text:
\section{Introduction}

We consider a so-called quantum graph; that is, is a metric graph $G$, equipped with a diffusion operator on each edge and certain vertex conditions. Our terminology follows \cite[Chap.~1]{BeKu}, we list here only the most important concepts. The graph $G$ consists of a finite set of vertices $\mathsf{V} = \{\mathsf{v}\}$ and a finite set $\mathsf{E} = \{\mathsf{e}\}$ of edges connecting the vertices. We denote by $m=|\mathsf{E}|$ the number of edges and by  $n=|\mathsf{V}|$ the number of vertices. In general, a metric graph is assumed to have directed edges; that is edges having an origin and a terminal vertex. In our case, dealing with self-adjoint operators, we can just consider undirected edges. Each edge is assigned a positive length $\ell_{\mathsf{e}}\in (0,+\infty)$, and we denote by $x\in [0,\ell_{\mathsf{e}}]$ a coordinate of $G$. We assume that $G$ is simple; that is, there are no multiple edges connecting two vertices, and there are no loops at any of the vertices in $G$.

The metric graph structure enables one to speak about functions $f$ on $G$, defined along the edges such that for any coordinate $x$, the function takes its value $f(x)$. If we emphasize that $x$ is taken from the edge $\mathsf{e}$, we write $f_{\mathsf{e}}(x)$.

To write down the vertex conditions in the form of equations, for a given function $f$ on $G$ and for each $\mathsf{v}\in\mathsf{V}$, we introduce the following notation. For any $\mathsf{v}\in\mathsf{V}$, we denote by $\mathsf{E}_{\mathsf{v}}$ the set of edges incident to the vertex $\mathsf{v}$, and by $d_{\mathsf{v}}=|\mathsf{E}_{\mathsf{v}}|$ the degree of $\mathsf{v}$. Let $f_{\mathsf{e}}(\mathsf{v})$ denote the value of $f$ in $\mathsf{v}$ along the edge $\mathsf{e}$ in the case $\mathsf{e}\in \mathsf{E}_{\mathsf{v}}$. Let $\mathsf{E}_{\mathsf{v}}=\{\mathsf{e}_1,\dots ,\mathsf{e}_{d_{\mathsf{v}}}\}$, and define
\begin{equation}\label{eq:Fv}
F(\mathsf{v})=\left(f_{\mathsf{e}}(\mathsf{v})\right)_{\mathsf{e}\in \mathsf{E}_{\mathsf{v}}}=\begin{pmatrix}
	f_{\mathsf{e}_1}(\mathsf{v})\\
	\vdots\\
	f_{\mathsf{e}_{d_{\mathsf{v}}}}(\mathsf{v})
\end{pmatrix}\in \mathbb{R}^{d_{\mathsf{v}}},
\end{equation}
the vector of the function values in the vertex $\mathsf{v}$.

Let $I_{\mathsf{v}}$ be the bi-diagonal matrix
\begin{equation}\label{eq:Iv}
I_{\mathsf{v}}=\begin{pmatrix}
	1 & -1 & & \\
	& \ddots & \ddots & \\
	& & 1 & -1
	\end{pmatrix}\in\mathbb{R}^{(d_{\mathsf{v}}-1)\times d_{\mathsf{v}}}.
\end{equation}
It is easy to see that if we set
\begin{equation}\label{eq:contv}
I_{\mathsf{v}}F(\mathsf{v})=0_{\mathbb{R}^{d_{\mathsf{v}}-1}},
\end{equation}
this means that all the function values coincide in $\mathsf{v}$. If it is satisfied for each vertex $\mathsf{v}\in\mathsf{V}$ then $f$ is continuous on $G$.

Similarly, for a differentiable function $f$ defined on $G$, we set
\begin{equation}\label{eq:Fvv}
F'(\mathsf{v})=\left(f'_{\mathsf{e}}(\mathsf{v})\right)_{\mathsf{e}\in\mathsf{E}_{\mathsf{v}}}=\begin{pmatrix}
	f'_{\mathsf{e}_1}(\mathsf{v})\\
	\vdots\\
	f'_{\mathsf{e}_{d_{\mathsf{v}}}}(\mathsf{v})
\end{pmatrix}\in \mathbb{R}^{d_{\mathsf{v}}},
\end{equation}
the vector of the function derivatives in the vertex $\mathsf{v}$. We will assume throughout the paper that derivatives are taken in the directions away from the vertex $\mathsf{v}$ (i.e.~into the edge), see \cite[Sec.~1.4.]{BeKu}.

We aim to analyse the existence, uniqueness and  regularity of solutions of the problem written formally as
\begin{equation}\label{eq:stochnet}
\left\{\begin{aligned}
\dot{u}_{\mathsf{e}}(t,x) & =  (c_{\mathsf{e}} u_{\mathsf{e}}')'(t,x)-p_{\mathsf{e}}(x) u_{\mathsf{e}}(t,x) &x\in (0,\ell_{\mathsf{e}}),\; t\in(0,T],\; \mathsf{e}\in\mathsf{E},\;\; & (a)\\
\dot{\Phi}_{\mathsf{v}}(t)& = I_{\mathsf{v}}U(t,\mathsf{v}),\;   &t\in(0,T],\; \mathsf{v}\in\mathsf{V},\;\;& (b)\\
\dot{\beta}_{\mathsf{v}}(t) & = C(\mathsf{v})^{\top}U'(t,\mathsf{v}), & t\in(0,T],\; \mathsf{v}\in\mathsf{V},\;\;& (c)\\
 u_{\mathsf{e}}(0,x) & =  u_{0,\mathsf{e}}(x), &x\in [0,\ell_{\mathsf{e}}],\; \mathsf{e}\in\mathsf{E}.\;\;& (d)
\end{aligned}
\right.
\end{equation}
Here $\dot{u}_{\mathsf{e}}$ and $u_{\mathsf{e}}'$ denotes the time and space derivative, respectively, of $u_{\mathsf{e}}$. The processes $(\Phi_{\mathsf{v}}(t))_{t\in [0,T]}$ are independent standard $\mathbb{R}^{d_{\mathsf{v}}-1}$-valued Brownian motions, while $(\beta_{\mathsf{v}}(t))_{t\in [0,T]}$ are independent standard scalar Brownian motions. We assume that $\beta_{\mathsf{v}}\neq 0$ at least for one $\mathsf{v}\in\mathsf{V}$, and we treat the case $\Phi_{\mathsf{v}}=0$ for all $\mathsf{v}\in\mathsf{V}$ separately in Section \ref{sec:stochKir}. The functions $c_{\mathsf{e}}$ are (variable) diffusion coefficients or conductances, and we assume that
\[0<c_{\mathsf{e}}\in C^1[0,\ell_{\mathsf{e}}],\quad \mathsf{e}\in\mathsf{E}.\]
The functions $p_{\mathsf{e}}$ are nonnegative, continuous functions, hence
\begin{equation
0\leq p_{\mathsf{e}}\in C[0,\ell_{\mathsf{e}}],\quad \mathsf{e}\in\mathsf{E}.
\end{equation}
Analogously to \eqref{eq:Fv}, for each $\mathsf{v}\in\mathsf{V}$ and $\mathsf{E}_{\mathsf{v}}=\{\mathsf{e}_1,\dots ,\mathsf{e}_{d_{\mathsf{v}}}\}$ the 1-row matrix in $(\ref{eq:stochnet}c)$ is defined by
\[C(\mathsf{v})^{\top}=\left(c_{\mathsf{e}_1}(\mathsf{v}), \dots ,c_{\mathsf{e}_{d_{\mathsf{v}}}}(\mathsf{v})\right)\in\mathbb{R}^{1\times d_{\mathsf{v}}}.\]
It is clear by definition, that $(\ref{eq:stochnet}b)$ consists of
\begin{equation}\label{eq:conteqnumber}
\sum_{\mathsf{v}\in\mathsf{V}}(d_{\mathsf{v}}-1)=2m-n
\end{equation}
equations. At the same time, $(\ref{eq:stochnet}c)$ consists of $n$ equations. Hence, we have altogether $2m$ (boundary or vertex) conditions in the vertices.

In equation~$(\ref{eq:stochnet}d)$ we pose the initial conditions on the edges.

The main motivation to consider \eqref{eq:stochnet} is to generalize the classical concept of boundary noise, see, for example, \cite{DPZ93, BZ06} for Gaussian noise and \cite[Sec.~15.1]{PeZa} for L\'evy noise, to quantum graphs.

The paper is organized as follows. In Section \ref{sec:determnetwork} we first investigate the deterministic version of \eqref{eq:stochnet}. We rewrite the system in the form of an abstract Cauchy problem governed by the operator $A$ and prove well-posedness by showing that $A$ generates a strongly continuous, analytic contraction semigroup, see Proposition \ref{prop:eaASttul}. The statement of Proposition \ref{prop:Diropletezik} is the existence and boundedness of the so-called Dirichlet-operator. Using this and results from \cite{DPZ93}, we can treat two versions of the stochastic problem. In Theorem \ref{theo:boundpertKircont} we prove the existence of the mild solution of \eqref{eq:stochnet} with continuous paths in the fractional domain space of $A$ of order $\alpha<-\frac{1}{4}$. In Section \ref{sec:stochKir} we modify \eqref{eq:stochnet} by setting $0$ on the left-hand-side of $(\ref{eq:stochnet}b)$, that is, assuming continuity in all vertices, cf.~\eqref{eq:contv}. In Theorem \ref{theo:boundpertKir} we prove the existence of the mild solution for this modified problem with continuous paths in the Hilbert space $\mathcal{H}$, see \eqref{eq:E2}. Assuming that the vertex values of the eigenfunctions of $A$ are uniformly bounded, we can show in Theorem \ref{theo:eigenfctest} that the mild solutions have continuous paths in the fractional domain space of order $\alpha<\frac{1}{4}$. This is the case, for example, if all diffusion coefficients are constant and equal $1$, see Corollary \ref{cor:Laplace}. These regularity results are the analogues of the ones obtained by Da Prato and Zabczyk in \cite{DPZ93} in case of a single interval and classical boundary Dirichlet or Neumann noise.

