# A non-autonomous $p$-Adic diffusion equation on time changing graphs

Patrick Erik Bradley and Ángel Morán Ledezma

###### Abstract

Motivated by the recently proven presence of ultrametricity in physical models
(certain spin glasses) and the very recent study of Turing patterns on locally
ultrametric state spaces, first non-autonomous diffusion operators on such
spaces, where finitely many compact $p$-adic spaces are joined by a graph
structure, are studied, including their Dirichlet and van Neumann eigenvalues.
Secondly, the Cauchy problem for the heat equation associated with these
operators is solved, its solution approximated by Trotter products, and
thirdly, the corresponding Feller property as well as the Markov property (a
Hunt process) is established.

##  1 Introduction

The Gibbs measure associated with certain mean-field spin glass models,
including the Sherrington-Kirkpatrick model, was conjectured to have
ultrametric support in the thermodynamic limit [11]. This conjecture, known as
the _Parisi ultrametricity conjecture_ , was proved by Panchenko under the
condition that the so-called _Ghirlanda-Guerra identities_ are satisfied [13].
This conjecture (and now result) has motivated up to this day the study of
random processes on ultrametric state spaces, the most studied type being
diffusion on the $p$-adic numbers, cf. e.g. [12, 8] and many other research
articles in the physical and mathematical literature.

Ultrametricity of a space means that it has a hierarchical structure, i.e. its
points can be viewed as the boundary of a tree, and therefore its topology, as
well as any physical process on it, are determined by this tree structure. The
$p$-adic numbers form a convenient mathematical framework for studying
ultrametric spaces by viewing those trees as part of the regular tree
structure of the $p$-adic numbers—as long as the vertex degrees do not exceed
the bound given by the $p$-adic regularity. However, if the state space is
only locally ultrametric, then it seemed unclear how to mathematically model
diffusion on such structures until Zúñiga-Galindo’s idea of modelling
interactions between those local ultrametric pieces appeared via an externally
imposed graph structure on this ensemble of pieces which then can be viewed as
disjoint compact open subsets of the $p$-adic number field $\mathbb{Q}_{p}$
[19]. He even argues that $p$-adic analysis is a natural tool for studying
Turing patterns on mean-field models. In his case, these local pieces were
modelled as $p$-adic balls, and thus he was lead to define a $p$-adic
Laplacian integral operator resembling the integral operator representation of
the well-studied Vladimirov (or Vladimirov-Taibleson) operator, cf. [16, 18].

The study of non-autonomous diffusion on $p$-adic domains seems to have been
given significantly less consideration in the literature. This motivates the
authors of this article to investigate diffusion processes on locally
hierarchical structures whose interactions (given as weighted edges) are time-
dependent. The vertices of the underlying time-dependent graph are fixed in
this setting. After clarifying the construction of $p$-adic operators
associated with finite graphs—the “dictionary” set up in [2] needs a
correction—properties of a time-dependent version of the Zúñiga operators from
[19] are studied, proving that the uniform continuity of the kernel function
implies the strong continuity of the new time-dependent operator (Theorem
2.11) which is necessary for proving the Feller property later. Dirichlet and
van Neumann eigenvalues are calculated (Theorem 2.13), whereby the
corresponding boundary conditions are generalised from graph theory.
Thereafter, the non-autonomous Cauchy problem for the heat equation associated
with these operators is solved (Theorem 3.9), thereby approximating the
solution using Trotter products, and then finding that the temporal
discretisation error is in first order
$O\left(\frac{1}{n}e^{\frac{1}{n}}\right)$ (Corollary 3.11). In the end the
Feller property (Theorem 4.6) and corresponding Markov property—a Hunt
process—is established (Theorem 4.8).

##  2 Non-autonomous $p$-adic equations for time-changing graphs

###  2.1 Preliminaries

In order to fix notation, denote the field of $p$-adic numbers as
$\mathbb{Q}_{p}$. The usual $p$-adic absolute value on $\mathbb{Q}_{p}$ is
written as $\left\lvert\cdot\right\rvert_{p}$. The additive group of
$\mathbb{Q}_{p}$ being a locally compact abelian group means that it has a
Haar measure, denoted as $dx$ (if the current variable is $x$). It is
invariant under translation by adding any fixed element of $\mathbb{Q}_{p}$,
used for integrating functions, and is normalised here by setting

$\int_{\mathbb{Z}_{p}}dx=1$

where $\mathbb{Z}_{p}$ is the subring of $\mathbb{Q}_{p}$ consisting of
elements with $p$-adic absolute value at most $1$, i.e. $\mathbb{Z}_{p}$ is
the $p$-adic unit ball.

