Asymptotic behaviour of fast diffusions on graphs

TL;DR

This paper studies the long-term behavior of fast diffusions on finite directed graphs, analyzing how the diffusion process evolves as the speed increases and the likelihood of particles passing vertices decreases.

Contribution

It provides new results on the asymptotic behavior of diffusion semigroups on graphs, extending previous duality approaches to $L^1$ and $L^2$ spaces.

Findings

Asymptotic behavior characterized for increasing diffusion speeds.

Diffusion semigroup convergence in $L^1$ and $L^2$ spaces.

Results extend duality methods to new settings.

Abstract

We investigate fast diffusions on finite directed graphs. We prove results in a way dual to presented in Bobrowski, A. Ann. Henri Poincar\'e (2012) 13(6): 1501-1510 and Bobrowski, A., Morawska, K. DCDS-B (2012), 17(7): 2313-2327, and obtain asymptotic behaviour of a diffusion semigroup on a graph in $ L^1 $ and $ L^2 $ as the diffusions' speed increases and the probability of a particle passing through a vertex decreases.

Equations

$$\mathcal{N}:=\left\{1,\dots ,N\right\},\mathbb{N}:=\{1,2,\dots \}.$$

$$Af=\sigma f^{\prime \prime },$$

$$f^{\prime }(V_i)=l_{i}f(V_i)-\sum _{j\ne i}{l}_{ij}f(V_j),$$

$$-f^{\prime }(V_i)=r_{i}f(V_i)-\sum _{j\ne i}{r}_{ij}f(V_j),$$

$$A_{n}f={\kappa }_n\sigma f^{\prime \prime },$$

$$\underset{n\to +\infty }{\lim }{\mathrm{e}}^{t{A}_n}f={\mathrm{e}}^{tQ}Pf$$

$$\Vert \phi {\Vert }_{L^1(I)}:={\int }_I|\phi (t)|\mathrm{d}t={\int }_I|\phi |,\phi \in L^1(I).$$

$$\Vert \phi {\Vert }_{W^{2,1}(I)}:=\Vert \phi {\Vert }_{L^1(I)}+\Vert {\phi }^{\prime \prime }{\Vert }_{L^1(I)},\phi \in W^{2,1}(I)$$

$$L^1(S):=\{\phi :\phi =({\phi }_i{)}_{i\in \mathcal{N}},{\phi }_i\in L^1(E_i)\}.$$

$$\Vert \phi {\Vert }_{L^1(S)}:=\sum _{i\in \mathcal{N}}\Vert {\phi }_i\circ {\ell }_i{\Vert }_{L^1(0,d_i)},\phi =({\phi }_i{)}_{i\in \mathcal{N}}\in L^1(S).$$

$$A_n^{\ast }\phi ={\kappa }_n\sigma {\phi }^{\prime \prime }$$

$${\kappa }_n{\sigma }_i{\phi }^{\prime }(L_i)$$

$${\kappa }_n{\sigma }_i{\phi }^{\prime }(R_i)$$

$${\kappa }_n{\phi }^{\prime }(L_i)=F_{L,i}\phi ,{\kappa }_n{\phi }^{\prime }(R_i)=F_{R,i}\phi ,i\in \mathcal{N},$$

$${\int }_S\phi A_{n}f={\int }_S(A_n^{\ast }\phi )f.$$

$$\begin{array}{rl}{\int }_{E_j}\phi f^{\prime \prime }& =\phi (R_j)f^{\prime }(R_j)-{\phi }^{\prime }(R_j)f(R_j)+{\phi }^{\prime }(L_j)f(L_j)-\phi (L_j)f^{\prime }(L_j)+{\int }_{E_j}{\phi }^{\prime \prime }f\end{array}$$

$$\sum_{j\in\mathcal{N}}\sigma_{j}\big{[}\varphi(R_{j})f^{\prime}(R_{j})-\varphi^{\prime}(R_{j})f(R_{j})+\varphi^{\prime}(L_{j})f(L_{j})-\varphi(L_{j})f^{\prime}(L_{j})\big{]}=0.$$

$$\displaystyle\kappa_{n}^{-1}\sum_{j\in\mathcal{N}}\sigma_{j}\varphi(R_{j})\bigg{[}\sum_{i\neq j}r_{ji}f(R_{ji})-r_{j}f(R_{j})\bigg{]}-\sum_{j\in\mathcal{N}}\sigma_{j}\varphi^{\prime}(R_{j})f(R_{j})$$

$$+$$

$$\displaystyle\kappa_{n}^{-1}\sum_{i\in\mathcal{N}}\sum_{j\neq i}\sigma_{j}r_{ji}\varphi(R_{j})f(R_{ji})-\sum_{i\in\mathcal{N}}\sigma_{i}f(R_{i})\big{[}\kappa_{n}^{-1}r_{i}\varphi(R_{i})+\varphi^{\prime}(R_{i})\big{]}$$

$$+$$

$$\sum_{i\in\mathcal{N}}\sigma_{i}f(R_{i})\big{[}\kappa_{n}^{-1}F_{R,i}\varphi-\varphi^{\prime}(R_{i})\big{]}+\sum_{i\in\mathcal{N}}\sigma_{i}f(L_{i})\big{[}\varphi^{\prime}(L_{i})-\kappa_{n}^{-1}F_{L,i}\varphi\big{]}=0,$$

$$G\phi :={\phi }^{\prime \prime }$$

$${\phi }^{\prime }(a)={\phi }^{\prime }(b)=0,$$

$$\Vert \lambda (\lambda -G{)}^{-1}{\Vert }_{\mathcal{L}(L^1(a,b))}⩽M$$

$$\lambda {\psi }_{\lambda }-{\psi }_{\lambda }^{\prime \prime }=\phi .$$

$${\psi }_{\lambda }=J_{\mu }+c_{\mu }{\mathrm{e}}_{-\mu }+d_{\mu }{\mathrm{e}}_{\mu },$$

$$J_{\mu }(x):=\frac{1}{2\mu }{\int }_a^b{\mathrm{e}}_{\mu }(|x-y|)\phi (y)\mathrm{d}y,x\in (a,b),$$

$$c_{\mu}=\frac{\xi_{\mu}\mathrm{e}^{-\mu b}+\zeta_{\mu}\mathrm{e}^{-\mu a}}{\mu\big{(}\mathrm{e}^{\mu(b-a)}-\mathrm{e}^{-\mu(b-a)}\big{)}},\qquad d_{\mu}=\frac{\xi_{\mu}\mathrm{e}^{\mu b}+\zeta_{\mu}\mathrm{e}^{\mu a}}{\mu\big{(}\mathrm{e}^{\mu(b-a)}-\mathrm{e}^{-\mu(b-a)}\big{)}},$$

$${\xi }_{\mu }:=\frac{1}{2}{\int }_a^b{\mathrm{e}}_{\mu }(y-a)\phi (y)\mathrm{d}y,{\zeta }_{\mu }:=\frac{1}{2}{\int }_a^b{\mathrm{e}}_{\mu }(b-y)\phi (y)\mathrm{d}y.$$

$$\underset{\lambda \to 0^{+}}{\lim }\lambda (\lambda -G{)}^{-1}\phi =\frac{1}{b-a}{\int }_a^b\phi ,$$

$$\Vert \lambda {\psi }_{\lambda }-C{\Vert }_{L^1(a,b)}⩽2^{-1}\mu \Vert \phi {\Vert }_{L^1(a,b)}+\Vert {\mu }^2{c}_{\mu }{\mathrm{e}}_{-\mu }+{\mu }^2{d}_{\mu }{\mathrm{e}}_{\mu }-C{\Vert }_{L^1(a,b)}.$$

Full text

1. Introduction

Assume that $\mathcal{G}$ is a directed graph in $\mathbb{R}^{3}$ without loops, and there is a Markov process on $\mathcal{G}$, which on each edge behaves like Brownian motion with given variance. Moreover, assume that each vertex is a semipermeable membrane with given permeability coefficients, that is for each vertex there are nonnegative numbers $p_{ij}$, describing the probability of a particle passing through membrane from the $i$-th to the $j$-th edge.

In [3] and [6] the authors prove that if the diffusion’s speed increases to infinity with the same rate as permeability coefficients decreases to zero, then we obtain a limit process which is a Markov chain on the vertices of the line graph of $\mathcal{G}$, see Figure 1.

The aim of this paper is to prove similar asymptotic result but in a different spaces. In [3, 6] the authors consider the process in the space of continuous functions on a graph $\mathcal{G}$. Here we consider $L^{1}$ and $L^{2}$-type spaces of Lebesgue integrable and square integrable functions.