\section{Heat equation on a network}\label{sec:determnetwork}

\subsection{The abstract Cauchy problem}\label{subsec:systeq}

We start with the deterministic problem
\begin{equation}\label{eq:netcp}
\left\{\begin{aligned}
\dot{u}_{\mathsf{e}}(t,x) & =  (c_{\mathsf{e}} u_{\mathsf{e}}')'(t,x)-p_{\mathsf{e}}(x) u_{\mathsf{e}}(t,x) &x\in (0,\ell_{\mathsf{e}}),\;t>0,\; \mathsf{e}\in\mathsf{E},\;\; & (a)\\
0 & = I_{\mathsf{v}}U(t,\mathsf{v}),\;   &t>0,\; \mathsf{v}\in\mathsf{V},\;\;& (b)\\
0 & =  C(\mathsf{v})^{\top}U'(t,\mathsf{v}), & t>0,\; \mathsf{v}\in\mathsf{V},\;\;& (c)\\
 u_{\mathsf{e}}(0,x) & =  u_{0,\mathsf{e}}(x), &x\in [0,\ell_{\mathsf{e}}],\;\; \mathsf{e}\in\mathsf{E}.\;\;& (d)
\end{aligned}
\right. ,
\end{equation}
where $0$ denotes the constant $0$ vector of dimension $d_{\mathsf{v}}$ on the left-hand-side of $(\ref{eq:netcp}b)$.

We would like to rewrite our system in the form of an abstract Cauchy problem. Recall that $m=|\mathsf{E}|$ the number of edges and $n=|\mathsf{V}|$ the number of vertices in $G$. First we consider the Hilbert space.
\begin{equation}\label{eq:E2}
\mathcal{H}\coloneqq \prod_{\mathsf{e}\in\mathsf{E}}L^2\left(0,\ell_{\mathsf{e}}\right)
\end{equation}
as the \emph{state space} of the edges, endowed with the natural inner product
\[\langle f,g\rangle_{\mathcal{H}}\coloneqq \sum_{\mathsf{e}\in\mathsf{E}} \int_0^{\ell_{\mathsf{e}}} f_{\mathsf{e}}(x)g_{\mathsf{e}}(x) dx,\qquad
f=\left(f_{\mathsf{e}}\right)_{\mathsf{e}\in\mathsf{E}},\;g=\left(g_{\mathsf{e}}\right)_{\mathsf{e}\in\mathsf{E}}\in \mathcal{H}.\]
On $\mathcal{H}$ we define the operator
\begin{equation}\label{eq:opAmax}
A_{\max}\coloneqq \mathrm{diag}\left(\frac{d}{dx}\left(c_{\mathsf{e}} \frac{d}{dx}\right)-p_{\mathsf{e}} \right)_{\mathsf{e}\in\mathsf{E}}
\end{equation}
with maximal domain
\begin{equation}\label{eq:domAmax}
D(A_{\max}) =\mathcal{H}^2\coloneqq \prod_{\mathsf{e}\in\mathsf{E}} H^2(0,\ell_{\mathsf{e}}).
\end{equation}
We also introduce the \emph{boundary space}
\begin{equation}\label{eq:Y}
\mathcal{Y}\coloneqq \ell^2(\mathbb{R}^{2m})\cong \mathbb{R}^{2m}.
\end{equation}

Notice that for fixed $u\in D(A_{\max})$, the boundary (or vertex) conditions can be written as
\begin{equation}\label{eq:contKir}
0_{\mathbb{R}^{d_{\mathsf{v}}}}=I_{\mathsf{v}}U(\mathsf{v}),\; 0=C(\mathsf{v})^{\top}U'(\mathsf{v}),\quad \mathsf{v}\in\mathsf{V},
\end{equation}
cf.~$(\ref{eq:netcp}b)$ and $(\ref{eq:netcp}c)$.

\begin{rem}
Define $A_{\mathsf{v}}$ as the square matrix that arises from $I_{\mathsf{v}}$ by inserting an additional row containing only $0$'s; that is,
\begin{equation}\label{eq:Av}
A_{\mathsf{v}}=\begin{pmatrix}
	1 & -1 & & \\
	& \ddots & \ddots & \\
	& & 1 & -1 \\
	0 & \hdots & 0 & 0
	\end{pmatrix}\in\mathbb{R}^{d_{\mathsf{v}}\times d_{\mathsf{v}}}.
\end{equation}
Furthermore, let $B_{\mathsf{v}}$ be the square matrix defined by
\begin{equation}\label{eq:Bv}
B_{\mathsf{v}}=\begin{pmatrix}
	0 & \hdots &  0\\
	\vdots   &  & \vdots\\
	0 & \hdots &  0\\
	c_{\mathsf{e}_1}(\mathsf{v}) & \hdots &  c_{\mathsf{e}_{d_{\mathsf{v}}}}(\mathsf{v}),
	\end{pmatrix}\in\mathbb{R}^{d_{\mathsf{v}}\times d_{\mathsf{v}}}.
\end{equation}
It is straghtforward that for a fixed $\mathsf{v}\in\mathsf{V}$, equations \eqref{eq:contKir} have the form
\begin{equation}\label{eq:AvBv}
A_{\mathsf{v}}U(\mathsf{v})+B_{\mathsf{v}}U'(\mathsf{v})=0_{\mathbb{R}^{d_{\mathsf{v}}}},
\end{equation}
where the $d_{\mathsf{v}}\times (2d_{\mathsf{v}})$ matrix $\left(A_{\mathsf{v}},B_{\mathsf{v}}\right)$ has maximal rank. Thus, our vertex conditions have the form as in \cite[Sec.~1.4.1]{BeKu}.
\end{rem}

We now define the feedback operator $B\colon D(A_{\max})\to \mathcal{Y}$ by
\begin{equation}\label{eq:opB}
\begin{split}
D(B) & = D(A_{\max});\\
B u & =\begin{pmatrix}
	\left(I_{\mathsf{v}}U(\mathsf{v})\right)_{\mathsf{v}\in\mathsf{V}}\\
	\left(C(\mathsf{v})^{\top}U'(\mathsf{v})\right)_{\mathsf{v}\in\mathsf{V}}
\end{pmatrix},
\end{split}
\end{equation}
where the first ''block'' of $Bu$ has $\sum_{\mathsf{v}\in\mathsf{V}}(d_{\mathsf{v}}-1)=2m-n$ coordinates (see also \eqref{eq:conteqnumber}), while the second ''block'' has $n$ coordinates. Hence, $B$ maps indeed into $\mathbb{R}^{2m}$.

With these notations, we can finally rewrite \eqref{eq:netcp} in form of an abstract Cauchy problem. Define
\begin{align}\label{eq:amain}
A &\coloneqq A_{\max}\\
D(A)&\coloneqq\{u\in D(A_{\max})\colon Bu=0_\mathcal{Y}\}.
\end{align}
Using this, \eqref{eq:netcp} becomes
\begin{equation}\label{eq:acp}
\left\{\begin{array}{rcll}
\dot{u}(t)&=& A u(t), &t> 0,\\
u(0)&=& u_0,
\end{array}
\right.
\end{equation}
with $u_0=(u_{0,1},\dots ,u_{0,m})^{\top}$.

\subsection{Well-posedness of the abstract Cauchy problem}

\begin{prop}\label{prop:Diropletezik}
The operators $A_{\max}$ and $B$ satisfy the conditions in \cite[Section 15.1]{PeZa} and \cite[Section 1]{DPZ93}, that is
\begin{enumerate}[(i)]
	\item the operator $A$ defined in \eqref{eq:amain} generates a $C_0$-semigroup on $\mathcal{H}$;
	\item for $\lambda\in\rho(A)$ there exists a bounded operator $D_{B,\lambda}\colon \mathcal{Y}\to \mathcal{H}$ such that
	\begin{equation}\label{eq:Dirop}
	D_{B,\lambda}=\left(B\mid_{\mathrm{Ker}(\lambda-A_{\max})}\right)^{-1}.
	\end{equation}
\end{enumerate}
\end{prop}
\begin{proof}$ $

(i) Follows directly by \cite[Prop.~2.5]{KS21}.

(ii) We will show that the assumptions (a)-(d) of \cite[(1.13)]{Gr87} are satisfied for $A_{\max}$ and $B$. Then, by \cite[Lemma 1.2]{Gr87} the existence of the bounded operator of $D_{B,\lambda}$ in \eqref{eq:Dirop} follows for every $\lambda\in\rho(A).$

The operator $\left(A_{\max},D(A_{\max})\right)$ is densely defined and closed on $\mathcal{H}$, hence assumption (a) is satisfied.
The boundary operator as a mapping
\[B\colon \left(D(A_{\max}),\|\cdot \|_{\mathcal{H}^{2}}\right)\to \mathcal{Y}\]
is bounded by Sobolev embedding, which is assumption (b). The statement of (d) is exactly (i) above.

