=\gamma_{\partial} r_\mathcal{O} \Lambda^{\nu} e_\mathcal{O}$ fulfils the Assumption \ref{ass2} 2) such that Theorem \ref{local} provides a local-in-time solution in this case. Moreover, since $F$ is linear and bounded, the same holds true for $G=A_{-\sigma} NF$. This means that $F$ satisfies the Assumption \ref{ass2} 3) and Theorem~\ref{thm:global} entails a global-in-time solution.
          \begin{comment}
        \textcolor{blue}{For the sake of completeness we provide an example in the Young case for Dirichlet and Neumann boundary noise. $F$ has to be only $C^2_b$ in this case, compare~\cite[Section 8.3]{FrizHairer} and need Assumption~\ref{ass2} 2) for $\theta\in\{0,\gamma\}$.}
        As pointed out in Remark \ref{rem:dirichlet} the crucial property to treat the noise on the boundary is that we can choose $\gamma > 1- \varepsilon$. In the case of Neumann conditions this is, as seen before, possible for $\gamma \in (\frac{1}{3},\frac{1}{2}]$. If we now wanna use Dirichlet conditions instead, than the Neumann map gets replaced by $D:B^{\alpha-\frac{1}{p}}_{p,p}\to D(A^{\varepsilon_D})$ for $\varepsilon_D < \frac{1}{2p}<\frac{1}{4}$. But that means $1-\varepsilon_D>\frac{3}{4}>\gamma$, so it is not possible to treat Dirichlet conditions for those $\gamma$-H\"older rough paths. But if we leave the rough path regime, and take a look at some path $X\in C^{\widetilde{\gamma}}([0,T];\mathbb{R})$ for $\widetilde{\gamma}\in(\frac{1}{2},1]$. Then for $\varepsilon_D:=\frac{1}{2p}-\widetilde{\delta}$ with $\widetilde{\delta}>0$ small, we can choose $\widetilde{\gamma}:=\frac{5}{6}+ \hat{\delta}<1$ with $\hat{\delta}>\widetilde{\delta}$ to satisfy $1-\varepsilon_D<\widetilde{\gamma}$. Since we are in the Young regime with this regularity we can define the Integral without a second order process like in the rough path case, recall \eqref{young}.
        \end{comment}
        \begin{remark}
            Before we conclude,
            we compare our results to the ones obtained in~\cite{DuncanMaslowski, DuncanMaslowski2} for {\em additive} infinite-dimensional fractional noise. To this aim, let $U$ and $V$ stand for two separable Hilbert spaces. For a $U$-cylindrical fractional Brownian motion $(B^H_t)_{t\geq 0}$ it is known (\cite[Corollary 3.1]{DuncanMaslowski} for $H>\frac{1}{2}$ and \cite[Corollary 11.9]{DuncanMaslowski2} for $H<\frac{1}{2}$) that the stochastic convolution
            $$\int_0^t S_{t-r}\Phi ~{\textnormal{d}} B^H_r ~~\text{ is well-defined if } ~~\|S(t)\Phi\|_{\mathcal{L}_2(U,V)}<t^{-\theta},$$
            where $\mathcal{L}_2(U,V)$ denotes the space of Hilbert-Schmidt operators from $U$ to $V$, $\Phi\in\mathcal{L}(U,V)$ and $\theta<H$.  Consequently, in order to incorporate boundary noise given by an  $U:=L^2(\partial\mathcal{O})$-cylindrical fractional Brownian motion with covariance operator $Q^{\frac{1}{2}}\in \mathcal{L}_2(U)$, one has to verify that
            \begin{align*}
                \| A S(t) N Q^{\frac{1}{2}}\|_{\mathcal{L}_2(U,V)}\leq t^{-\theta},
            \end{align*}
            where $V:=L^2(\mathcal{O})$ and $\theta<H$. Setting $\mathcal{B}_\varepsilon:=D(A^\varepsilon)$ as in Section~\ref{main}, one infers that
            \begin{align*}
                \| A S(t) N Q^{\frac{1}{2}}\|_{\mathcal{L}_2(U,V)}\leq \|A S(t)\|_{\mathcal{L}(\mathcal{B}_\varepsilon,V)}\|N\|_{\mathcal{L}(U,\mathcal{B}_\varepsilon)}\|Q^{\frac{1}{2}}\|_{\mathcal{L}_2(U)} \leq c t^{\varepsilon-1} .
            \end{align*}
            This leads to the condition $\varepsilon>1-H$. Since $\varepsilon<\frac{3}{4}$ (recall Corollary~\ref{cor:RoughConvInDomain}), this means that it is possible to deal with fractional Neumann boundary noise for $H>\frac{1}{4}$.
            The condition $\varepsilon>1-H$ is consistent with the result of Corollary~\ref{cor:RoughConvInDomain}, which is established in the more general setting of rough path theory. Moreover, due to the multiplicative structure of the noise, the theory of extrapolation operators is additionally required.
        \end{remark}

redpajama_set_name: RedPajamaArXiv


Actress Raai Laxmi injured her leg severely while shooting for her upcoming movie, Julie 2. The team had to take a break from the shooting schedule due to this. The actress tweeted a picture of her swollen leg and expressed her unhappiness over the unfortunate incident. Julie 2 is a suspense thriller directed by Deepak Shivdasani. The movie will be Raai Laxmi's 50th film and also her first full-length Hindi film. We wish her a speedy recovery and hope to see her back in action!