For each $t\in[0,\infty)$, we take a matrix $A(t)\in\mathbb{C}^{n\times n}$.
If for every $t\in[0,\infty)$ we assume that $A(t)$ is a symmetric matrix,
then we can view this matrix as an adjacency matrix of a weighted graph
$\mathcal{G}_{t}$ with $n$ nodes in a natural way. For every $t$, the vertex
set of $\mathcal{G}_{t}$ can be embedded into $\mathbb{Q}_{p}$. The fact that
$\mathcal{G}_{t}$ has an unchanging set of nodes allows to fix an embedding
for all $t>0$. In other words the only difference between $\mathcal{G}_{t}$
and $\mathcal{G}_{s}$ for $s\neq t$ are the edge weights. The set of vertices
being constant, allows to write: $V(\mathcal{G}):=V(\mathcal{G}_{t})$, for any
$t>0$. Following [19], we fix the embedding

$V(\mathcal{G})\rightarrow\mathbb{Q}_{p},$

and fix $N$ such that each vertex $I\in V(G)$ lies in a distinct $p$-adic ball
of radius $r=p^{-N}$

$B_{r}(I)=\\{x\in\mathbb{Q}_{p}:|x-I|\leq r\\}$

sufficiently small such that all these balls are disjoint in $\mathbb{Q}_{p}$.
Let

$K_{N}=\bigsqcup_{I\in V(\mathcal{G})}B_{p^{-N}}(I).$

Each vertex of $\mathcal{G}$ can be assumed to have the form

$I=I_{0}+I_{1}p+...+I_{N-1}p^{N-1},$

where each $I_{i}\in\\{0,\dots,p-1\\}$ for $i=0,1,2,\dots$ is a $p$-adic
digit.

###  2.2 Basis of $L^{2}(K_{N},\mathbb{C})$

We are interested in the Hilbert space $L^{2}(K_{N},\mathbb{C})$ with inner
product

$\langle f,g\rangle=\int_{K_{N}}f(x)\ \overline{g(x)}\,dx$

Let $\Omega(p^{r}|x-a|_{p})$ denote the indicator function of the ball
$B_{-r}(a)$. If $r=N$ and $a=I\in V(G)$, we write

$\Omega(p^{N}|x-I|_{p})=:\Omega_{I,N}(x)$

The space of functions
$X_{N}=\operatorname{Span}_{\mathbb{C}}\\{\Omega_{I,N}(x)\\}\cong\mathbb{C}^{n}$
is an $n$-domensional vector space and it will be denoted by $X_{N}$.

######  Definition 2.1.

The functions

$\Psi_{j,n,r}(x)=p^{\frac{r}{2}}\chi_{p}(p^{-r-1}jx)\Omega(p^{r}|x-p^{-r}n|_{p})$

where $r\in\mathbb{Z}$, $j\in\\{1,...,p-1\\}$, and $n$ runs through a fixed
set of representatives of $\mathbb{Q}_{p}/\mathbb{Z}_{p}$ are called Kozyrev
functions. ion. If $Supp(\Psi_{j,n,r})=B_{-r}(p^{-r}n)\subset B_{-N}(I)$, then
$|p^{-r}n|_{p}=|I|_{p}$, with $N\leq r$.

Proof. We have that $p^{-r}\leq p^{-N}$, and

$|p^{-r}n-I|_{p}<p^{-N}\implies|p^{-N-r}n-p^{-N}I|_{p}<1,$

where $p^{-N}I\in\mathbb{Q}_{p}/\mathbb{Z}_{p}$, and therefore
$|p^{-N}I|_{p}>1$. Hence, if $p^{-N-r}n\neq p^{-N}I$, we have

$1>|p^{-N-r}n-p^{-N}I|_{p}=\max\\{|p^{-N-r}n|_{p},|p^{-N}I|_{p}\\}>1.$

This shows that we must have $|p^{-N-r}n|_{p}=|p^{-N}I|_{p}$, implying the
assertion. ∎

Therefore an orthonormal basis of

$\mathcal{L}_{0}^{2}(K_{N})={\left\\{f\in
L^{2}(K_{N})\mid\int_{K_{N}}f(x)\,dx=0\right\\}}$

is the set of functions of the form

$\Psi_{j,I,r}(x)=p^{\frac{r}{2}}\chi_{p}(p^{-r-1}jx)\Omega(p^{r}|x-I|_{p}),$
(1)

with $r\geq N$, $j=0,...p-1$ and $I\in V(\mathcal{G})$.