It remains only to prove (c); that is, $\mathrm{Im}\,B=\mathbb{R}^{2m}$, which is the assertion of Proposition \ref{prop:Bsurj}.
\end{proof}

\begin{prop}\label{prop:eaASttul}
The operator $(-A,D(A))$, where $A$ is defined in \eqref{eq:amain}, is the operator associated with the form
\begin{equation}\label{eq:ea}
\begin{split}
\EuFrak{a}(u,v)&=\sum_{\mathsf{e}\in\mathsf{E}}\int_0^{\ell_{\mathsf{e}}}c_{\mathsf{e}}(x)u_{\mathsf{e}}'(x)v_{\mathsf{e}}'(x)\, dx+\sum_{\mathsf{e}\in\mathsf{E}}\int_0^{\ell_{\mathsf{e}}} p_{\mathsf{e}}(x)u_{\mathsf{e}}(x)v_{\mathsf{e}}(x)\, dx,\\
D(\EuFrak{a})&=\left\{u\in \mathcal{H}^1\colon I_{\mathsf{v}}U(\mathsf{v})=0,\; \mathsf{v}\in\mathsf{V}\right\},
\end{split}
\end{equation}
where
\begin{equation
\mathcal{H}^1\coloneqq \prod_{\mathsf{e}\in\mathsf{E}}H^1(0,\ell_{\mathsf{e}}),
\end{equation}
in the following sense:
\begin{equation
\begin{split}
D(A)&=\left\{u\in D(\EuFrak{a}) \colon \exists\, h\in \mathcal{H}\text{ s.t. }\EuFrak{a}(u,v)=\langle h,v\rangle_{\mathcal{H}}\text{ for all }v\in D(\EuFrak{a})\right\},\\
-A u&=h.
\end{split}
\end{equation}
The form $(\EuFrak{a},D(\EuFrak{a}))$ is symmetric, densely defined, continuous, closed and accretive. The operator $(A,D(A))$ is densely defined, dissipative, sectorial and self-adjoint with $(0,+\infty)\subset\rho(A)$.
The strongly continuous semigroup $(S(t))_{t\geq 0}$ generated by $(A,D(A))$ is analytic, positive and contractive.
\end{prop}
\begin{proof}
All the statements follow by \cite[Prop.~2.3--2.6]{KS21}.
\end{proof}
Notice that $D(\EuFrak{a})$ contains such $H^1$-functions on the edges that satisfy the continuity condition in all vertices.

\section{Stochastic perturbation of the vertex conditions}\label{sec:stochKircont}

Define
\[\Theta\coloneqq\begin{pmatrix}
	\left(\Phi_{\mathsf{v}}\right)_{\mathsf{v}\in\mathsf{V}}\\
	\left(\beta_{\mathsf{v}}\right)_{\mathsf{v}\in\mathsf{V}}
\end{pmatrix}.\]
Then for arbitrary $\lambda>0$,
\begin{equation}\label{eq:stochD}
X(t)=S(t)u_0+\int_0^t(\lambda-A)S(t-s)D_{B,\lambda}\, d\Theta (s)
\end{equation}
is the mild solution of \eqref{eq:stochnet} in the sense of \cite[Sec.~15.1]{PeZa} (and also \cite[Sec.~1]{DPZ93}) provided that the stochastic convolution is well-defined. We remark that somewhat informally, $X$ solves the stochastic equation
\begin{equation
dX=AX\, dt+ (\lambda-A)D_{B,\lambda}\, d\Theta(t).
\end{equation}
For the sake of simplicity we fix now $\lambda>0$, denote $D\coloneqq D_{B,\lambda}$, and define
\begin{equation}\label{eq:Zt}
Z(t)\coloneqq \int_0^t (\lambda-A) S(t-s)D\, d\Theta(t) .
\end{equation}
Since $A$ generates a contractive analytic semigroup, we can define its fractional powers for $\lambda>0$ and $\alpha\in (0,1)$. In particular, the fractional domain spaces
\begin{equation}\label{eq:Halphapos}
\mathcal{H}_{\alpha}\coloneqq D((\lambda-A)^{\alpha}),\quad \|u\|_{\alpha}\coloneqq \|(\lambda-A)^{\alpha}u\|,\quad u\in D((\lambda-A)^{\alpha})
\end{equation}
are Banach spaces. We fix $\mathcal{H}_0\coloneqq \mathcal{H}$.

For $\alpha\in(0,1)$ we define the extrapolation spaces $\mathcal{H}_{-\alpha}$ as the completion of $\mathcal{H}$ under the norms $\|u\|_{-\alpha}\coloneqq \|(\lambda-A)^{-\alpha}u\|$, $u\in \mathcal{H}$.

It is well-known (see e.g.~\cite[\textsection II.4--5.]{EN00}) that up to equivalent norms, these spaces are independent of the choice of $\lambda>0$.

\begin{rem}\label{rem:omega0}
Since the semigroup $(S(t))_{t\geq 0}$ is contractive, hence bounded, then by \cite[Prop.~3.1.7]{Haase06}
we can choose $\lambda=0$ in \eqref{eq:Halphapos}. That is,
\[\mathcal{H}_{\alpha}\cong D((-A)^{\alpha}),\quad \alpha\in [0,1),\]
when $D((-A)^\alpha)$ is equipped with the graph norm.
\end{rem}

\begin{theo}\label{theo:boundpertKircont}
If $\alpha<-\frac{1}{4}$ and $u_0\in \mathcal{H}_{\alpha}$, equation \eqref{eq:stochnet} has a mild solution of the form \eqref{eq:stochD} with continuous paths in $\mathcal{H}_{\alpha}$.
\end{theo}
\begin{proof}
It is enough to show that the stochastic process $Z(\cdot)$ defined in \eqref{eq:Zt} has a continuous version in $\mathcal{H}_{\alpha}$. Let $\delta>\frac{1}{4}$ arbitrary fixed and we will prove the result for $\alpha\coloneqq -\delta$. Without loss of generality we may assume that $\delta$ is close to $\frac{1}{4}$.

By \cite[Thm.~2.3]{DPZ93}, we have to show that for some $\gamma>0$,
\begin{equation}\label{eq:proof1}
\int_0^{T}t^{-\gamma}\left\|(\lambda-A)S(t)D\right\|^2_{HS(\mathcal{Y},\mathcal{H}_{-\delta})}\, dt <+\infty,
\end{equation}
where $HS$ denotes the Hilbert--Schmidt-norm between the appropriate spaces.