The described model is a special case of an evolution operator acting on a graph. For more such models see [14].

1.1. Continuous case

As in [3], let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be a finite geometric graph (see e.g. [14, p. 65]) without loops, where $\mathcal{V}\subset\mathbb{R}^{3}$ is the set of verices, and $\mathcal{E}$ is the set of edges of finite length. The edges are seen as $C^{1}$ curves connecting vertices. Let $N$ be the number of edges and denote

[TABLE]

For each $i\in\mathcal{N}$, by convention, we call the initial and terminal vertices of the $i$-th edge $E_{i}$ its “left” and “right” endpoints. We denote them by $L_{i}$ and $R_{i}$, respectively. Moreover, for $i\in\mathcal{N}$ let $V_{i}$ denote vertex $V\in\mathcal{V}$ as an endpoint of the $i$-th edge. If $V$ is not an endpoint of this edge, we leave $V_{i}$ undefined.

Let $S=\bigcup_{i\in\mathcal{N}}E_{i}$ be the disjoint union of the edges. Notice that there can be many “copies” of the same vertex in $S$, treated as an endpoint of different edges, since by convention $V_{i}\neq V_{j}$ in $S$ for $i,j\in\mathcal{N}$, $i\neq j$. Then $S$ is a disconnected compact topological space, and we denote by $C(S)$ the space of continuous functions on $S$ with standard supremum norm. We may identify $f\in C(S)$ with $(f_{i})_{i\in\mathcal{N}}$, where $f_{i}$ is a member of $C(E_{i})$, the space of continuous functions on the edge $E_{i}$. The latter space is isometrically isomorphic to the space $C[0,d_{i}]$ of continuous functions defined on the interval $[0,d_{i}]$, where $d_{i}$ is the length of the $i$-th edge.

Let $\sigma\in C(S)$ be defined by $\sigma(p)=\sigma_{i}$ for $i\in\mathcal{N}$ and $p\in E_{i}$, where $\sigma_{i}$’s are given positive numbers. Define the operator $A$ in $C(S)$ by

[TABLE]

on the domain composed of twice continuously differentiable functions, satisfying the transmission conditions described below.

For each $i\in\mathcal{N}$, let $l_{i}$ and $r_{i}$ be nonnegative real numbers describing the possibility of passing through the membrane from the $i$-th edge to the edges incident in the left and right endpoints, respectively. Also, for $i,j\in\mathcal{N}$ such that $i\neq j$ let $l_{ij}$ and $r_{ij}$ be nonnegative real numbers satisfying $\sum_{j\neq i}l_{ij}\leqslant l_{i}$ and $\sum_{j\neq i}r_{ij}\leqslant r_{i}$. The summation here is taken over all $j\in\mathcal{N}$ such that $j\neq i$. These numbers determine the probability that after filtering through the membrane from the $i$-th edge, a particle will enter the $j$-th edge.

By default, if $E_{j}$ is not incident with $L_{i}$, we put $l_{ij}=0$. In particular, by convention $l_{ij}f(V_{j})=0$ for $f\in C(S)$, if $V_{j}$ is not defined. The same remark concerns $r_{ij}$. With these notations, the transmission conditions mentioned above are as follows: if $L_{i}=V$, then

[TABLE]

where $f^{\prime}(V_{i})$ is the right-hand derivative of $f$ at $V_{i}$, and if $R_{i}=V$, then

[TABLE]

where $f^{\prime}(V_{i})$ is the left-hand derivative of $f$ at $V_{i}$.

It is showed in [3] that the operator $A$ generates a Feller semigroup $\{\mathrm{e}^{tA}\}_{t\geqslant 0}$ in $C(S)$. This means that $\{\mathrm{e}^{tA}\}_{t\geqslant 0}$ is a strongly continuous semigroup of nonnegative contractions, that is for all $t\geqslant 0$ we have $\lVert\mathrm{e}^{tA}\rVert_{\mathcal{L}(C(S))}\leqslant 1$, and $\mathrm{e}^{tA}f\geqslant 0$, provided that $f\in C(S)$ is nonnegative. Here, $\lVert\,\cdot\,\rVert_{\mathcal{L}(C(S))}$ is the operator norm related to the standard supremum norm in $C(S)$. Moreover, the semigroup is conservative, that is $\mathrm{e}^{tA}\mathbbm{1}_{S}=\mathbbm{1}_{S}$, where $\mathbbm{1}_{S}\equiv 1$ on $S$, if and only if $\sum_{j\neq i}l_{ij}=l_{i}$ and $\sum_{j\neq i}r_{ij}=r_{i}$ for $i\in\mathcal{N}$.

Let $(\kappa_{n})_{n\in\mathbb{N}}$ be a nondecreasing sequence of positive numbers converging to infinity, and let operators $A_{n}$ be defined by (1.1) with $\sigma$ replaced by $\kappa_{n}\sigma$, that is

[TABLE]

with domain $D(A_{n})$ composed of twice continuously differentiable functions on $S$ satisfying transmission conditions (1.2) and (1.3) with permeability coefficients (that is all $l_{i}$, $r_{i}$, $l_{ij}$ and $r_{ij}$’s) divided by $\kappa_{n}$. The following is proved in [3, Theorem 2.2].

Theorem 1.1**.**

For every $t>0$ and $f\in C(S)$ it follows that

[TABLE]

in $C(S)$, where $P$ is the projection of $C(S)$ onto the space $C_{0}(S)$ of functions that are constant on each edge, given by Pf=\big{(}d_{i}^{-1}\int_{E_{i}}f\big{)}_{i\in\mathcal{N}}, while $Q$ is the operator in $C_{0}(S)$ which may be identified with the matrix $(q_{ij})_{i,j\in\mathcal{N}}$ with $q_{ij}=\sigma_{i}d_{i}^{-1}(l_{ij}+r_{ij})$ for $i\neq j$ and $q_{ii}=-\sigma_{i}d_{i}^{-1}(l_{i}+r_{i})$. The convergence is uniform on compact subsets of $(0,\infty)$. For $f\in C_{0}(S)$, the formula holds also for $t=0$, and the convergence is uniform on compact subsets of $[0,\infty)$.

The aim of this paper is to prove “dual” version of Theorem 1.1. Loosely speaking, the main result is as follows (see Theorems 2.4, 2.11, 3.2, and 3.6 for precise formulation).

Main Theorem**.**

For each $n\in\mathbb{N}$ the part $A_{n}^{*}$ of the adjoint operator of $A_{n}$ in the space of Lebesgue integrable (or square integrable) functions on $S$, generates a strongly continuous semigroup. Moreover, the semigroups generated by $A_{n}^{*}$’s converge strongly to $\mathrm{e}^{tQ}P$ as $n$ goes to infinity, for the projection $P$ given by the same formula as in $C(S)$ and some “matrix” $Q$.

We give an explicit formula for $Q$, and it is slightly different from $Q$ of Theorem 1.1.

One may wish to mimic the proof of the continuous case but this is not fully possible. In particular, in the space of continuous functions on $S$ there exists an isomorphism transforming boundary conditions associated with the original process to much simpler homogeneous Neumann boundary conditions. Because of that, we can obtain limit for the isomorphic semigroups which leads to required asymptotics. What is crucial, in the Lebesgue-type space of integrable or square integrable functions such isomorphism does not exists. However, there is an isomorphism of the Sobolev space $W^{2,1}$ or $W^{2,2}$ in a way similar to the isomorphism in the space of continuous functions. This leads to a different approach via Kurtz’s convergence theorem [8, Theorem 1.7.6] in $L^{1}$-type space or, in $L^{2}$-type space, Ouhabaz’s monotone convergence theorem for sesquilinear forms [15, Theorem 5], which generalizes Simon’s theorem [18, Theorem 3.1].

For generation results in $C$, $L^{1}$ or $W^{1,1}$-type space, concerning a diffusion operator with generalized transmission conditions see also [1].

2. Analysis in $L^{1}(S)$

We consider a model that is in a way dual to that described in Section 1.1 by investigating the restriction of the adjoint of $A_{n}$ to $L^{1}$-type space.