In about three years from now, Bhutan Table Tennis Federation (BTTF) expects to come up with a group of Para table tennis player in the country.
The federation trained five children with disabilities (CWDs), all students, during two-day Para table tennis festival held in Thimphu.
The festival, which was held for the first time in the country, saw some junior table tennis player participate along with CWDs.
The national Paralympic committee is in the process of exploring persons living with disabilities (PWDs).
The committee was affiliated with Bhutan Olympic Committee last year. Since then, two Paralympic athletes took part in third Para Asian games in Indonesia last year.
Coach and administration assistant with BTTF, Namgay Dorji, said that the festival focused on bringing in the CWDs into sports and to provide them an opportunity to represent the country in the future.
"With the start of national Paralympic committee, the physically challenged person have the opportunities like the normal athletes to be at international competitions," said Namgay Dorji.
An official from national Paralympic Committee said that the committee was planning inclusive sports that would enable training and development of normal and sportspersons with disabilities together.
Today there are only four Para-athletes in the country.
There are about 22,000 Bhutanese living with disabilities, according to 2017 study by the Gross National Happiness Commission.
The federation in collaboration with Bhutan Paralympic Committee conducted the festival that ended yesterday.


Working from your home has started becoming a legitimate option for millions to earn money even though it used to be an idea that no one took seriously. There many unique ways to make money at home, but this article is going to look closely at 6 unique home business ideas.
Interior designers choose a refinished floor versus a painted floor because refinished hardwood floors have a more classic look. A dining room can be made more formal with a stained floor. Rooms that need to appear more spacious do well with a finish over natural wood. People sometimes choose a darker finish because it does not show scuff marks as easily as lighter and painted floors.
No Prima Donna's – It's tempting to get a swelled head when things are busy and you're in demand. But when things are slow, your clients have choices as to who they want to work with. And you may not be their first choice.
The last thing you could do is to purchase a mirror that is foldable and place it on one corner of your living room. You could generate a sort of triptych out of this. In the old times, a triptych is a set of three writing tablets hinged or coupled together. This is a beautiful decoration once situated properly.
A new house is always sterile and characterless; so about this time the average person starts thinking about colour charts and carpets and long conversations with decorators and builders. It doesn't generally take long before the new house starts to look like a home but invariably there is something very important missing.
B.: Well, fortunately the trends in interiors don't move as quickly and as in your face as they do in fashion-if they did we'd be redecorating our projects before they're even finished! But that said, I feel trends in home design should be injected exactly the same way they do in a wardrobe: If you choose your key pieces (large upholstery + rugs) in classic lines and a neutral color you can throw in some trendy elements in a small way (pillows, paint, accessories) and look current without looking like a trend victim.
As you can see from this article, interior design isn't just for professionals. The value of research and time used on the designs cannot be stressed enough. With a few simple tools at your disposal, armed with your imagination, happen to be on your way to designing the right space for you!


Are your employees still enrolling in benefits with paper forms? Simplify the process across your organization with secure, electronic enrollment. Integrated with your HR cloud modules, open enrollment and new employee enrollment is now easier than ever before.
Help employees get the most out of their benefits…anytime, from any device. Whether on a desktop or mobile device, selecting, comparing or changing to benefit elections during enrollment is both intuitive and simple.
Managing benefits enrollment can be time consuming and costly, but with employee notifications and enrollment status tracking, benefits enrollment can be completed in record time.
With benefits enrollment, everyone wins… from employees to the business owner. Employees want to take full advantage of earned benefits, and employers need to retain employees while reducing administrative costs.
Whether it's saving time or better understanding available benefits, your staff will love the convenience of electronic benefits enrollment. Simplify how your employees discover, compare and select their benefits.
You already know how challenging it can be to manage benefits during open enrollment and for new hires, especially if you're still using paper forms. With electronic enrollment, employees can be guided step-by-step through the process for the specific benefits they are eligible for.
What's more, the entire process from onboarding to benefits enrollment and payroll is integrated, minimizing the time involved and potential for errors.
Whether it's managing dependents or comparing plan options, your employees will be empowered to understand and more fully take advantage of employer-sponsored and voluntary benefits.
And without the stress and disruption of paper-based enrollment, your employees can focus on making the right selections and returning to work.