First we will prove that the Dirichlet operator $D$ maps $\mathcal{Y}$ into $\mathcal{H}_{\rho}$ for $0<\rho<\frac{1}{4}$. We obtain that the form
\begin{equation}\label{eq:eaomega}
\EuFrak{a}_{\lambda}(u,v)\coloneqq \EuFrak{a}(u,v)+\lambda\cdot \langle u,v\rangle_E,\quad u,v\in D(\EuFrak{a})
\end{equation}
is coercive, symmetric and continuous, see Proposition \ref{prop:eaASttul} and \cite[Rem.~7.3.3]{Haase06}. It is straightforward that the operator associated with $\EuFrak{a}_{\lambda}$ is $\lambda-A$. For the form-domain $D(\EuFrak{a}_{\lambda})=D(\EuFrak{a})$, see \eqref{eq:ea}, equipped with the usual $\mathcal{H}^1$-norm, we have that
\begin{equation
D((\lambda-A)^{\frac{1}{2}})\cong D(\EuFrak{a})
\end{equation}
holds with equivalence of norms, see e.g.~\cite[Prop.~5.5.1]{Ar04}. We also have
\begin{equation}\label{eq:DA12Dea}
D(\EuFrak{a})\cong D((\lambda-A)^{\frac{1}{2}})\cong D((-A)^{\frac{1}{2}})
\end{equation}
with equivalent norms, see Remark \ref{rem:omega0}. Using \eqref{eq:DA12Dea}, similarly as in \cite[Lem.~3.6]{KS21} one can show that
\begin{equation
D((-A)^{\frac{1}{2}})\cong \prod_{\mathsf{e}\in\mathsf{E}}H_0^1(0,\ell_{\mathsf{e}})\times\mathbb{R}^n.
\end{equation}
By \cite[Sec.~4.3.3]{Triebel78} we have that if $\theta<\frac{1}{2}$, for the complex interpolation space
\begin{equation}\label{eq:DA12interpol}
\left[D((-A)^{\frac{1}{2}}),\mathcal{H}\right]_{\theta}\cong \prod_{\mathsf{e}\in\mathsf{E}}H^{\theta}(0,\ell_{\mathsf{e}}) \times\{0_{\mathbb{R}^n}\}
\end{equation}
holds, where we have used $\mathcal{H}\cong\mathcal{H}\times \{0_{\mathbb{R}^n}\}$ and \cite[Thm.~4.2.2]{BeLo}.
Furthermore, \cite[Thm.~6.6.9]{Haase06} implies that for $0<\theta<1$,
\begin{equation}\label{eq:interpol1}
\left[D((-A)^{\frac{1}{2}}),\mathcal{H}\right]_{\theta}\cong \left[D(A),\mathcal{H}\right]_{\frac{\theta}{2}}.
\end{equation}
Using \cite[Thm.~in $\mathsection$4.7.3]{Ar04} and \cite[Prop.~in $\mathsection$4.4.10]{Ar04}, we obtain that for any $0<\rho<1$,
\begin{equation}\label{eq:interpol2}
\left[D(A),\mathcal{H}\right]_{\rho}\cong  D((-A)^{\rho})=\mathcal{H}_{\rho}.
\end{equation}
Combining \eqref{eq:DA12interpol}, \eqref{eq:interpol1} and \eqref{eq:interpol2} yields that for $0<\rho<\frac{1}{4}$
\begin{equation}\label{eq:Hrhoizom}
\prod_{\mathsf{e}\in\mathsf{E}}H^{2\rho}(0,\ell_{\mathsf{e}}) \times \{0_{\mathbb{R}^n}\}\cong \mathcal{H}_{\rho}
\end{equation}
holds. Hence, by \eqref{eq:domAmax} and \eqref{eq:Hrhoizom} we have that for $0<\rho<\frac{1}{4}$
\begin{equation}\label{eq:DmapsDA14}
D\colon \mathcal{Y}\to D(A_{\max})\hookrightarrow \prod_{\mathsf{e}\in\mathsf{E}}H^{2\rho}(0,\ell_{\mathsf{e}}) \times \{0_{\mathbb{R}^n}\}\cong \mathcal{H}_{\rho}.
\end{equation}
Now we are in the position to prove \eqref{eq:proof1}. We take a small $\varepsilon>0$, to be specified later, and estimate the integral in the following way, where we use the analyticity of the semigroup $(S(t))_{t\geq 0}$ from Proposition \ref{prop:eaASttul},
\begin{align
&\int_0^{T}t^{-\gamma}\left\|(\lambda-A)S(t)D\right\|^2_{HS(\mathcal{Y},\mathcal{H}_{-\delta}))}\, dt \leq c\cdot \int_0^{T}t^{-\gamma}\left\|(\lambda-A)^{1-\delta}S(t)D\right\|^2_{HS(\mathcal{Y},\mathcal{H})}\, dt\notag\\
& = c\cdot\int_0^T t^{-\gamma}\left\|(\lambda-A)^{\varepsilon}S(t)(\lambda-A)^{1-\delta-\varepsilon}D\right\|^2_{HS(\mathcal{Y},\mathcal{H})}\, dt\notag\\
&\leq C_T\cdot\int_0^T t^{-\gamma-2\varepsilon}\, dt\cdot \left\|(\lambda-A)^{1-\delta-\varepsilon}D\right\|^2_{HS(\mathcal{Y},\mathcal{H})}.\label{eq:intbecsles}
\end{align}
In the last expression,
\begin{equation}\label{eq:tkitevo}
\int_0^T t^{-\gamma-2\varepsilon}\, dt<\infty \Longleftrightarrow \gamma+2\varepsilon<1 \Longleftrightarrow \varepsilon<\frac{1}{2}-\frac{\gamma}{2}.
\end{equation}
Since the range of $D$ is finite dimensional, if we take a suitable $\varepsilon>0$ such that $(\lambda-A)^{1-\delta-\varepsilon}D$ is a bounded operator, then the Hilbert-Schmidt norm in \eqref{eq:intbecsles} is finite. To satisfy this, by \eqref{eq:DmapsDA14},
\begin{equation}\label{eq:deltaepszilon}
0<1-\delta-\varepsilon<\frac{1}{4}
\end{equation}
should hold. On the other hand, by \eqref{eq:tkitevo},
\[1-\delta-\varepsilon>\frac{1}{2}+\frac{\gamma}{2}-\delta.\]
Hence, we have to find $\gamma>0$ such that
\[\frac{1}{2}+\frac{\gamma}{2}-\delta<\frac{1}{4}\Longleftrightarrow  \frac{1}{4}+\frac{\gamma}{2}<\delta\]
holds true. Since, by assumption, $\delta>\frac{1}{4}$, an appropriate $0<\gamma<1$ can be chosen. Taking any $\varepsilon>0$ satisfying \eqref{eq:tkitevo} and \eqref{eq:deltaepszilon}, that is
\[\frac{3}{4}-\delta<\varepsilon<\frac{1}{2}-\frac{\gamma}{2},\]
the estimate \eqref{eq:intbecsles} yields a finite bound for the left-hand-side of \eqref{eq:proof1}. This means that for $\gamma$ chosen this way, \eqref{eq:proof1} is satisfied, and the proof is complete.
\end{proof}

\section{Stochastic perturbation of the Kirchhoff--Neumann vertex conditions}\label{sec:stochKir}

In this section we will slightly change the original problem \eqref{eq:stochnet} and consider
\begin{equation}\label{eq:stochnetK}
\left\{\begin{aligned}
\dot{u}_{\mathsf{e}}(t,x) & =  (c_{\mathsf{e}} u_{\mathsf{e}}')'(t,x)-p_{\mathsf{e}}(x) u_{\mathsf{e}}(t,x) &x\in (0,\ell_{\mathsf{e}}),\; t\in(0,T],\; \mathsf{e}\in\mathsf{E},\;\; & (a)\\
0 & = I_{\mathsf{v}}U(t,\mathsf{v}),\;  &t\in(0,T],\; \mathsf{v}\in\mathsf{V},\;\;& (b)\\
\dot{\beta}_{\mathsf{v}}(t) & = C(\mathsf{v})^{\top}U'(t,\mathsf{v}), & t\in(0,T],\; \mathsf{v}\in\mathsf{V},\;\;& (c)\\
 u_{\mathsf{e}}(0,x) & =  u_{0,\mathsf{e}}(x), &x\in [0,\ell_{\mathsf{e}}],\; \mathsf{e}\in\mathsf{E}.\;\;& (d)
\end{aligned}
\right.
\end{equation}
Namely, in equations $(\ref{eq:stochnet}b)$ we set $\dot{\Phi}_{\mathsf{v}}(\cdot)=0$, $\mathsf{v}\in\mathsf{V}$, that is we assume continuity of the function $u(t,\cdot)$ on the metric graph $G$, cf.~\eqref{eq:contv}.

To treat this new problem, on $\mathcal{H}$ we define the maximal operator $(A_{K,\max},D(A_{K,\max}))$ in a different way than it has been done in \eqref{eq:opAmax} and \eqref{eq:domAmax}. Namely, we put the continuity conditions in its domain (see also \cite{KMS07}), that is, we set
\begin{align}\label{eq:opAKmax}
A_{K,\max}&\coloneqq \mathrm{diag}\left(\frac{d}{dx}\left(c_{\mathsf{e}} \frac{d}{dx}\right)-p_{\mathsf{e}} \right)_{\mathsf{e}\in\mathsf{E}},\\
D(A_{K,\max})&\coloneqq \left\{u\in\mathcal{H}^2\colon I_{\mathsf{v}}U(\mathsf{v})=0,\; \mathsf{v}\in\mathsf{V} \right\}.
\end{align}

Accordingly, we have to modify the boundary space as
\begin{equation}\label{eq:YK}
\mathcal{Y}_{K}\coloneqq\ell^2(\mathbb{R}^n)\cong \mathbb{R}^n,
\end{equation}
where $n=|\mathsf{V}|$ the number of vertices in $G$, cf.~\eqref{eq:Y}. The feedback operator becomes $B_K\colon D(A_{K,\max})\to \mathcal{Y}_K$,

\begin{align}\label{eq:opBK}
D(B_K) & = D(A_{K,\max});\\
B_K u & \coloneqq \left(C(\mathsf{v})^{\top}U'(\mathsf{v})\right)_{\mathsf{v}\in\mathsf{V}},
\end{align}
see \eqref{eq:opB}.

Defining
\begin{align}\label{eq:aKmain}
A &\coloneqq A_{K,\max}\\
D(A)&\coloneqq\{u\in D(A_{K,\max})\colon B_Ku=0_{\mathcal{Y}_K}\}
\end{align}
we obtain the same operator as in \eqref{eq:amain}.