######  Proposition 2.2.

We have $L^{2}(K_{N},\mathbb{C})=X_{N}\oplus\mathcal{L}_{0}^{2}(K_{N})$.

Proof. This is shown in [19, (10.8)], where orthogonality uses the well-known
result:

$\langle\Omega_{I,N},\Psi_{j,J,r}\rangle=0,$ (2)

for every $r\geq N$, $j=0,...,p-1$ and $I,J\in V(\mathcal{G})$, cf. e.g. [10,
Thm. 3.9] or [1, Thm. 9.4.2]. ∎

By the above proposition, the set of functions consisting of the functions
$p^{-\frac{N}{2}}\Omega_{I,N}(x)$ and the Kozyrev functions of the form (1)
form an orthonormal basis of $L^{2}(K_{N},\mathbb{C})$.

###  2.3 Two operators over $L^{2}(K_{N},\mathbb{C})$

For every matrix $A\in\mathbb{C}^{n\times n}$ we can define a bounded linear
operator in the following way. We define

$A(x,y)=p^{N}\sum_{I,J\in
V(\mathcal{G})}A_{I,J}\Omega_{I,N}(x)\Omega_{J,N}(y).$

This function is a test function on $K_{N}\times K_{N}$, and hence, $A(x,y)\in
L^{2}(K_{N}\times K_{N},\mathbb{C})$. This function gives rise to an integral
operator in the usual way:

$\mathcal{A}u(x)=\int_{K_{N}}A(x,y)u(y)dy.$ (3)

This operator is a bounded linear operator from $L^{2}(K_{N},\mathbb{C})$ to
itself.

######  Proposition 2.3.

Let $A\in\mathbb{C}^{n\times n}$ and $\mathcal{A}$ its integral operator of
the form (3). Then $X_{N}^{\perp}\subset Ker(\mathcal{A})$, where
$X_{N}^{\perp}$ is the orthogonal complement of $X_{N}$.

Proof. Note that

$\mathcal{A}u(x)=p^{N}\sum_{I,J\in V(\mathcal{G})}A_{I,J}\langle
u,\Omega_{J,N}\rangle\Omega_{I,N}(x)\in X_{N}.$

If $u\in X_{N}^{\perp}$, then $\langle u,\Omega_{I,N}\rangle=0$, therefore by
the above equality $\mathcal{A}u=0$. ∎

######  Corollary 2.4.

For every matrix $A\in\mathbb{C}^{n\times n}$, and every
$u\in\mathcal{L}_{0}^{2}(K_{N})$, $\mathcal{A}u=0$. In particular the set
Kozyrev functions of the form (1) are contained in $Ker(\mathcal{A})$.

Proof. By Proposition 2.3 and equality (2), the result follows. ∎

Recall that the Laplacian matrix $L$ is defined as

$L=A-(\gamma_{I}\delta_{IJ})$

where $A$ is the adjacency matrix of a graph $\mathcal{G}$,
$\gamma_{I}=\sum_{J}A_{I,J}$ is the degree of the vertex $I$, and
$\delta_{I,J}$ is the identity matrix. We can associate to $L$ an operator of
the form (3) denoted by $\mathcal{L}$. By Corollary 2.3, the operator
$\mathcal{L}$ only ”acts” on the space $X_{N}$, that is the operator
$\mathcal{L}$ defined similarly as in (3):

$\mathcal{L}u(x)=\int_{K_{N}}L(x,y)u(y)\,dy$

with

$L(x,y)=p^{N}\sum\limits_{I,J\in
V(\mathcal{G})}(A_{IJ}-\gamma_{I}\delta_{IJ})\Omega_{I,N}(x)\Omega_{J,N}(x)$

is just a rewrite of the Laplacian matrix in a space of dimension $n$, namely
$X_{N}\cong\mathbb{C}^{n}$ and does not give an outcome of functions in
$\mathcal{L}_{0}^{2}(K_{N})$. In particular, the Kozyrev functions of the form
(1) are not associated to any non zero eigenvalue of the operator
$\mathcal{L}$. Therefore the operator $\mathcal{L}$ is not an adequate
candidate for the $p$-adic Laplacian, because it gives the same information as
the Laplacian matrix, but not more.