In order to set up notations, for an interval $I\subset\mathbb{R}$ equipped with the Lebesgue measure, let $L^{1}(I)$ be the real space of (equivalence classes of) Lebesgue integrable real functions defined on $I$. By $\lVert\,\cdot\,\rVert_{L^{1}(I)}$ we denote the standard norm

[TABLE]

Moreover, let $W^{2,1}(S)$ be the Sobolev space of (equivalence classes of) functions $\varphi\in L^{1}(I)$ such that $\varphi$ and $\varphi^{\prime}$ are weakly differentiable with $\varphi^{\prime},\varphi^{\prime\prime}\in L^{1}(I)$. The space $W^{2,1}(I)$ equipped with the norm

[TABLE]

is a Banach space. Moreover, if $(\mathbb{X},\lVert\,\cdot\,\rVert_{\mathbb{X}})$ is a Banach space, then by $\lVert\,\cdot\,\rVert_{\mathcal{L}(\mathbb{X})}$ we denote the operator norm in $\mathbb{X}$.

2.1. Adjoint of the operator $A_{n}$

Using the same identification as in Section 1.1, we consider the space

[TABLE]

Here $L^{1}(E_{i})$ is the space of (equivalence classes of) Lebesgue integrable functions on $E_{i}$, identified with $L^{1}(0,d_{i})$. More precisely, if $\ell_{i}(t)$ is the unique point on the edge $E_{i}$, whose distance from $L_{i}$ (along the edge) is $t\in(0,d_{i})$, then the function $\varphi\in L^{1}(E_{i})$ is identified with $\varphi\circ\ell_{i}\in L^{1}(0,d_{i})$. Such identification is an isometric isomorphism. In particular, for $\varphi=(\varphi_{i})_{i\in\mathcal{N}}\in L^{1}(S)$, we have $\int_{E_{i}}\varphi_{i}=\int_{0}^{d_{i}}\varphi_{i}\circ\ell_{i}$ and $\int_{S}\varphi=\sum_{i\in\mathcal{N}}\int_{E_{i}}\varphi_{i}$. We introduce the norm $\lVert\,\cdot\,\rVert_{L^{1}(S)}$ in $L^{1}(S)$ by

[TABLE]

Let $W^{2,1}(S)$ be the Sobolev-type space on $S$, that is the subspace of $L^{1}(S)$ composed of (equivalence classes of) functions $\varphi\in L^{1}(S)$ such that $\varphi$ and $\varphi^{\prime}$ are weakly differentiable and $\varphi^{\prime},\varphi^{\prime\prime}\in L^{1}(S)$.

Let $\sigma$, $(\kappa_{n})_{n\in\mathbb{N}}$, and all $l_{i}$, $r_{i}$, $l_{ij}$, $r_{ij}$’s be as in Section 1.1. For each $n\in\mathbb{N}$ we define the operator $A_{n}^{*}$ in $L^{1}(S)$ by

[TABLE]

with domain $D(A_{n}^{*})$ composed of members of $W^{2,1}(S)$ satisfying the transmission conditions

[TABLE]

for all $i\in\mathcal{N}$. Here, $I_{i}^{L}$ and $I_{i}^{R}$ are the sets of indexes $j\neq i$ of edges incident in $L_{i}$ and $R_{i}$, respectively. The prime in the sums denotes the fact that, since there are no loops, at most one of the terms $\sigma_{j}l_{ji}\varphi(L_{j})$ and $\sigma_{j}r_{ji}\varphi(R_{j})$ is taken into account. Denoting the right-hand sides of (2.2) and (2.3) by, respectively, $\sigma_{i}F_{L,i}\varphi$ and $\sigma_{i}F_{R,i}\varphi$, we may rewrite these conditions in the form

[TABLE]

and consider $F_{L,i}$, $F_{R,i}$ as linear functionals in $W^{2,1}(S)$.

Keeping in mind the Riesz representation theorem, the following lemma shows that the operator $A_{n}^{*}$ is in a way adjoint to $A_{n}$ defined in Section 1.1. More precisely, $A_{n}^{*}$ is the part (see [7, p. 60]) of the adjoint of $A_{n}$ in $L^{1}(S)$.

Lemma 2.1**.**

Let $n\in\mathbb{N}$. If $f\in D(A_{n})$ and $\varphi\in D(A_{n}^{*})$, then

[TABLE]

Proof.

Integrating by parts we obtain

[TABLE]

for every $j\in\mathcal{N}$. Hence, equality (2.5) holds if and only if

[TABLE]

Since $f$ belongs to $D(A_{n})$, transmission conditions (1.2) and (1.3), with left-hand sides multiplied by $\kappa_{n}$, are satisfied. Thus (2.6) holds if and only if

[TABLE]

where $L_{ji}$ and $R_{ji}$ are, by definition, respectively left and right ends of $E_{j}$, seen as members of $E_{i}$. Changing the order of summation, the last equality becomes

[TABLE]

Notice that $L_{ji}$ is either $L_{i}$ or $R_{i}$, or is left undefined, and the same holds for $R_{ji}$. Thus we can rewrite the last condition in the form

[TABLE]

which is true, since $\varphi$ satisfies the transition conditions (2.4). ∎

2.2. One-dimensional Laplacian in

$L^{1}(a,b)$ and $W^{2,1}(a,b)$

Let $a,b\in\mathbb{R}$ be such that $a<b$, and consider the one-dimensional Laplacian $G$ in $L^{1}(a,b)$ with homogeneous Neumann boundary conditions, that is

[TABLE]

with the domain $D(G)$ composed of functions $\varphi\in W^{2,1}(a,b)$ satisfying

[TABLE]

It is easy to check that $D(G)$ is dense in $L^{1}(a,b)$. Moreover, standard calculations show, see e.g. [13, Proposition 2.1.2 and Exercise 2.1.3.4], that the resolvent set of $G$ contains the interval $(0,+\infty)$, and there exists $M>0$ such that

[TABLE]

for every $\lambda>0$. Consequently, by the Hill-Yosida theorem, the operator $G$ generates a strongly continuous semigroup of contractions in $L^{1}(a,b)$.

In the following two propositions we also need an explicit formula for the resolvent of $G$. For $\nu\in\mathbb{R}$ let $\mathrm{e}_{\nu}$ be the function defined by $\mathrm{e}_{\nu}(x)\coloneqq\mathrm{e}^{-\nu x}$ for $x\in\mathbb{R}$. Fix $\varphi\in L^{1}(a,b)$ and $\lambda>0$. The function $\psi_{\lambda}\coloneqq(\lambda-G)^{-1}\varphi\in D(G)$ satisfies the resolvent equation

[TABLE]

Hence, letting $\mu\coloneqq\sqrt{\lambda}$, we may write $\psi_{\lambda}$ in the form

[TABLE]

where

[TABLE]

and $c_{\mu}$, $d_{\mu}$, depending merely on $\mu$ and $\varphi$, are chosen so that $\psi^{\prime}(a)=\psi^{\prime}(b)=0$. Precisely,

[TABLE]

for

[TABLE]

Proposition 2.2**.**

For every $\varphi\in L^{1}(a,b)$ we have

[TABLE]

in $L^{1}(a,b)$, where $\int_{a}^{b}\varphi$ is identified with the constant function on $(a,b)$.

Proof.

For fixed $\varphi\in L^{1}(a,b)$ and $\lambda>0$, let $\psi_{\lambda}\coloneqq(\lambda-G)^{-1}\varphi$ be given by (2.8). Observe that $\lVert J_{\mu}\rVert_{L^{1}(a,b)}\leqslant(2\mu)^{-1}\lVert\varphi\rVert_{L^{1}(a,b)}$. Consequently, letting $C\coloneqq(b-a)^{-1}\int_{a}^{b}\varphi$,

[TABLE]

Since

[TABLE]

it follows that $\mu^{2}c_{\mu}$ and $\mu^{2}d_{\mu}$ both converge to $C/2$ as $\mu\to 0^{+}$, by the Lebesgue dominated convergence theorem. Using the Lebesgue dominated convergence theorem again, we see that

[TABLE]

Therefore

[TABLE]

which completes the proof. ∎

Let $\tilde{G}$ be the part of $G$ in $W^{2,1}(a,b)$, that is

[TABLE]

with domain

[TABLE]

Proposition 2.3**.**

The resolvent set of $\tilde{G}$ contains the interval $(0,+\infty)$, and there exists $\tilde{M}>0$ such that

[TABLE]

for every $\lambda>0$.

Proof.

For simplicity, we assume that $a=0$ and $b=1$. The general case follows in the same way.