Mimicking the proof of Proposition \ref{prop:Diropletezik} one obtains the following result.

\begin{prop}\label{prop:DiropKirletezik}
The operators $A_{K,\max}$ and $B_K$ satisfy the conditions in \cite[Section 15.1]{PeZa} and \cite[Section 1]{DPZ93}, that is
\begin{enumerate}[(i)]
	\item the operator $A$ defined in \eqref{eq:aKmain} -- that is, in \eqref{eq:amain} --	generates a $C_0$-semigroup on $\mathcal{H}$;
	\item for $\lambda>0$ there exists a bounded operator $D_{B_K,\lambda}\colon \mathcal{Y}\to \mathcal{H}$ such that
	\begin{equation}\label{eq:DiropK}
	D_{B_K,\lambda}=\left(B_K\mid_{\mathrm{Ker}(\lambda-A_{K,\max})}\right)^{-1}.
	\end{equation}
\end{enumerate}
\end{prop}

Next, we define
\[\beta=\left(\beta_{\mathsf{v}}\right)_{\mathsf{v}\in\mathsf{V}},\]
and consider
\begin{equation}\label{eq:stochDK}
X_K(t)=S(t)u_0+\int_0^t(\lambda-A)S(t-s)D_{B_K,\lambda}\, d\beta(s)
\end{equation}
as the mild solution of the problem \eqref{eq:stochnetK}, see \eqref{eq:stochD}. For the sake of simplicity we fix again $\lambda>0$, denote $D_K\coloneqq D_{B_K,\lambda}$, and define
\begin{equation}\label{eq:ZKt}
Z_K(t)\coloneqq \int_0^t (\lambda-A) S(t-s)D_K\, d\beta (s).
\end{equation}

In general, the mild solution has continuous trajectories in $\mathcal{H}$ as the next result shows.

\begin{theo}\label{theo:boundpertKir}
For $u_0\in \mathcal{H}$ arbitrary, equation \eqref{eq:stochnetK} has a mild solution of the form \eqref{eq:stochDK} with continuous paths in $\mathcal{H}$.
\end{theo}
\begin{proof}
Again, it is enough to show that the stochastic process $Z_K(\cdot)$ defined in \eqref{eq:ZKt} has a continuous version in $\mathcal{H}$. First we prove that the stochastic process $Z_K(\cdot)$ in \eqref{eq:ZKt} is well-defined in $\mathcal{H}$, that is,
\begin{equation}\label{eq:ZKtcontpf}
\int_0^T\left\|(\lambda-A) S(t)D_K\right\|^2_{HS(\mathcal{Y}_K,\mathcal{H})}\, dt<+\infty.
\end{equation}
Using a standard energy argument (or, alternatively, Parseval's formula), it follow that
\begin{equation}\label{eq:DKbecsles}
\int_0^T\left\|(\lambda-A) S(t)D_K\right\|^2_{HS(\mathcal{Y}_K,\mathcal{H})}\, dt\leq c_T\cdot \left\|(\lambda-A)^{\frac{1}{2}}D_K\right\|^2_{HS(\mathcal{Y}_K,\mathcal{H})}.
\end{equation}
We have
\[D_K\colon \mathcal{Y}_K\to D(A_{K,\max})\]
and
\[D(A_{K,\max})=\left\{u\in\mathcal{H}^2\colon I_{\mathsf{v}}U(\mathsf{v})=0,\; \mathsf{v}\in\mathsf{V} \right\}\hookrightarrow D(\EuFrak{a}),\]
see \eqref{eq:ea}. By \eqref{eq:DA12Dea},
\[D((\lambda-A)^{\frac{1}{2}})=D(\EuFrak{a})\]
with equivalence of norms, hence $D_K$ is a bounded linear operator into $D((\lambda-A)^{\frac{1}{2}})$. Since the range of $D_K$ is finite dimensional,
\[\left\|(\lambda-A)^{\frac{1}{2}}D_K\right\|_{HS(\mathcal{Y}_K,\mathcal{H})}<+\infty\]
holds. Thus, using \eqref{eq:DKbecsles}, the  assertion in \eqref{eq:ZKtcontpf} holds, the stochastic convolution  $Z_K(\cdot)$ is well-defined in $\mathcal{H}$.
Finally, since the semigroup $(S(t))_{t\geq 0}$ is contractive, \cite[Rem.~1]{HauSei01} implies the continuity of the trajectories.
\end{proof}

\begin{rem}\label{rem:Aeigen}
Observe that $D(A)\subset \mathcal{H}^1$, and by the Rellich-Kondrachov theorem, $\mathcal{H}^1\hookrightarrow \mathcal{H}$ is a continuous, compact embedding. If $\lambda\in\rho(A)$ is arbitrary, for the resolvent operator $\mathrm{ran} (R(\lambda,A))\subset D(A)$ holds, and we obtain that $R(\lambda,A)\colon \mathcal{H}\to \mathcal{H}$ is bounded and compact. That is, $A$ has compact resolvent and thus $A$ has only point spectrum. Since $A$ is self-adjoint and dissipative, its eigenvalues $(\lambda_k)_{k\in\mathbb{N}}$ form a sequence of negative real numbers and
\begin{equation}\label{eq:lambdak}
\lambda_k\to-\infty,\quad k\to\infty.
\end{equation}
We may then choose a set $(f_k)_{k\in\mathbb{N}}\subset D(A)$ of eigenfunctions such that
\begin{equation}\label{eq:fk}
Af_k=\lambda_k f_k,\quad k\in\mathbb{N},
\end{equation}
and the functions $(f_k)_{k\in\mathbb{N}}$ form a complete orthonormal system in $\mathcal{H}$.
\end{rem}

It turns out that if we assume uniform boundedness for the vertex values of the eigenfunctions $(f_k)$, the mild solution has a continuous version in $\mathcal{H}_{\alpha}$ for $\alpha<\frac{1}{4}$.

To this aim, we introduce the boundary operator $L\colon D(L)\to \mathcal{Y}_K$ defined by
\begin{equation}\label{eq:L}
\begin{split}
D(L) & = \left\{u\in \prod_{\mathsf{e}\in\mathsf{E}}C[0,\ell_{\mathsf{e}}]: I_{\mathsf{v}} U(\mathsf{v})=0,\; \mathsf{v}\in\mathsf{V}\right\},\\
L u & \coloneqq\left(u(\mathsf{v})\right)_{\mathsf{v}\in\mathsf{V}}\in \mathcal{Y}_K.
\end{split}
\end{equation}
That is, $L$ assigns to each function $u$ that is continuous on $G$ the vector of the vertex values of $u$. Observe that $D(A)\subset D(L)$ holds.

\begin{theo}\label{theo:eigenfctest}
Suppose that there exists $c>0$ such that
\begin{equation}\label{eq:fkunifbound}
\|Lf_k\|^2_{\mathcal{Y}_K} \leq c,\quad k\in\mathbb{N},
\end{equation}
where $(f_k)_{k\in\mathbb{N}}$ is the complete orthonormal system consisting of eigenfunctions of $A$ (see Remark \ref{rem:Aeigen}) and $L$ is the operator \eqref{eq:L}.

Then, for $\alpha<\frac{1}{4}$ the stochastic convolution process $Z_K(\cdot)$ defined in \eqref{eq:ZKt} has a continuous version in $\mathcal{H}_{\alpha}$.
\end{theo}
\begin{proof}
By \cite[Thm.~2.3]{DPZ93}, we have to show that if $\alpha<\frac{1}{4}$, then for a fixed $\lambda>0$ and $D_K= D_{B_K,\lambda}$, there exists $\gamma>0$ such that
\begin{equation}\label{eq:Kproof1}
\int_0^{T}t^{-\gamma}\left\|(\lambda-A)S(t)D_K\right\|^2_{HS(\mathcal{Y}_K,\mathcal{H}_{\alpha})}\, dt <+\infty,
\end{equation}
where $HS$ denotes the Hilbert--Schmidt-norm between the appropriate spaces.

Using Remark \ref{rem:omega0} and proceeding as in \cite[Sec.~3]{DPZ93}, we obtain that \eqref{eq:Kproof1} holds if
\begin{equation}\label{eq:sorosszeg}
\sum_{k=1}^{\infty}(\lambda-\lambda_k)^{2\alpha+\gamma+1}\left\|D_K^*f_k\right\|^2_{\mathcal{Y}_K}<\infty.
\end{equation}
We will show that under the assumptions,
\begin{equation}\label{eq:DKcsillagbecsles}
\left\|D_K^*f_k\right\|_{\mathcal{Y}_K}^2 \leq \frac{c}{(\lambda-\lambda_k)^2},\quad k\in\mathbb{N},
\end{equation}
where $c\in\mathbb{R}$ is the constant from \eqref{eq:fkunifbound}. By Proposition \ref{prop:szigmaA}, there exist constants $l_1,l_2>0$ such that
\begin{equation}\label{eq:lambdakaszimpt}
l_1\cdot k^2\leq \lambda-\lambda_k\leq l_2\cdot k^2,\quad k\in\mathbb{N}.
\end{equation}
Hence, if \eqref{eq:DKcsillagbecsles} holds, by \eqref{eq:lambdakaszimpt} the terms of the series \eqref{eq:sorosszeg} can be estimated as
\[(\lambda-\lambda_k)^{2\alpha+\gamma+1}\left\|D_K^*f_k\right\|^2_{\mathcal{Y}_K}\leq C \cdot \frac{k^{4\alpha+2\gamma+2}}{k^4} \]
for an appropriate constant $C>0.$ Thus, the series \eqref{eq:sorosszeg} converges if and only if
\[4\alpha+2\gamma+2-4<-1 \Longleftrightarrow 4\alpha+2\gamma<1.\]
This means that there exists appropriate $\gamma>0$ if and only if
\[\alpha<\frac{1}{4}.\]

Now we turn to the proof of \eqref{eq:DKcsillagbecsles}. First notice that
\begin{equation}\label{eq:DKfk}
\left\|D_K^*f_k\right\|^2_{\mathcal{Y}_K}=\sum_{i=1}^n \langle e_i,D_K^*f_k \rangle_{\mathcal{Y}_K}^2,
\end{equation}
where
\[e_i=\left(\begin{smallmatrix}
0\\
\vdots\\
1\\
\vdots\\
0
\end{smallmatrix}\right)\leftarrow i^{\rm th}\hbox{ row}\]
the usual $i$th basis vector in $\mathbb{R}^n$ and $\langle \cdot,\cdot\rangle_{\mathcal{Y}_K}$ denotes the scalar product in $\mathcal{Y}_K$.