######  Remark 2.5.

This is important in order to define an analogous operator to the Laplacian
operator in the $p$-adic case. In [2, Lemma 2.7] it is stated that the
operator $\mathcal{L}$ coincides with the operator defined by the integral

$\mathbb{L}u(x)=\int_{K_{N}}A(x,y)(u(y)-u(x))dy$

in the space of test functions, but this is not true as the following
computations shows: Note that

$\displaystyle\mathbb{L}u(x)$ $\displaystyle=\int_{K_{N}}A(x,y)(u(y)-u(x))dy$
$\displaystyle=p^{N}\sum_{I,J}A_{I,J}\langle
u,\Omega_{J,N}\rangle\Omega_{I,N}(x)-u(x)\gamma(x),$

where

$\gamma(x)=\int_{K_{N}}A(x,y)dy=\sum_{I,J}A_{I,J}\Omega_{I,N}(x).$

Then let $f=\Omega(p^{r}|x-a|_{p})$ be the indicator function of a ball
$B_{-r}(a)\subset B_{-N}(J_{0})$, for some $J_{0}\in V(\mathcal{G})$. Then

$\begin{split}\mathcal{L}f(x)&=p^{N}\sum_{I,J}L_{I,J}\langle
f,\Omega_{J,N}\rangle\Omega_{I,N}(x)\\\
&=p^{N}\sum_{I,J}L_{I,J}p^{-r}\delta_{J_{0},J}\Omega_{I,N}(x)\\\
&=p^{N-r}\sum_{I}L_{I,J_{0}}\Omega_{I,N}(x)\\\
&=p^{N-r}\sum_{I}A_{I,J_{0}}\Omega_{I,N}(x)-p^{N-r}\gamma_{J_{0}}\Omega_{J_{0},N}(x),\end{split}$

On the other hand, we have

$\begin{split}\mathbb{L}f(x)&=\int_{K_{N}}A(x,y)(f(y)-f(x))dy\\\
&=p^{N}\sum_{I,J}A_{I,J}\langle
f(y),\Omega_{J,N}(y)\rangle\Omega_{I,N}(x)-f(x)\gamma(x),\\\
&=p^{N}\sum_{I,J}A_{I,J}p^{-r}\delta_{J_{0},J}\Omega_{I,N}(x)-f(x)\gamma(x)\\\
&=p^{N-r}\sum_{I}A_{I,J_{0}}\Omega_{I,N}(x)-\Omega(p^{r}|x-a|_{p})\gamma(x),\end{split}$

If $B_{-r}(a)\subset B_{-N}(J_{0})$ is a proper inclusion, then we can take an
$x_{0}\in B_{-N}(J_{0})\setminus B_{-r}(a)$. Then

$\mathcal{L}f(x_{0})=-p^{N-r}\gamma_{J_{0}},$

and

$\mathbb{L}f(x_{0})=0.$

Here we supposed that $A_{I,I}=0$ for all $I$, i.e. that $\mathcal{G}$ is a
simple graph. It is clear that, if $r=N$ and $a=I$, then equality holds, that
is, the two operators coincide in the space $X_{N}$. Moreover, since
$\mathcal{L}$ and $\mathcal{A}$ (the operator associated to the adjacency
matrix of $\mathcal{G}$) are zero for all $u\in X_{N}^{\perp}$, we have for
such $u$ that

$\mathcal{L}u(x)=0,$

and

$\mathbb{L}u(x)=\mathcal{A}u(x)-u(x)\gamma(x)=-u(x)\gamma(x).$

There is another fundamental difference between these two operators, as the
following results shows.

######  Proposition 2.6.

The operator $\mathcal{L}$ is compact.

Proof. It is clear that $\mathcal{L}$ is a Hilbert-Schmidt operator, since it
is zero for all the Kozyrev functions supported in $K_{N}$. It is known that
all Hilbert-Schmidt operators are compact. ∎

######  Proposition 2.7.

The operator $\mathbb{L}:L^{2}(K_{N},\mathbb{C})\rightarrow
L^{2}(K_{N},\mathbb{C})$ defined by

$\mathbb{L}u(x)=\mathcal{A}u(x)-u(x)\gamma(x),$

where

$\gamma(x)=\int_{K_{N}}A(x,y)dy=\sum_{I\in
V(\mathcal{G})}\gamma_{I}\Omega_{I,N}(x),$

is a bounded linear operator, and all the Kozyrev functions of the form (1)
are eigenfunctions.