For fixed $\varphi\in W^{2,1}(0,1)$ and $\lambda>0$, let $\psi_{\lambda}\coloneqq(\lambda-G)^{-1}\varphi$ be given by (2.8). Since $\lambda\psi_{\lambda}-\psi_{\lambda}^{\prime\prime}=\varphi$, and $\varphi\in W^{2,1}(0,1)$, it follows that $\psi_{\lambda}^{\prime\prime}\in W^{2,1}(0,1)$. Let $D_{\mu}\coloneqq 1-\mathrm{e}^{-2\mu}$, and rewrite (2.9) in the form

[TABLE]

and

[TABLE]

Then (2.8) takes the form

[TABLE]

for $x\in(0,1)$. Expanding $D_{\mu}^{-1}$ into the geometric series $\sum_{k\geqslant 0}\mathrm{e}^{-2\mu k}$ we note that

[TABLE]

and

[TABLE]

for $t\in[0,2]$. Similarly,

[TABLE]

Finally, we may rewrite $\psi_{\lambda}$ in the form

[TABLE]

Changing variables in the integral we see that

[TABLE]

for $x\in(0,1)$. Differentiating this twice leads to

[TABLE]

where $\Psi_{\lambda}$ is given by the right-hand side of (2.10) with $\varphi$ replaced by $\varphi^{\prime\prime}$, that is $\Psi_{\lambda}=(\lambda-G)^{-1}\varphi^{\prime\prime}$, and

[TABLE]

In order to estimate the norm of $\Phi_{\lambda}$, observe that

[TABLE]

which implies

[TABLE]

Therefore, by the Sobolev embedding theorem,

[TABLE]

Finally, by (2.7),

[TABLE]

which completes the proof. ∎

2.3. Generation theorem in $L^{1}(S)$

As we said before, we know from [3, Proposition 2.1] that for each $n\in\mathbb{N}$ the operator $A_{n}$ generates a Feller semigroup in $C(S)$. We prove that the operator $A_{n}^{*}$ defined in Section 2.1 generates a sub-Markov semigroup $\{\mathrm{e}^{tA_{n}^{*}}\}_{t\geqslant 0}$ in $L^{1}(S)$, that is a semigroup of operators such that for every nonnegative $\varphi\in L^{1}(S)$ we have $\mathrm{e}^{tA_{n}^{*}}\varphi\geqslant 0$ and $\int_{S}\mathrm{e}^{tA_{n}^{*}}\varphi\leqslant\int_{S}\varphi$ for all $t\geqslant 0$. Moreover, if the semigroup $\{\mathrm{e}^{tA_{n}}\}_{t\geqslant 0}$ is conservative, we show that $\{\mathrm{e}^{tA_{n}^{*}}\}_{t\geqslant 0}$ is Markov, that is $\int_{S}\mathrm{e}^{tA_{n}^{*}}\varphi=\int_{S}\varphi$ for all nonnegative $\varphi\in L^{1}(S)$. The main theorem of this section is as follows.

Theorem 2.4**.**

For each $n\in\mathbb{N}$ the operator $A_{n}^{*}$ generates a sub-Markov semigroup in $L^{1}(S)$. Moreover, if the semigroup generated by $A_{n}$ is conservative, then $A_{n}^{*}$ generates a Markov semigroup.

Before we prove the theorem, we need auxiliary results. In what follows in this section we fix $n\in\mathbb{N}$.

Lemma 2.5**.**

The resolvent set of $A_{n}^{*}$ contains the interval $(0,+\infty)$. Moreover, for each $\lambda>0$ we have

[TABLE]

and

[TABLE]

Observe that if the resolvent of $A_{n}^{*}$ exists, then equality (2.11) becomes obvious (see [17, Lemma 1.10.2]), since $A_{n}^{*}$ is the part of the adjoint of $A_{n}$ in $L^{1}(S)$. However, the existence of $(\lambda-A_{n}^{*})^{-1}$ is not obvious. We prove Lemma 2.5 in a moment. First we show that the part of $\lambda-A_{n}^{*}$ in $W^{2,1}(S)$ satisfies the range condition.

Lemma 2.6**.**

For sufficiently large $\lambda>0$ the image of $D(A_{n}^{*})$ under the operator $\lambda-A_{n}^{*}$ contains $W^{2,1}(S)$.

Before proving the lemma we introduce some notations. It is crucial in our analysis to consider $L^{1}(S)$ with the Bielecki-type norm, see [2] or [10, p. 56]. For $i\in\mathcal{N}$ and $\omega>0$ let $\lVert\,\cdot\,\rVert_{\omega}$ be the norm in $L^{1}(0,d_{i})$ given by

[TABLE]

Naturally, see the beginning of Section 2.1, we set

[TABLE]

Such norm in $L^{1}(S)$ is equivalent to the standard norm $\lVert\,\cdot\,\rVert_{L^{1}(S)}$. It is also clear that

[TABLE]

for every $\varphi\in L^{1}(S)$. Furthermore, by $\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}\,\cdot\,\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{\omega}$ we denote the related norm in $W^{2,1}(S)$, that is

[TABLE]

Finally, for simplicity of notation, also by $\lVert\,\cdot\,\rVert_{\omega}$ and $\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}\,\cdot\,\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{\omega}$ we denote the operator norms corresponding to defined above Bielecki’s norms in, respectively, $L^{1}(S)$ and $W^{2,1}(S)$.

For $i\in\mathcal{N}$ let $G_{i}$ be the version of $G$, see Section 2.2, in $L^{1}(E_{i})$, and let $B$ be the operator in $L^{1}(S)$ defined by

[TABLE]

with the domain $D(B)$ composed of functions $\varphi=(\varphi_{i})_{i\in\mathcal{N}}$ such that $\varphi_{i}\in D(G_{i})$. Since $B$ is equal to $G_{i}$ on each $E_{i}$, operator $B$ generates strongly continuous semigroup $\{\mathrm{e}^{tB}\}_{t\geqslant 0}$.

The main idea of the proof of Lemma 2.6 is to consider the isomorphic image of the part of $A_{n}^{*}$ in $W^{2,1}(S)$. It turns out that it is possible to choose an isomorphism such that the image is a perturbation of the Laplacian with homogeneous Neumann boundary conditions.

For each $i\in\mathcal{N}$ choose smooth functions $h_{L,i}$, $h_{R,i}$ defined on $E_{i}$, and such that

[TABLE]

and

[TABLE]

Let $J$ be the linear operator in $W^{2,1}(S)$ given by

[TABLE]

where $F_{L,i}$ and $F_{R,i}$ are linear functionals in $W^{2,1}(S)$ defined in Section 2.1. By the Sobolev embedding theorem the functionals are bounded and there exists $M>0$, depending merely on permeability coefficients, such that

[TABLE]

Hence the operator $J$ is bounded and we estimate its Bielecki’s norm, obtaining

[TABLE]

for all $\omega>0$. Let $I_{n}\colon W^{2,1}(S)\to W^{2,1}(S)$ be the bounded linear operator given by

[TABLE]

Here, $I_{W^{2,1}(S)}$ is the identity operator in $W^{2,1}(S)$. Then, the choice of $h_{L,i}$, $h_{R,i}$ guarantees (see [3, Lemma 3.1]) that $I_{n}$ is an isomorphism of $W^{2,1}(S)$ with the inverse

[TABLE]

What is crucial, note that

[TABLE]

or equivalently, $I_{n}$ is an isomorphism between functions in $W^{2,1}(S)$ satisfying conditions (2.4), and those satisfying homogeneous Nuemann boundary conditions on each edge.

Furthermore, let $K\colon W^{2,1}(S)\to W^{2,1}(S)$ be defined as $J$ with $h_{L,i}$, $h_{R,i}$ replaced by their second derivatives, that is

[TABLE]

Proof of Lemma 2.6.