By definition, for each $i=1,\dots, n$,
\begin{equation}\label{eq:eiDKcsillag}
\langle e_i,D_K^*f_k \rangle_{\mathcal{Y}_K} =\langle D_K e_i, f_k \rangle_{\mathcal{H}}=\langle u^i, f_k \rangle_{\mathcal{H}},
\end{equation}
where $u^i \in D(A_{K,\max})$
\begin{equation}\label{eq:ui}
A_{K,\max}u^i=\lambda u^i,\quad B_Ku^i=e_i,
\end{equation}
see \eqref{eq:DiropK}. Integration by parts then yields
\begin{align}
\langle u^i, f_k \rangle_{\mathcal{H}}&=\frac{1}{\lambda}\langle A_{K,\max}u^i, f_k \rangle_{\mathcal{H}}\\
& =\frac{1}{\lambda}\sum_{\mathsf{e}\in\mathsf{E}} \int_0^{\ell_{\mathsf{e}}} \left(c_{\mathsf{e}} (u^i_{\mathsf{e}})'\right)'(x)\cdot f_{k,{\mathsf{e}}}(x)\, dx-\frac{1}{\lambda}\sum_{\mathsf{e}\in\mathsf{E}} \int_0^{\ell_{\mathsf{e}}} p_{\mathsf{e}}(x)\cdot u^i_{\mathsf{e}}(x)\cdot f_{k,{\mathsf{e}}}(x)\, dx\\
& =\frac{1}{\lambda}\sum_{\mathsf{e}\in\mathsf{E}}\left[c_{\mathsf{e}}(x)\cdot (u^i_{\mathsf{e}})'(x)\cdot f_{k,{\mathsf{e}}}(x)\right]_0^{\ell_{\mathsf{e}}}\\
&-\frac{1}{\lambda}\sum_{\mathsf{e}\in\mathsf{E}}\left( \int_0^{\ell_{\mathsf{e}}} c_{\mathsf{e}}(x)\cdot (u^i_{\mathsf{e}})'(x)\cdot f_{k,{\mathsf{e}}}'(x)\, dx+\int_0^{\ell_{\mathsf{e}}} p_{\mathsf{e}}(x)\cdot u^i_{\mathsf{e}}(x)\cdot f_{k,{\mathsf{e}}}(x)\, dx\right)\\
& =\frac{1}{\lambda}\left( \sum_{\mathsf{e}\in\mathsf{E}}\left[c_{\mathsf{e}}(x)\cdot (u^i_{\mathsf{e}})'(x)\cdot f_{k,{\mathsf{e}}}(x)\right]_0^{\ell_{\mathsf{e}}}-\EuFrak{a} (u^i, f_{k})\right)\\
&=\frac{1}{\lambda}\left( \sum_{\mathsf{e}\in\mathsf{E}}\left[c_{\mathsf{e}}(x)\cdot (u^i_{\mathsf{e}})'(x)\cdot f_{k,{\mathsf{e}}}(x)\right]_0^{\ell_{\mathsf{e}}}+ \langle u^i, Af_{k}\rangle_{\mathcal{H}} \right)\\
&=\frac{1}{\lambda}\left( \sum_{\mathsf{e}\in\mathsf{E}}\left[c_{\mathsf{e}}(x)\cdot (u^i_{\mathsf{e}})'(x)\cdot f_{k,{\mathsf{e}}}(x)\right]_0^{\ell_{\mathsf{e}}}+ \lambda_k\cdot \langle u^i, f_{k}\rangle_{\mathcal{H}} \right),
\end{align}
where we have used Proposition \ref{prop:eaASttul} and \eqref{eq:fk}. This implies that
\begin{equation
\langle u^i, f_k \rangle_{\mathcal{H}}=\frac{1}{\lambda-\lambda_k}\sum_{\mathsf{e}\in\mathsf{E}} \left[c_{\mathsf{e}}(x)\cdot (u^i_{\mathsf{e}})'(x)\cdot f_{k,{\mathsf{e}}}(x)\right]_0^{\ell_{\mathsf{e}}}.
\end{equation}
Since $f_k \in D(L)$, we have by similar calculations as in the proof of \cite[Prop.~A.1]{KS21JEE}
\begin{equation}\label{eq:uifk}
\langle u^i, f_k \rangle_{\mathcal{H}} =\frac{1}{\lambda-\lambda_k}\langle B_K u^i, Lf_k \rangle_{\mathcal{Y}_K}=\frac{1}{\lambda-\lambda_k}\langle e_i, Lf_k \rangle_{\mathcal{Y}_K},
\end{equation}
where we used \eqref{eq:ui}.

Hence by \eqref{eq:fkunifbound}, \eqref{eq:DKfk}, \eqref{eq:eiDKcsillag} and \eqref{eq:uifk} we obtain
\begin{equation
\left\|D_K^*f_k\right\|_{\mathcal{Y}_K}^2=\frac{1}{(\lambda-\lambda_k)^2}\sum_{i=1}^n \langle e_i, Lf_k \rangle^2_{\mathcal{Y}_K}=\frac{\|Lf_k\|_{\mathcal{Y}_K}^2}{(\lambda-\lambda_k)^2}\leq \frac{c}{(\lambda-\lambda_k)^2},
\end{equation}
which is exactly \eqref{eq:DKcsillagbecsles} and the proof is complete.
\end{proof}

\begin{cor}\label{cor:Laplace}
Let $c_{\mathsf{e}}\equiv 1$ and $p_{\mathsf{e}}\equiv 0$, $\mathsf{e}\in\mathsf{E}$ in \eqref{eq:stochnetK}. Then \eqref{eq:stochnetK} has a mild solution of the form \eqref{eq:stochDK} with continuous paths in $\mathcal{H}_{\alpha}$ for $\alpha<\frac14$
\end{cor}
\begin{proof}
By \cite[(13)]{HaMa20}, for the constants
\begin{equation
M\coloneqq 2\max_{x\in [\pi,+\infty)}\frac{1}{1-\frac{\sin x}{x}}\approx 2.29456
\end{equation}
and
\begin{equation}\label{eq:lmin}
l\coloneqq\min_{\mathsf{e}\in\mathsf{E}}\ell_{\mathsf{e}}>0
\end{equation}
we have
\begin{equation
\|Lf_k\|^2_{\mathcal{Y}_K}\leq \frac{2\cdot n}{l}\cdot M \eqqcolon c.
\end{equation}
Hence, if the diffusion coefficients are all equal to $1$ and there is no potential term, \eqref{eq:fkunifbound} holds with an appropriate positive $c$. Thus, by Theorem \ref{theo:eigenfctest}, problem \eqref{eq:stochnetK} has a mild solution of the form \eqref{eq:stochDK} with continuous paths in $\mathcal{H}_{\alpha}$ for $\alpha<\frac14$.

\end{proof}

redpajama_set_name: RedPajamaArXiv


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The Dolomites region in Northern Italy is a mountain region of sheer natural beauty. I have not been there often but last summer I visited it for the first time. It is in the South Tyrol area of Italy where German is widely spoken. I covered the area from a hiking and scenery point of view. For those food lovers it's a must visit. I was so impressed with it all that I had to revisit the area and its food again in winter. This time I focused on the Alta Badia area which is famous for its skiing in Corvara where one of the world cup runs take place. I was going to cover the Gourmet Skisafari, which was a day of skiing with Michelin starred food served in 5 different mountain huts.
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The XC90 is almost 5 meters long.
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Mountain driving at its best. Notice the Bower & Wilkins speaker in the centre of the dashboard. The Dashboard was full leather and had a classy feel about it.
Picking the skis and boots up was easy and with the middle part of the rear seat folded down you can fit even the largest skis in easily. With the MenStyleFashion magnets on the doors I arrived at the Rosa Alpina hotel in San Cassiano. The sheer size of the car made it noticed as soon as I pulled up the XC90 in front of the hotel. The car was the talking point in a positive way for our three day visit. We picked up some extra guest with skis and 4 sets of skis boots and other gear, and it easily fitted in the XC90. The rear passengers were impressed with their heated rear seats to be driven in comfort to the nearest ski station. The Gourmet Skisafari was a great success as well as the whole skiing experience in Alta Badia, the Volvo XC90 just matched it well.
The return road to the airport was different as I took the road via Cortina d'Ampezzo. I stopped the XC90 lots of times just to soak up the amazing Dolomites mountain scenery. The two and half hour drive to the airport was again super relaxing, I was sad to leave the car behind.
The XC90 is a luxurious SUV with an interior quality to a very high standard. It is a very comfortable highway cruiser and therefor an excellent car to take around your next European skiing trip. The boot can swallow all your luggage and ski gear with ease. For those in the market for a German SUV alternative and want an SUV that looks different and is luxurious, take a look at the Volvo XC90.


text:
Q: How to properly remove bootstrap CDN from `assets.precompile` and isolate controller stylesheets I was trying to segregate my CSS to specific controllers by compiling my assets and rendering them in my layouts via <%= stylesheet_link_tag "application", params[:controller] %>
I am using the Bootstrap (4.3.1) CDN and have this in my application.scss file:
@import "bootstrap-sprockets";
@import "bootstrap";

The problem is that when I do Rails.application.config.assets.precompile += %w( *.css ) it must be compiling bootstrap as it gives me this error:

Sass::SyntaxError in RecipeCategories#index
Undefined variable: "$alert-padding".