text:
Satish Sharma, MD, FACS
Chairman & Chief Executive Officer,
You First Services Group of Companies
Contact You First Service, Inc.
"Achievement and success are meaningful only if they can help bring hope and comfort to lives of those who need them the most."
Welcome to You First Services Group of Companies (YFS). Since our company's founding, our chief objective has been to enhance and improve lives through research and development.
Unique and innovative concepts have proven vital for the sustenance of our organization, leading to the state of art healthcare related inventions and accessible technologies for individuals and organizations across the world.
YFS derives its energy from the contributions of our creative staff, skillful expertise, high-profile partnerships, scientific ventures and collaborative research. We value the contribution of each individual team member along with our partners. With these fundamentals, we are committed to growing our platform and creating diverse employment opportunities.
Constructed on a foundation of shared values, YFS seeks to provide its employees and partners with ways to explore their potential while empowering them to further refine their strengths for mutual growth. With world-renowned physicians, scientists, engineers and other technocrats contributing to it's projects, YFS in a short span has been able to introduce alternatives that will improve well-being of our community, decrease the cost of healthcare and reduce the impact of negative environmental factors. We have established global partnerships with like-minded organizations at national and international front to achieve our mutual objectives.
We are firmly committed to support child education and development of research as they can lay a strong foundation for a vibrant growth of the society and economic development at multiple levels. We will work closely with academic institutions to support scientific research and create engaging opportunities for its students.
While fulfilling our expansive range of responsibilities, YFS will establish platforms where the world's best talents will have the opportunity to find expression of their thoughts in a tangible way.

redpajama_set_name: RedPajamaCommonCrawl


text:
How to Use Community Q&A Hubs to Boost App Downloads
0 comments, 25/05/2017, by Pingler, in General
If you're like most app developers, the end-goal of every product is to generate plenty of downloads and installs. With so many apps on each of the marketplaces today, it can be difficult to compete with other, more established developers. Fortunately, the internet provides developers with all of the tools needed to boost exposure for their apps and ensure people are paying attention – if you know how to do it properly. Websites like Reddit, Yahoo Answers and Quora can be fertile ground for app developers in this regard: today, we'll walk you through how you can get the most out of these community hubs when it comes to downloading apps.
Fluff Your Profile
When providing insight on Q&A platforms that allow for customization, it's important to establish your credibility at the beginning. In the case of platforms like LinkedIn and Quora, you have the ability to flourish your profile by adding relevant information that establishes authority. Be sure to link back to your website, social media profiles and provide descriptions where possible of your expertise. This way, you're not pinging noise at people who might otherwise think you're just some random advertiser trying to pad their app downloads.
Peruse Relevant Discussions
On sites like Reddit and Quora, it's very easy to find topics and discussions that pertain to your app's particular topic. By providing genuine insight and addressing what benefits/features can be expected from the app, you can both provide useful information and increase app downloads. On specific Q&A community sites like Quora and Yahoo Answers, it is much easier to answer peoples' questions and gain credibility as a community expert. In the case of platforms like Reddit, however, the plugging of your app will have to be much more subtle in order to be successful. In either case, you will be building links back to your app, convincing readers and ultimately transforming those reads and clicks into downloads and installs.
Keep Tabs on Your Progress
In order to get the most value of your app promotion on community hubs, you need to always be tracking your progress. This will help you quickly determine which promotions and platforms are paying off and which ones are not. Through the use of UTM codes, you can track each specific link you post on each reply and platform. With a bit of spreadsheet work, you can keep track of who is coming from where and how many visits/downloads are being generated by each. In the initial stages, this can be a great component of broader A/B testing: you'll figure out which audiences, platforms and approaches work best and can reform your pitch for the future.
Finding ways to promote your app in organic situations can be difficult. Too much promotion and it's like pinging noise or static at people. However, through the use of organic advice, a bit of preparation and careful refinements, you can use multiple community-based platforms to dramatically increase exposure for your app and ensure its long-term success. If you've already built an app, leave us a comment below and tell us what it does!
Quick and Easy Ways to Drum Up Content
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How to Build Community and Interaction on Your Website
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AppsDownloadsQ&A Hubs
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Black Friday 2015 Sale – 50% OFF!


While on Prince Eric's boat, Ariel met a wonderful kitten and she named her Treasure. The two bonded instantly and Ariel decided to turn the lovely kitty into a Palace Pet. Help her groom Treasure into a true royalty. Take a pair of tweezers and one by one remove all the bugs from her fur. Then take a brush and gently remove the pebbles from her hair. Play a fun memory game to unlock a sponge and wash Treasure up and remove the ink stains on her and then play another mini game, a hidden object this time in the bathroom in order to unlock a soap. Wash the kitten and rinse with water. Then you can use a towel to wipe her clean and finally disinfect her wounds with an antiseptic solution. Wash her clothes as well. Have a great time playing this amazing game!


text:
require 'spec_helper'

describe MarkLogic::Queries::CollectionQuery do

  describe "#to_xqy" do
    it "should handle a single collection" do
      q = MarkLogic::Queries::CollectionQuery.new("foo")
      expect(q.to_xqy).to eq(%Q{cts:collection-query(("foo"))})
    end

    it "should handle multiple collections" do
      q = MarkLogic::Queries::CollectionQuery.new(["foo", "bar", "baz"])
      expect(q.to_xqy).to eq(%Q{cts:collection-query(("foo","bar","baz"))})
    end
  end
end

redpajama_set_name: RedPajamaGithub