We consider $W^{2,1}(S)$ as a Banach space with the Bielecki norm $\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}\,\cdot\,\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{\omega}$. We define $\tilde{B}_{n}\coloneqq I_{n}\tilde{A}_{n}^{*}I_{n}^{-1}$, where $\tilde{A}_{n}^{*}$ is the part of $A_{n}^{*}$ in $W^{2,1}(S)$. We have

[TABLE]

where $\tilde{B}$ is the part of $B$ in $W^{2,1}(S)$. Moreover,

[TABLE]

where the last equality is a consequence of (2.18). Furthermore, by (2.16),

[TABLE]

Denoting $C\coloneqq-J\tilde{B}$, $D\coloneqq\sigma K-\kappa_{n}^{-1}\sigma JK$, we have

[TABLE]

By Proposition 2.3, there exists $\tilde{M}>0$ such that $\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}\lambda(\lambda-\kappa_{n}\tilde{B})^{-1}\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{\omega}\leqslant\tilde{M}$ for all $\lambda>0$ (recall that the standard norm and the Bielecki norm are equivalent). Hence, using the fact that

[TABLE]

we have

[TABLE]

Choose $\omega>0$ large enough, so that $\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}J\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{\omega}<\kappa_{n}(\tilde{M}+1)^{-1}$. Such $\omega$ exists by (2.12) and (2.15). Then

[TABLE]

which implies that $\lambda\in\rho(\kappa_{n}\tilde{B}+C)$. Therefore we have Neumann series expansion

[TABLE]

and consequently

[TABLE]

This means that $\kappa_{n}\tilde{B}+C$, being densely defined, generates a strongly continuous semigroup in $W^{2,1}(S)$. What is more, the operator $D$ in (2.20) is bounded, since $J$ and $K$ are. Hence, by the bounded perturbation theorem (see e.g. [7, Proposition III.1.12]), the operator $\tilde{B}_{n}$ generates a strongly continuous semigroup in $W^{2,1}(S)$, and so does its isomorphic image $\tilde{A}_{n}^{*}$. In particular $(\lambda-\tilde{A}_{n}^{*})(D(\tilde{A}_{n}^{*}))=W^{2,1}(S)$ for sufficiently large $\lambda>0$. ∎

Remark 2.7**.**

We showed in the proof of Lemma 2.6 that $\tilde{A}_{n}^{*}$ is the generator of a strongly continuous semigroup in $W^{2,1}(S)$. In particular, the domain $D(\tilde{A}_{n}^{*})$ of $\tilde{A}_{n}^{*}$ is dense in $W^{2,1}(S)$ equipped with the norm $\mathopen{|\mkern-1.5mu|\mkern-1.5mu|}\,\cdot\,\mathclose{|\mkern-1.5mu|\mkern-1.5mu|}_{\omega}$. The norm is stronger than the $L^{1}$-type norm in $W^{2,1}(S)$. Therefore, since $W^{2,1}(S)$ is dense in $L^{1}(S)$, the domain of $A_{n}^{*}$, which contains $D(\tilde{A}_{n}^{*})$, is dense in $L^{1}(S)$.

We are now ready to show that the resolvent of $A_{n}^{*}$ exists, as claimed in Lemma 2.5.

Proof of Lemma 2.5.

By the remark stated after the lemma, it is enough to show that $\lambda-A_{n}^{*}$ is invertible, and that the norm of the inverse is bounded by $\lambda^{-1}$.

First we show that the operator $A_{n}^{*}$ is dissipative, that is

[TABLE]

for all $\lambda>0$. Let $A_{n}^{\prime}$ be the adjoint operator of $A_{n}$ in the dual space of $C(S)$. That is $A_{n}^{\prime}$ acts in the space $M_{b}(S)$ of regular Borel measures on $S$. As we said in Section 1.1, $A_{n}$ generates a Feller semigroup in $C(S)$, hence $\lVert\mathrm{e}^{tA_{n}}\rVert_{\mathcal{L}(C(S))}\leqslant 1$ for all $t\geqslant 0$. Therefore $\lVert(\lambda-A_{n})^{-1}\rVert_{\mathcal{L}(C(S))}\leqslant\lambda^{-1}$ for all $\lambda>0$. However, we know (see e.g. [17, Theorem 1.10.2]) that for each $\lambda\in\rho(A_{n})$ it follows that $\lambda\in\rho(A_{n}^{*})$ and the adjoint of $(\lambda-A_{n})^{-1}$ equals $(\lambda-A_{n}^{\prime})^{-1}$. Consequently, since the norm of an operator is the same as the norm of its adjoint,

[TABLE]

Thus $\lVert(\lambda-A_{n}^{\prime})\mu\rVert_{M_{b}(S)}\geqslant\lambda\lVert\mu\rVert_{M_{b}(S)}$ for all $\mu\in D(A_{n}^{\prime})\subset M_{b}(S)$. Let $\varphi\in D(A_{n}^{*})$ and denote by $\mu_{\varphi}\in M_{b}(S)$ the measure corresponding to $\varphi$, that is the measure defined by $\mu_{\varphi}(E)\coloneqq\int_{E}\varphi$ for any Borel measurable set $E\subset S$. We have

[TABLE]

where in the last equality we used Lemma 2.1. Hence we may write, with slight abuse of notation, $A_{n}^{\prime}\mu_{\varphi}=A_{n}^{*}\varphi$. This means that $(\lambda-A_{n}^{\prime})\mu_{\varphi}=(\lambda-A_{n}^{*})\varphi$ for all $\varphi\in D(A_{n}^{*})$, and (2.21) follows by (2.22).

Since $A_{n}^{*}$ is dissipative, we are left with proving that $\lambda-A_{n}^{*}$ is surjective for some (hence all) $\lambda>0$. Since $A_{n}^{\prime}$ is closed and $L^{1}(S)$ is a closed subspace of $M_{b}(S)$, the operator $A_{n}^{*}$ is also closed. Hence, see e.g. [7, Proposition II.3.14(iii)], the range of $\lambda-A_{n}^{*}$ is closed in $L^{1}(S)$. However, by Lemma 2.6, for sufficiently large $\lambda>0$ the range contains $W^{2,1}(S)$, which is dense in $L^{1}(S)$. Hence the range equals $L^{1}(S)$. ∎

Proof of Theorem 2.4.

The domain of $A_{n}^{*}$ is dense in $L^{1}(S)$ (see Remark 2.7), hence by Lemma 2.5 it follows that $A_{n}^{*}$ is the generator of a strongly continuous semigroup in $L^{1}(S)$.

It is well known, see e.g. [12, Corollary 7.8.1], that $\{\mathrm{e}^{tA_{n}^{*}}\}_{t\geqslant 0}$ is sub-Markov, provided that the operator $\lambda(\lambda-A_{n}^{*})^{-1}$ is sub-Markov for all $\lambda>0$.

We prove that if $\varphi\in L^{1}(S)$ and $\varphi\geqslant 0$, then $(\lambda-A_{n}^{*})^{-1}\varphi\geqslant 0$ for every $\lambda>0$. Let $m$ be the Lebesgue measure on $S$, and suppose, contrary to our claim, that there exists a function $\varphi\geqslant 0$, a set $\Gamma\subset S$ with $m(\Gamma)>0$, and a real number $\delta>0$ such that for some $\lambda_{0}>0$ we have $(\lambda_{0}-A_{n}^{*})^{-1}\varphi\leqslant-\delta$ almost everywhere on $\Gamma$. Without loss of generality, we may assume that $\Gamma$ is a subset of some edge $E_{i}$. Then, for a given $\varepsilon>0$, we choose an open set $G\subset E_{i}$ and a closed set $\Gamma^{\prime}$ such that $\Gamma^{\prime}\subset\Gamma\subset G$ and $m(G\setminus\Gamma^{\prime})<\varepsilon$. By the Urysohn lemma, there exists a continuous real function $0\leqslant f\leqslant 1$ with $f\equiv 1$ on $\Gamma^{\prime}$ and $f\equiv 0$ outside $G$. Then

[TABLE]

Since $\varepsilon$ is arbitrary small, it follows that the left-hand side is strictly negative. However, by Lemma 2.5,

[TABLE]

where the inequality is a consequence of the fact that $A_{n}$ generates a Feller semigroup. This leads to contradiction and proves that $(\lambda-A_{n}^{*})^{-1}$ is a positive operator for each $\lambda>0$.

In order to prove the sub-Markov property, let $\varphi\in L^{1}(S)$. Since $A_{n}$ generates a Feller semigroup, we have

[TABLE]

where $\mathbbm{1}_{S}\equiv 1$ on $S$. Thus, by Lemma 2.5,

[TABLE]

which completes the first part of the proof.

If we assume that the semigroup generated by $A_{n}$ is conservative, then inequality in (2.23) becomes equality, and $\int_{S}\lambda(\lambda-A_{n}^{*})^{-1}\varphi=\int_{S}\varphi$ for all $\lambda>0$ and $\varphi\in L^{1}(S)$. ∎

2.4. Convergence in $L^{1}(S)$

To prove a convergence result that resembles Theorem 1.1, we begin with a theorem due to Kurtz (see [8, Theorem 7.6] or [4, Theorem 42.2]).