So I'm using this to "solve" it:
Rails.application.config.assets.precompile = [ Proc.new{ |path| !File.extname(path).in?('.css') }, /bootstrap.css$/ ]

I don't know if this is effective or best practice to prevent bootstrap from being precompiled.
I also run into the problem where a style defined in one controller's stylesheet is being loaded in the view of another until the page is refreshed. (I put the class on an element in multiple controller views to see if it was really isolated to a specific controller.)
Basically, the style in one controller's CSS file is able to be used in other controllers views until the page is refreshed, then the style is "removed" from the element in the controller view that doesn't have the style in it's CSS file.

A: You seem to mistakenly be precompiling the bootstrap assets via the gem and Rails' asset pipeline AND getting bootstrap from a CDN. You only need one or the other, so you can probably remove the gem and just use the CDN.

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Data Pack 5.00: Includes latest PES 2014 DP 4.00 with winter transfers and many new players for most club teams.
Known error: Brazil kits are not working atm, because they are apparently blocked by Konami, in relation to the World Challange Data Pack (not available for PC). We will provide a fix as soon as possible.


text:
Occupy Madison's 'tiny house' village plan embraced by some neighbors, cops not sold
Pat Schneider
Neighbors of the proposed site of a village of tiny houses for the homeless are trying to stop approval of the plan with a zoning protest petition.
MICHELLE STOCKER -- Capital Times
This sketch shows how Occupy Madison plans to arrange 11 "little houses" around a former auto body shop on East Johnson St.
Occupy Madison's plan to put tiny homes for the homeless on the site of an east-side auto repair shop is picking up support from some residents of the Emerson East neighborhood, but Madison police are coming out against it.
Some of the neighbors of the proposed site at 2050 E. Johnson Ave. who packed a meeting Wednesday night said the planned cluster of eleven 99-square-foot tiny homes would lower property values, endanger safety and pose sanitation problems.
Residents living adjacent to the site were especially concerned. They objected to the lack of screening of residents of the tiny homes and social services to help the homeless people curb undesirable behavior.
But many of the more than 100 neighborhood residents who gathered at James Reeb Unitarian Universalist Congregation welcomed the proposed construction workshop and village of tiny homes as a way to diversify and enrich the area.
Responding to others' worries over drug and alcohol use, violence and possible child molestation, Kate Sandretto said annoying or illegal behavior is a risk with any new neighbor.
She and her family have dealt with noise, drug use, gun violence and other problems with a neighbor on East Dayton Street, she said.
"Nobody is vetting anyone over there. No one is providing social services there," she said. "There are no assurances when you buy property what neighbors you're going to get in the future."
Sandretto said she would be delighted to have the enterprise in the neighborhood, especially to broaden her children's humanity.
"I want them to understand that you don't need a mortgage and a bank account to have worth and dignity," she said.
Small houses are an alternative form of transitional housing for the homeless being tried in a number of cities. Occupy Madison, Inc., a nonprofit organization that grew out of the political protests of 2011, has been public for months about its goal of developing a village of tiny homes for the homeless in Madison.
The proposed development would convert the auto shop to a workshop to build tiny homes which would then be placed on the property. Each tiny home would have a composting toilet, and that waste would be removed by a porta-potty company, according to the plan. In addition, existing plumbing at the auto shop would be used to form four flush-toilet bathrooms, as well as add showers.
But Jay Lengfeld, captain of the Madison Police Department's north district, said the proposed site is the wrong place for a village of tiny homes.
"We are opposed to this development at this time," Lengfeld told the crowd. The site is too close to East High School, Emerson Elementary School, as well as Demetral Park, where there are already problems with drug and alcohol use, he said. "We believe this will increase calls for (police) service."
After some speakers accused police of reacting to the kind of stereotypes of the homeless that worried neighbors, Lengfeld said that his opinion is based on the track record of the Occupy Madison tent encampments in 2012.
The group camped in three locations that year: the 800 block of East Washington Avenue, on the lawn of Dane County Human Services headquarter on Northport Drive, and on private property on Portage Road on the east side.
At each location, calls for police service increased, there were problems and the group was asked leave, Lengfeld said.
"It's important to know this is based on data and experience," he said.
Brenda Konkel, a vocal homeless services advocate and board member of Occupy Madison, told the group that she was surprised by the police opposition. Occupy Madison members participating in the project to build the tiny houses have had good relations with police officers and administrators and no problems at their current rented east-side workshop, she said. She said in follow-up email messages that they had been "blindsided" by police.
Konkel told the crowd Wednesday that it was hard to hear stereotypes about who and what homeless people are. It's hard to find affordable housing in Madison, where a 2 percent vacancy rate drives up rents and landlords require significant deposits to secure housing and move in, she said.
"The system is not working for them," said Konkel. "They're trying to do something a little bit outside the system."
Pat Schneider joined The Capital Times in 1989 and has written on a wide variety of topics including neighborhoods, minority communities and the nonprofit sector.
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Occupy Madison Inc.
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* @return an IR for m, built according to the specified options. null if m is abstract or native.
This is intended as an internal interface; clients probably shouldn't be using this directly. If you have a call graph in hand, to get the IR for a CGNode, use node.getIR(); Otherwise, look at SSACache.
Does this factory always return the same IR for a method, regardless of context?


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The diploma thesis is focused on the evaluation of the impact of innovation activities and their impact on labor productivity in the selected company. The theoretical part is dedicated to the literary review of terms like innovation, human capital, and efficiency. The practical part contains a description of the chosen company, financial analysis, and Human Resources management. The most significant innovations in the past seven years were selected for the examination of the influence on labor productivity.
The Diploma thesis deals with a period of a glass company which had to declare an insolvency proceedings and consequently went into bankruptcy. The aim of the thesis is to determine the financial situation of the company before its collapse by using a financial analysis. The work contains causes and resulting consequences of this situation and the insolvency proceedings process.
This thesis focuses on financial analysis of a small company, specifically a small retail store that specialises on children´s footwear. The aim of the thesis is to evaluate the financial health of the retail store. Methods of financial analysis are applied to data obtained from the financial statements from 2005 to 2017, while the change in the accounting method is reflected in the period. Horizontal and vertical analysis, analysis of differential and ratio indicators are used to perform financial analysis.
The topic of this thesis is financial analalysis of both, a company and an investment. It also contains description of software for its execution, which was implemented within the thesis. The first part describes financial analysis in general -- definitions, motivation, input data set definition, enumeration of common methods and indicators, their meaning and iterpretation of their values. The second part contains description of design, implementation and functionality of a tool, which purpose is automatization of chosen domains from financial analysis.
The master thesis deals with performance evaluation of Imos Brno a.s. using the Balanced Scorecard method. The theoretical part is the basis for measuring the performance of the company, it contains the theoretical knowledge gained from professional literature. The next part of the thesis is dedicated to the presentation of the selected company and then financial and strategic analysis is performed. The results of these analyzes become the basis for the last part of the work. This section discusses the individual steps of implementing the Balanced Scorecard model to the selected business to improve business performance.


We designed the LF1 to be easily mounted to almost any projector.
After unboxing your LF1, you are ready to set it up to work with your projector. In this article we go over the initial mounting and setup of the LF1 on your projector.
If your projector is not available you can skip this article and continue to networking setup.
The LF1 works with most projectors and auto-calibrates to it during the scan process. Check the full projector compatibility requirements here but the most important thing to remember is to make sure both the Lightform and projector are powered on and connected to each other via HDMI before you can begin any project.
Before mounting the LF1 to your projector, make sure that the projector is functioning. Power on your projector and check that the lamp is on. When working with Lightform, always check that your projector is on — you will not be able to take a scan if the projector is off. You may shut off your projector when mounting the LF1, but it must be turned on to take a scan and to project content.
Before taking a Scan, you will want to mount the LF1 securely to your projector so it does not move during or after the Scan. We recommend doing this with the adhesive mount provided, or alternatively with the universal mounting system (additional hardware required).
When you first unbox the LF1, do not immediately apply the adhesive mount to your projector. The adhesive mount is intended for a single application — please proceed with caution.
The LF1 comes fully assembled to a hinge mount with an adhesive base (the two combined being the Adhesive Mount), ready for placement and mounting on the projector. There is an extra adhesive base, an alternative universal mounting bracket, and additional mounting tools found in the box labeled Mounting.
The adhesive mount is very difficult to remove. We do not recommend using the adhesive mount if you are renting or borrowing a projector, please follow this article instead.
When setting up the LF1 and your projector, make sure there is properventilation in your setup to prevent the LF1 from overheating.
Lightform mounting brackets are compatible with most common action camera (i.e. GoPro) mounting accessories. Always take proper safety precautions when mounting anything in a hanging or overhead location.
Be sure to avoid covering any of your projector buttons and make sure to leave enough room to adjust the focus on your projector lens.
This is intended for a single application, ensure the placement is correct.
You can fine-tune the placement of the LF1 after the adhesive on the mount is applied to the projector.
If you are using a curved projector and want to level your LF1, use the roll adjustment of the rear right screw. Roll adjustment helps align the camera image in rare or more advanced cases.
Connect the LF1 power adapter to a power outlet and the LF1. Check that the LF1 LED has turned on. Read more about LED messages.
Connect the LF1 HDMI cable from the LF1 to the projector. Check that projector is on the correct HDMI input. The LF1 will now control the projector for scanning and content playback.
Remove the protective plastic film from the front window of the LF1 before scanning.
Your LF1 hardware is now setup! The next step is to connect it to your network.


text:
Read Next Black Sabbath Prep 'Vinyl Collection' With Bonus Rarities	Send Us a Tip	Subscribe
May 29, 2018 5:59PM ET
How BTS Are Breaking K-Pop's Biggest Taboos
The chart-topping Korean group has spoken out on LGBTQ rights, mental health and more
Jae Ha Kim
Jae Ha Kim's Most Recent Stories
Tiffany Young Steps Into the Solo Spotlight
In modern South Korea, pop stars and politics don't often mix. While some hip-hop acts (such as Epik High) address controversial topics, many idol groups stick to a slick, apolitical formula with a proven record of success.
BTS, who just became the first K-pop act ever to top the Billboard 200 album sales chart, have become a record-setting success story in part because of their willingness to buck this convention. The seven young men who make up the group have been speaking their minds since their debut, openly discussing LGBTQ rights, mental health and the pressure to succeed – all taboo subjects in South Korea. Their stance is particularly bold given the Korean government's history of keeping an eye on controversial themes in pop music. By straddling the line between maintaining a respectable image and writing critical lyrics, BTS have offered a refreshing change from what some critics and fans dislike about the K-Pop machine.
BTS' 'Love Yourself: Tear': Inside the K-Pop Sensations' New LP
Inside BTS-mania: A Day in the Life of the K-Pop Superstars
The group's history of open-mindedness dates back to 2013, when they were still rookies and talking about same-sex relationships could have been a career killer. The group's leader, RM, tweeted out his admiration for Macklemore & Ryan Lewis' inclusive hit "Same Love" nonetheless: "This is a song about homosexuality. The song is twice as good when I listen to the lyrics." At the time, he was a relatively unknown musician who had little to lose. But earlier this year, RM once again took a stance on Twitter, recommending "Strawberries and Cigarettes," a song by the openly gay musician Troye Sivan, from the gay coming-of-age film Love, Simon.
These are small gestures, but they mean a lot to BTS' fans. LGBTQ rights certainly aren't perfect in the United States, but in South Korea, the situation is so abysmal that Amnesty International got involved last year after a South Korean soldier was sentenced to six months in prison for having a consensual homosexual relationship.
Behind the screams that fill any venue where BTS performs are millions of devotees who span generations, genders, religions, nationalities and sexualities. And these fans – collectively known as ARMY – listen intently to the group's self-penned lyrics and message, before disseminating and translating them into many languages.
For BTS and their fans, acts of generosity often bear a political message. After the Sewol Ferry sank off the coast of South Korea in April 2014, killing almost 300 teenage students, Korea's politicians tried to distance themselves from the tragedy. Grieving parents participated in hunger strikes, while conservative supporters of then-President Park Geun-Hye downplayed the deaths, saying it was time to put the calamity behind them. Reports later exposed a government-sanctioned blacklist of celebrities who appeared to criticize Park's regime; while BTS were likely unaware of this, they demonstrated which side they supported by donating $100,000 to benefit the victims' families.
Several years ago, BTS member Suga told his fans that when he became rich, he would buy them beef – an expensive treat in today's Korea due to high import duties. For his 25th birthday this year, he fulfilled his promise. But instead of handpicking fans to join him for dinner, he donated $19,000 worth of beef to feed orphans on behalf of ARMY. His fans were delighted and followed suit by making their own charitable donations.
When BTS partnered with the United Nations Children's Fund (UNICEF) for its "Love Myself" anti-violence campaign, their fans asked how they could help. This May, ARMY played a crucial role in raising $1 million in less than two days for a UNICEF USA campaign to feed malnourished children.
As the group has matured from teenagers into young men, their lyrics have gained more gravitas. So, too, has their outspokenness about the issues that matter to them. They are aware that their fame gives them a huge platform.
Thirty-six South Koreans commit suicide every day, according to a report from Korea's Ministry of Health and Welfare. While South Korea's government has been slow to address this problem, some BTS members have been vocal about their own bouts with depression. In their docu-series "Burn the Stage," a fan said, "Sometimes, BTS is the only reason I wake up in the morning." In that same episode, RM talked about how depression had previously made him afraid to go back on stage: "I was unable to enjoy things like I used to." And in their annual New Year's message to their fans, Suga told them, "It's okay to live without a dream."
The group's members know firsthand about the intense pressures many of their countrymen place on education and the negative impact this can have on mental health. With their songs, BTS tell their fans to fight back and remain alive. "Live as you like, it's your life anyway," they sing in the anthemic "Fire." They continue the theme of survival on their new track, "Paradise," telling listeners it's okay to have modest goals: "We deserve a life."
South Korea's leadership is listening. After BTS' historic No. 1 placement on the Billboard 200 this week, President Moon Jae-In said, "BTS has a magical ability to turn sadness into hope, and differences into similarity."
In This Article: K-Pop,	LGBT


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text:
in: France articles missing geocoordinate data, All articles needing coordinates, Articles incorporating text from Wikipedia,
Battles of the War of the Sixth Coalition
Battles involving Russia
Battles of the Six Days' Campaign
Battle of Champaubert
Part of the War of the Sixth Coalition
The French Campaign, 1814 by Jean Meissonier (1815-1891)
Champaubert, east of Paris.
French victory
French Empire Russian Empire
Emperor Napoleon
Auguste Marmont General Olssufiev
15,000, 120 guns 3,700, 24 guns
600 2,400, 9 guns
Six Days Campaign
The Battle of Champaubert (now Giffaumont-Champaubert) was the opening engagement of the Six Days Campaign. It was fought on February 10, 1814 by a French force under Napoleon I against Russians and Prussians under General Olssufiev. The battle was a French victory.
The battle of Champaubert was one of the few times during the War of the Sixth Coalition that France was able to take to the field with a considerable numerical advantage. Napoleon Bonaparte moved against an over-extended Prussian army in the hope of whittling it down by a series of battles. On 10 February, he caught General Olssufiev's five thousand Russians just south of Champaubert, a town located in the valley of the Marne, east of Paris.
French strength consisted of 30,000 hungry and tired men, including many raw conscripts, and 120 cannons.[1] The French, nonetheless, enjoyed a six-to-one advantage. They were commanded in the field by the marshal, Auguste Marmont, under the direction of Napoleon himself.
Badly outnumbered, Olssufiev decided to fight rather than retreat. His decision was based on the mistaken hope that he would get reinforcements from Field Marshal Blücher in time to prevent a disaster. He was wrong, and Marmont crushed him. After five hours of fighting, the Russians were surrounded by French cavalry. They suffered three thousand killed, wounded, and captured. One of the prisoners was Olsufiev himself, who dined that very evening with Napoleon.
The French lost about three hundred men, among whom was General Joseph Lagrange, wounded.
Historian Digby Smith wrote that the French lost 600 killed and wounded out of the 13,300 infantry and 1,700 cavalry that were engaged in the action. The Russians lost 2,400 men and nine guns out of the 3,700 soldiers and 24 guns that were present. Captured were General-Leutnant Olssufiev and General-major Poltaratzky, who led a brigade. The second brigade under General-major Kornieloff fought its way out.[2]
↑ Chandler, David. Dictionary of the Napoleonic wars. Wordsworth editions, 1999. p.87
↑ Smith, p.494
Smith, Digby (1998). The Napoleonic Wars Data Book. London: Greenhill. ISBN 1-85367-276-9.
La bataille de Champaubert, Montmirail-Marchais et Vauchamps
France articles missing geocoordinate data


text:
Q: pass arguments to a function call in a directive via angularjs I have a custom directive. In my html i write this:
<uploader action="/rest/file/create_from_form.json" success="writeFid()"></uploader>

What I need is to exec "success" attribute function passing some data that I get from my directive. I can exec "success" attribute function via
$scope.$eval($scope.success)

and in my "controller" I have this:
$scope.writeFid = function (data) {
        console.log("Into writeFid");
        console.log(data);  //this is my problem: it is always undefined.
    }

I can see (via console.log() messages) that "success" function is called but without passing "data".
I have tried to use
<uploader action="/rest/file/create_from_form.json" success="writeFid(data)"></uploader>

but it does not work.
So: how can i pass some type of $scope.data ?
Thanks.

A: I'm not sure whether your directive uses an isolate scope or not, so I'll cover both cases:
Without an isolate scope:
scope.$eval(attrs.success, { data: 'Some value from foo' });

With an isolate scope:
scope: { success: '&' },
...
scope.success({ data: 'Some value from bar' });

Regardless of the type of the scope, your markup must be like this:
<uploader success="writeFid(data)" ...></uploader>

And here's a plunk script showing both approaches.

A: You could use an isolate scope with a bi-directional binding in your uploader directive to somewhat achieve what you're after.
uploader directive
...
scope: {
    data: '=',
    success: '&'
}
...

Sample parent controller
app.controller('MainCtrl', function($scope) {
   $scope.dataAttribFromParent = { data: [] };

   $scope.writeFid = function () {
       console.log($scope.dataAttribFromParent);
   };
});

markup
...
<uploader data="dataAttribInParent" success="writeFid()">
...

Here's a plunk to elaborate:
http://plnkr.co/edit/7Kzy6D7cwxlATEBKRcuT