For each $n\in\mathbb{N}$ let $\mathcal{A}_{n}$ be the generator of a strongly continuous semigroup $\{\mathrm{e}^{t\mathcal{A}_{n}}\}_{t\geqslant 0}$ in a Banach space $\mathbb{X}$. Assume that the semigroups are equibounded, that is

[TABLE]

for some $C>0$. Denote by $\mathcal{A}_{\textnormal{ex}}$ the extended limit of $(\mathcal{A}_{n})_{n\in\mathbb{N}}$, that is the multivalued operator in $\mathbb{X}$ with the domain $D(\mathcal{A}_{\textnormal{ex}})$ composed of all $x\in\mathbb{X}$ such that there exists a sequence $(x_{n})_{n\in\mathbb{N}}$ in $\mathbb{X}$ that converges to $x$ while the limit of $\mathcal{A}_{n}x_{n}$ exists as $n\to+\infty$. By $(x,y)\in\mathcal{A}_{\textnormal{ex}}$ we mean that $x\in D(\mathcal{A}_{\textnormal{ex}})$ and $\lim_{n\to+\infty}\mathcal{A}_{n}x_{n}=y$ for some sequence $(x_{n})_{n\in\mathbb{N}}$ in $D(\mathcal{A}_{\textnormal{ex}})$ converging to $x$. Moreover, assume that $(\varepsilon_{n})_{n\in\mathbb{N}}$ is a sequence of positive real numbers converging to [math], and denote by $\mathcal{B}_{\textnormal{ex}}$ the extended limit of $(\varepsilon_{n}\mathcal{A}_{n})_{n\in\mathbb{N}}$.

Suppose also that an operator $\mathcal{B}$ with domain $D(\mathcal{B})$ generates a strongly continuous semigroup $\{\mathrm{e}^{t\mathcal{B}}\}_{t\geqslant 0}$ in $\mathbb{X}$ such that $\lVert\mathrm{e}^{t\mathcal{B}}\rVert_{\mathcal{L}(\mathbb{X})}\leqslant C$, and that for every $x\in\mathbb{X}$ the limit

[TABLE]

exists. The operator $\mathcal{P}$ is a bounded projection, hence its range, which we denote by

[TABLE]

is a closed subspace of $\mathbb{X}$. With this setup we use a special case of Kurtz’s theorem (for a general version see [8, Theorem 7.6]).

Theorem 2.8**.**

Let $\mathcal{A}$ be an operator in $\mathbb{X}$ such that $\mathbb{Y}$ is a subset of its domain. Assume that

1. (i)

if $x\in\mathbb{Y}$, then $(x,\mathcal{A}x)\in\mathcal{A}_{\textnormal{ex}}$, 2. (ii)

if $y\in D(\mathcal{B})$, then $(y,\mathcal{B}y)\in\mathcal{B}_{\textnormal{ex}}$, 3. (iii)

the operator $\mathcal{P}\mathcal{A}$ with domain $\mathbb{Y}$ generates a strongly continuous semigroup in $\mathbb{Y}$.

Then for every $x\in\mathbb{X}$ and $t>0$,

[TABLE]

in $\mathbb{X}$, and the convergence is uniform on compact subsets of $(0,\infty)$. If $x\in\mathbb{Y}$, then the formula holds also for $t=0$, and the convergence is uniform on compact subsets of $[0,+\infty)$.

In order to verify conditions (i)-(iii) of Kurtz’s theorem we need some lemmas. Recall that for each $n\in\mathbb{N}$ the operator $A_{n}^{*}$ defined by (2.1) with transmission conditions (2.4), generates a strongly continuous semigroup in $L^{1}(S)$. By $A_{\textnormal{ex}}$ we denote the extended limit of $(A_{n}^{*})_{n\in\mathbb{N}}$. Moreover, for $B$ defined by (2.13), it follows from Proposition 2.2 that the limit

[TABLE]

exists for every $\varphi\in L^{1}(S)$, and that

[TABLE]

The range of $P$ is the closed subspace of $L^{1}(S)$ consisting of all functions that are constant on each edge. We denote this subspace by $L_{0}^{1}(S)$, and note that it is isometrically isomorphic to $\mathbb{R}^{N}$ equipped with the appropriate norm.

Lemma 2.9**.**

The domain $D(A_{\textnormal{ex}})$ contains $L_{0}^{1}(S)$, and for the operator $K$ defined by (2.19) we have

[TABLE]

Proof.

Fix $\varphi\in L_{0}^{1}(S)$ and set (see (2.17))

[TABLE]

Since the operator $J$ is bounded in $W^{2,1}(S)$ and $\kappa_{n}^{-1}\to 0$ as $n\to+\infty$, the sequence $(\varphi_{n})_{n\in\mathbb{N}}$ converges to $\varphi$ in $L^{1}(S)$ as $n\to+\infty$. What is more $\varphi_{n}\in D(A_{n}^{*})$ for every $n\in\mathbb{N}$ by (2.18), and

[TABLE]

Hence $(\varphi,\sigma K\varphi)\in A_{\textnormal{ex}}$, which completes the proof. ∎

For the next lemma let $B_{\textnormal{ex}}$ be the extended limit of $(\kappa_{n}^{-1}A_{n}^{*})_{n\in\mathbb{N}}$.

Lemma 2.10**.**

For every $\varphi\in D(B)$ we have

[TABLE]

Proof.

Let $\varphi\in D(B)$ and set $\varphi_{n}\coloneqq I_{n}^{-1}\varphi$. Then $\varphi_{n}\in D(A_{n}^{*})$ for all $n\in\mathbb{N}$ by (2.18). As in the previous lemma $(\varphi_{n})_{n\in\mathbb{N}}$ converges to $\varphi$ in $L^{1}(S)$ as $n\to+\infty$, and

[TABLE]

Since $\lim_{n\to+\infty}\kappa_{n}^{-1}=0$ and since $J$ is bounded, it follows that $(\varphi,B\varphi)\in B_{\textnormal{ex}}$, as claimed. ∎

We are now ready to apply Theorem 2.8. In $L_{0}^{1}(S)$ we define the operator $Q$ by

[TABLE]

where $K$ is given by (2.19). Observe that for all $\varphi=(\varphi_{i})_{i\in\mathcal{N}}\in W^{2,1}(S)$, we have

[TABLE]

The last equality follows by (2.14). Denoting by $I_{i}^{E}$ the set of indexes $j\neq i$ of edges incident to $E_{i}$, it follows by (2.2)–(2.4), that for every $\varphi\in L_{0}^{1}(S)$ we have

[TABLE]

where $\varphi_{j}$ is the value of $\varphi$ on the edge $E_{j}$. We introduce the matrix $(q_{ij})_{i,j\in\mathcal{N}}$ by

[TABLE]

and

[TABLE]

Then

[TABLE]

and the operator $Q$ may be identified with the matrix $(q_{ij})_{i,j\in\mathcal{N}}$. (Notice the difference between the matrix defined here and the matrix from Theorem 1.1.) The operator $Q$, since the matrix $(q_{ij})_{i,j\in\mathcal{N}}$ is finite, generates strongly continuous semigroup $\{\mathrm{e}^{tQ}\}_{t\geqslant 0}$ in $L_{0}^{1}(S)$.

Theorem 2.11**.**

For each $n\in\mathbb{N}$ let the operator $A_{n}^{*}$ be defined by (2.1) with domain composed of functions $\varphi\in W^{2,1}(S)$ satisfying boundary conditions (2.4). Then, for $P$ and $Q$ defined by (2.25) and (2.27), respectively, we have

[TABLE]

in $L^{1}(S)$. The convergence is uniform on compact subsets of $(0,\infty)$. If $\varphi\in L_{0}^{1}(S)$, then (2.28) holds also for $t=0$, and the convergence is uniform on compact subsets of $[0,+\infty)$.

Proof.

Let $\mathbb{X}\coloneqq L^{1}(S)$, $\mathcal{A}_{n}\coloneqq A_{n}^{*}$, $\mathcal{A}\coloneqq\sigma K$, $\mathcal{B}\coloneqq B$, and $\varepsilon_{n}\coloneqq\kappa_{n}^{-1}$. Then $\mathcal{P}$ defined by (2.24) equals $P$. By Lemma 2.9 and Lemma 2.10 condistions (i) and (ii) from Kurtz’s theorem are satisfied. Moreover, $\mathcal{P}\mathcal{A}$ with domain $\mathbb{Y}$ equals $Q$. Therefore, the claim follows by Theorem 2.8. ∎

3. Analysis in $L^{2}(S)$

Here we consider a similar problem as in Section 2, however we change the space $L^{1}(S)$ to $L^{2}(S)$. Naturally,

[TABLE]

where $L^{2}(E_{i})$ is the complex Hilbert space of (equivalence classes of) Lebesgue square integrable complex functions on $E_{i}$, and the latter space is isometrically isomorphic to the standard $L^{2}(0,d_{i})$ (see remarks at the beginning of Section 2.1). In contradistinction to $L^{1}(S)$, we denote elements of $L^{2}(S)$ by $u$ and $v$. The space $L^{2}(S)$ equipped with the scalar product

[TABLE]

is a complex Hilbert space. By $H^{1}(S)$ we denote the Sobolev space $W^{1,2}(S)\subset L^{2}(S)$, that is $u\in H^{1}(S)$ if and only if $u\in L^{2}(S)$, $u$ is weakly differentiable and $u^{\prime}\in L^{2}(S)$. Similarly we define $H^{2}(S)=W^{2,2}(S)$ as the space of $u\in L^{2}(S)$ such that $u$ and $u^{\prime}$ are weakly differentiable, and $u^{\prime},u^{\prime\prime}\in L^{2}(S)$.

For each $n\in\mathbb{N}$ we define the operator $A_{n}^{*}$ in $L^{2}(S)$ similarly as in Section 2.4, that is

[TABLE]

where $D(A_{n}^{*})$ is the set of function $u\in H^{2}(S)$ such that transmission conditions (2.4) hold. Here we consider $F_{L,j}$ and $F_{R,j}$ as functionals on $H^{2}(S)$. We prove in Theorem 3.2 that $A_{n}^{*}$’s generate holomorphic semigroups in $L^{2}(S)$ and, in Theorem 3.6, investigate their asymptotics.

3.1. Sesquilinear forms

In what follows we extensively use the theory of sesquilinear forms, see for example [11, Chapter 6] or [16, Chapter 1]. We recall that a sesquilinear form (or simply form) in a complex Hilbert space $(H,\langle\cdot,\cdot\rangle_{H})$ is a mapping $\mathfrak{a}\colon D(\mathfrak{a})\times D(\mathfrak{a})\to\mathbb{C}$ such that $\mathfrak{a}(\cdot,u)$ is linear and $\mathfrak{a}(u,\cdot)$ is antilinear for all $u\in D(\mathfrak{a})$. The set $D(\mathfrak{a})$ is a linear subspace of $H$ and is called the domain of $\mathfrak{a}$. We say that $\mathfrak{a}$ is densely defined if $D(\mathfrak{a})$ is a dense set in $H$, accretive if $\operatorname{Re}\mathfrak{a}(u,u)\geqslant 0$ for each $u\in D(\mathfrak{a})$, and closed if $D(\mathfrak{a})$ is a Hilbert space with respect to the inner product $\langle u,v\rangle_{\mathfrak{a}}\coloneqq\operatorname{Re}\mathfrak{a}(u,v)+\langle u,v\rangle_{H}$, $u,v\in D(\mathfrak{a})$. Moreover, we call $\mathfrak{a}$ sectorial if there exists $M>0$ such that

[TABLE]

If $(\mathfrak{a}_{n})_{n\in\mathbb{N}}$ is a sequence of forms in $H$, then we say that forms $\mathfrak{a}_{n}$’s are uniformly sectorial if there exists $M>0$ (independent of $n$) such that (3.1) holds with $\mathfrak{a}$ replaced by $\mathfrak{a}_{n}$ for $n\in\mathbb{N}$. Also, to shorten notation, we write $\mathfrak{a}(u)$ for $\mathfrak{a}(u,u)$.

For a densely defined form $\mathfrak{a}$ we define the associated operator $A$ in the following way. The domain $D(A)$ of $A$ is the set of $u\in D(\mathfrak{a})$ such that there exists $f\in D(\mathfrak{a})$ satisfying

[TABLE]

For $u\in D(\mathfrak{a})$ we set

[TABLE]

This definition is correct since by the density of $D(\mathfrak{a})$ the element $f$ is unique. It turns out, see [11, Theorem VI.2.1] or [16, Theorem 1.52], that the operator associated with a densely defined, accretive, closed and sectorial form $\mathfrak{a}$ is the generator of a bounded holomorphic semigroup in $H$ denoted $\{\mathrm{e}^{-t\,\mathfrak{a}}\}_{t\geqslant 0}$.

In order to state Ouhabaz’s result (see [15, Theorem 5]), which is our main tool in this section, we need to introduce the notion of the degenerate semigroup related to a non densely defined form. Let $\mathfrak{a}$ be a form in $H$. If the domain $D(\mathfrak{a})$ is not dense in $H$, then there is no operator associated with the form $\mathfrak{a}$. However, we may consider the form in the closure $H_{0}$ of $D(\mathfrak{a})$ in $H$. Then $H_{0}$ is a Hilbert space and there is the operator $A_{0}$ associated with $\mathfrak{a}$ as restricted to $H_{0}$. If the form $\mathfrak{a}$ is accretive, closed and sectorial, then $A_{0}$ generates a bounded, holomorphic semigroup $\{\mathrm{e}^{tA_{0}}\}_{t\geqslant 0}$ in $H_{0}$. We extend this semigroup to the degenerate semigroup $\{\mathrm{e}^{-t\,\mathfrak{a}}\}_{t\geqslant 0}$ in $H$, by setting

[TABLE]

where $P_{H_{0}}$ is the orthogonal projection of $H$ onto $H_{0}$.

In our particular setup, we use the following special case of Ouhabaz’s theorem (see [5, Theorem 3.2 and Corollary 3.3] for the general version).

Theorem 3.1**.**

Let $(\mathfrak{a}_{n})_{n\in\mathbb{N}}$ be a sequence of accretive, closed and uniformly sectorial forms defined on the same domain $D$ in a Hilbert space $H$. Assume that

1. (i)

$\operatorname{Re}\mathfrak{a}_{n}(u)\leqslant\operatorname{Re}\mathfrak{a}_{n+1}(u)$* for every $u\in D$,* 2. (ii)

for each $u\in D$ the imaginary part $\operatorname{Im}\mathfrak{a}_{n}(u)$ does not depend on $n\in\mathbb{N}$.

Then the form $\mathfrak{a}$ defined by

[TABLE]

with domain

[TABLE]

is accretive, closed and sectorial. Moreover, for every $u\in H$ and $t>0$,

[TABLE]

in $H$, and the convergence is uniform on compact subsets of $(0,\infty)$. If $u$ is in the closure of $D(\mathfrak{a})$, then (3.2) holds also for $t=0$, and the convergence is uniform on compact subsets of $[0,+\infty)$.

3.2. Generation theorem in $L^{2}(S)$

We prove a generation result analogous to Theorem 2.4.

Theorem 3.2**.**

For each $n\in\mathbb{N}$ the operator $A_{n}^{*}$ in $L^{2}(S)$ generates a holomorphic semigroup $\{\mathrm{e}^{tA_{n}^{*}}\}_{t\geqslant 0}$ in $L^{2}(S)$. Furthermore, there exists $\gamma>0$ such that

[TABLE]

Throughout this section fix $n\in\mathbb{N}$. We begin by finding a form $\mathfrak{a}_{n}$ in $L^{2}(S)$ such that $A_{n}^{*}$ is the operator associated with $\mathfrak{a}_{n}$. Define the form $\mathfrak{b}_{n}$ in $L^{2}(S)$ by

[TABLE]

with domain $D(\mathfrak{b}_{n})\coloneqq H^{1}(S)$. Let $u\in D(A_{n}^{*})$ and $v\in H^{1}(S)$. Integration by parts gives

[TABLE]

Hence, since $u$ satisfies transmission conditions (2.4),

[TABLE]

where $\mathfrak{c}$ is the form in $L^{2}(S)$ given by

[TABLE]

with domain $D(\mathfrak{c})\coloneqq H^{1}(S)$. Note that $\mathfrak{c}$ does not depend on $n$. Formula (3.4) suggests that we should set $D(\mathfrak{a}_{n})\coloneqq H^{1}(S)$ and define

[TABLE]

The space $H^{1}(S)$ is dense in $L^{2}(S)$, therefore, in order to prove that the operator associated with $\mathfrak{a}_{n}$ generates a holomorphic semigroup, we are left with proving that the form $\mathfrak{a}_{n}$ is accretive, closed and sectorial.

For the proofs of Lemma 3.3 and Proposition 3.4 it is useful to denote

[TABLE]

Lemma 3.3**.**

The form $\mathfrak{b}_{n}$ is accretive and closed.

Proof.

For $\sqrt{\sigma}\coloneqq(\sqrt{\sigma_{i}})_{i\in\mathcal{N}}$ we have

[TABLE]

which proves accretivity. Observe that

[TABLE]

Hence the norm $\lVert\,\cdot\,\rVert_{\mathfrak{b}_{n}}$ associated with $\mathfrak{b}_{n}$ is equivalent to the standard norm in $H^{1}(S)$ (which is a Hilbert space), and the claim follows. ∎

Proposition 3.4**.**

The form $\mathfrak{a}_{n}$ is closed and there exists $\gamma>0$ such that the form $\mathfrak{a}_{n}+\gamma$ is sectorial with

[TABLE]

Here, by the form $\mathfrak{a}_{n}+\gamma$ we mean the form defined by $(\mathfrak{a}_{n}+\gamma)(u,v)=\mathfrak{a}_{n}(u,v)+\gamma\langle u,v\rangle_{L^{2}(S)}$.

Proof.

Observe that for some $c>0$ we have

[TABLE]

where $\lVert\,\cdot\,\rVert_{L^{\infty}(S)}$ is the standard (essential) supremum norm. By the Gagliardo-Nirenberg interpolation (see e.g. [9, Theorem 12.83]) there exists $C>0$ such that

[TABLE]

Hence, by Young’s inequality,

[TABLE]

for $\gamma\coloneqq c^{2}C^{2}/(\kappa_{n}\sigma_{\min})$. Therefore,

[TABLE]

This means that $\mathfrak{c}$ is $\mathfrak{b}_{n}$-form bounded with $\mathfrak{b}_{n}$-bound $1/2$ (see [16, Definition 1.17]). Using [11, Theorem VI.3.4] or [16, Theorem 1.19], it follows that the form $\mathfrak{a}_{n}=\mathfrak{b}_{n}+\mathfrak{c}$ is closed as a relatively bounded perturbation of the closed form $\mathfrak{b}_{n}$.

To show the second part of the lemma notice that by (3.5) and (3.8) we have

[TABLE]

and

[TABLE]

Combining these two inequalities we obtain

[TABLE]

which proves (3.7). ∎

Proposition 3.5**.**

The operator associated with $\mathfrak{a}_{n}$ is $A_{n}^{*}$.

Proof.

Let $B_{n}$ be the operator associated with $\mathfrak{a}_{n}$. We claim that $B_{n}$ is exactly $A_{n}^{*}$. Formula (3.4) shows that $D(A_{n}^{*})\subseteq D(B_{n})$ and $B_{n}u=A_{n}^{*}u$ for $u\in D(A_{n}^{*})$. On the other hand let $u\in D(B_{n})$. There exists $f\in H^{1}(S)$ such that $\mathfrak{a}_{n}(u,v)=-\langle f,v\rangle_{L^{2}(S)}$ for all $v\in H^{1}(S)$. Choose $v\in H^{1}(S)$ that on each edge is compactly supported smooth function. Then $\mathfrak{c}(u,v)=0$ and consequently

[TABLE]

Therefore $\langle f,v\rangle=-\kappa_{n}\langle\sigma u^{\prime},v^{\prime}\rangle$, which proves that $u\in H^{2}(S)$ and $f=B_{n}u=\kappa_{n}\sigma u^{\prime\prime}$. Now for fixed $i\in\mathcal{N}$ choose $v\in H^{1}(S)$ with

[TABLE]

Then

[TABLE]

and, integrating by parts,

[TABLE]

Hence

[TABLE]

This equality, since $\mathfrak{a}_{n}(u,v)=-\kappa_{n}\langle\sigma u^{\prime\prime},v\rangle$, is equivalent to

[TABLE]

In the same way we prove that

[TABLE]

This means that transmission conditions (2.4) are satisfied and, since $u\in H^{2}(S)$, it follows that $u\in D(A_{n}^{*})$. Finally $D(B_{n})=D(A_{n}^{*})$ and $B_{n}u=A_{n}^{*}u$ for all $u\in D(A_{n}^{*})$. ∎

Proof of Theorem 3.2.

Let $\gamma>0$ be as in Proposition 3.4. Then the form $\mathfrak{a}_{n}+\gamma$ is densely defined, accretive, closed and sectorial. Moreover, by Proposition 3.5, the operator associated with $\mathfrak{a}_{n}+\gamma$ is $A_{n}^{*}-\gamma$. Therefore, by [16, Theorem 1.52], it follows that $A_{n}^{*}-\gamma$ generates a holomorphic contraction semigroup in $L^{2}(S)$. Hence, $A_{n}^{*}$ generates a holomorphic semigroup in $L^{2}(S)$, and since

[TABLE]

inequality (3.3) holds. ∎

3.3. Convergence result in $L^{2}(S)$

Let $L_{0}^{2}(S)$ be the closed subspace of $L^{2}(S)$ consisting of complex functions that are constant on each edge. Similarly as for $L_{0}^{1}(S)$ defined in Section 2.4, the space $L_{0}^{2}(S)$ is isometrically isomorphic to $\mathbb{C}^{N}$ equipped with the appropriate scalar product.

Let $Q$ be the operator in $L_{0}^{2}(S)$ defined as in $L_{0}^{1}(S)$ by formula (2.27). Similarly, let $P$ be the projection of $L^{2}(S)$ onto $L_{0}^{2}(S)$ given by (2.25). Then, for the operators $A_{n}^{*}$’s defined in the beginning of Section 3, the following analogous result to Theorem 2.11 holds.

Theorem 3.6**.**

For every $u\in L^{2}(S)$ we have

[TABLE]

in $L^{2}(S)$. The convergence is uniform on compact subsets of $(0,+\infty)$. If $\varphi\in L_{0}^{2}(S)$, then (3.9) holds also for $t=0$, and the convergence if uniform on compact subsets of $[0,+\infty)$.

For $n\in\mathbb{N}$ let $\mathfrak{a}_{n}$ be the form in $L^{2}(S)$ defined by (3.5). Fix $\gamma>0$ as in Proposition 3.4 and define

[TABLE]

with domain

[TABLE]

Lemma 3.7**.**

The sequence $(\mathfrak{a}_{n}^{\gamma})_{n\in\mathbb{N}}$ consists of accretive, closed and uniformly sectorial forms. Moreover,

[TABLE]

and

[TABLE]

for all $n\in\mathbb{N}$, $u\in H^{1}(S)$.

Proof.

The first part is a consequence of Proposition 3.4. For the second observe that

[TABLE]

The claim follows from the fact that $\kappa_{n}\leqslant\kappa_{n+1}$ and $\mathfrak{b}_{n}(u)=\operatorname{Re}\mathfrak{b}_{n}(u)$ for all $n\in\mathbb{N}$ and $u\in H^{1}(S)$. ∎

Let $\mathfrak{a}^{\gamma}$ be the form in $H$ defined by

[TABLE]

with domain

[TABLE]

This definition makes sense because the limit of $\mathfrak{a}_{n}^{\gamma}(u)$ as $n\to+\infty$ exists, and we may define $\mathfrak{a}^{\gamma}$ by the polarization equality.

Lemma 3.8**.**

We have

[TABLE]

and

[TABLE]

Proof.

Let $u\in H^{1}(S)$ and observe that $u\in D(\mathfrak{a}^{\gamma})$ if and only if

[TABLE]

By (3.6), the last condition holds if and only if $u^{\prime}=0$ in $L^{2}(S)$, since $\kappa_{n}\to+\infty$ as $n\to+\infty$. This completes the proof, because $u^{\prime}=0$ is equivalent to $u\in L_{0}^{2}(S)$, and the formula (3.12) follows now immediately from (3.11) and (3.10). ∎

Let $\mathfrak{c}_{0}$ be the restriction of $\mathfrak{c}$ to $L_{0}^{2}(S)$, that is the form in $L_{0}^{2}(S)$ given by

[TABLE]

Lemma 3.9**.**

The operator associated with $\mathfrak{c}_{0}$ equals $Q$.

Proof.

Let $u,v\in L_{0}^{2}(S)$. Then, calculating as in (2.26),

[TABLE]

where $u_{j}$ is the value of $u$ on the edge $E_{j}$. Therefore, see (2.27),

[TABLE]

and the claim follows. ∎

Corollary 3.10**.**

The operator associated with $\mathfrak{a}^{\gamma}$, as a form in $L_{0}^{2}(S)$, equals $Q-\gamma$.

Proof.

The claim is a consequence of (3.12) and Lemma 3.9. ∎

Finally, we are ready to prove our convergence result in $L^{2}(S)$.

Proof of Theorem 3.6.

By Lemma 3.7 the assumptions of Theorem 3.1 hold for the sequence $(\mathfrak{a}_{n}^{\gamma})_{n\in\mathbb{N}}$. Hence,

[TABLE]

By Proposition 3.5, $A_{n}^{*}-\gamma$ is associated with $\mathfrak{a}_{n}^{\gamma}$, and hence by Corollary 3.10 we can rewrite the above relation in the form

[TABLE]

which is equivalent to (3.9). ∎