Proof. In [19, Lem. 10.1] it is shown that $\mathbb{L}$ is compact. The
eigenfunction property and corresponding eigenvalues of the Kozyrev wavelets
are shown in [19, Thm. 10.1]. Notice that the claim in _loc. cit_. about
$\mathbb{L}$ being compact is false, cf. Corollary 2.8 below. ∎

######  Corollary 2.8.

$\mathbb{L}$ is not compact.

###### Proof.

This is an immediate consequence of Proposition 2.7. ∎

Notice that the non-compactness of $\mathbb{L}$ was also observed in [3, Ex.
1].

######  Definition 2.9.

The operator $\mathcal{A}$ is called the _$p$ -adic Zúñiga operator_
associated with the finite graph $\mathcal{G}$, and $\mathbb{L}$ is called the
corresponding _$p$ -adic Zúñiga Laplacian_.

###  2.4 Non-autonomous $p$-adic Zúñiga operators

For a time-changing graph $\mathcal{G}_{t}$ we define its $p$-adic Laplacian
as the function $t\mapsto\mathbb{L}(t)$, where

$\mathbb{L}(t)=\mathcal{A}(t)-\Gamma(t)$

where

$\mathcal{A}(t)u(x)=\int_{K_{N}}A(x,y,t)u(y)dy,$

and

$\Gamma(t)u(x)=u(x)\gamma(t)(x),$

where the kernel function $A(x,y,t)$ and the degree function $\gamma(t)(x)$
are given by

$A(x,y,t)=p^{N}\sum_{I,J\in
V(\mathcal{G})}A_{IJ}(t)\Omega_{I,N}(x)\Omega_{J,N}(y),$

and

$\gamma(t)(x)=\sum_{I\in V(\mathcal{G})}\gamma_{I}(t)\Omega_{I,N}(x)$

with $\gamma_{I}(t)=\sum_{J\in V(\mathcal{G})}A_{IJ}(t).$

######  Definition 2.10.

The operators $\mathcal{A}(t)$ and $\mathbb{L}(t)$ are called _non-autonomous
$p$-adic Zúñiga operators_. Operator $\mathbb{L}(t)$ is also called _non-
autonomous $p$-adic Zúñiga Laplacian_. They are associated with the time-
changing graph $\mathcal{G}_{t}$.

######  Theorem 2.11.

If $t\mapsto A_{I,J}(t)$ is uniformly continuous for each $I,J\in
V(\mathcal{G})$ then the $p$-adic Laplacian $t\mapsto\mathbb{L}(t)$ of
$\mathcal{G}_{t}$ is strongly continuous, that is $t\mapsto\mathbb{L}(t)u$ is
a continuous function from $[0,\infty)$ to $L^{2}(K_{N},\mathbb{C})$ for every
$u\in L^{2}(K_{N},\mathbb{C})$.

Proof. Let $u\in L^{2}(K_{N},\mathbb{C})$ then

$||\mathcal{A}(t)u-\mathcal{A}(s)u||_{L^{2}}^{2}=\int_{K_{N}}\left|\int_{K_{N}}(A(x,y,t)-A(x,y,s))u(y)dy\right|^{2}dx.$

For $\varepsilon>0$, there is a positive real number $\delta>0$ such that

$|t-s|<\delta\implies|A_{IJ}(t)-A_{IJ}(s)|<\frac{\varepsilon}{n^{2}}.$

Hence

$\begin{split}|A(x,y,t)-A(x,y,s)|&\leq\sum_{IJ}|A_{IJ}(t)-A_{IJ}(t)|\\\
&\leq\varepsilon.\end{split}$

Therefore

$\begin{split}||\mathcal{A}(t)u-\mathcal{A}(s)u||_{L^{2}}^{2}&\leq\int_{K_{N}}\left(\int_{K_{N}}|A(x,y,t)-A(x,y,s)||u(y)|dy\right)^{2}dx\\\
&\leq\varepsilon^{2}\int_{K_{N}}\left(\int_{K_{N}}|u(y)|dy\right)^{2}dx\\\
&\leq\varepsilon^{2}\operatorname{Vol}(K_{N})||u||_{L^{2}}^{2},\end{split}$

where the last inequality is due to Hölder’s inequality. On the other hand,

$\displaystyle|\gamma(t)(x)-\gamma(s)(x)|$ $\displaystyle=\left|\sum_{I\in
V(\mathcal{I})}(\gamma_{I}(t)-\gamma_{I}(s))\Omega_{I,N}(x)\right|$
$\displaystyle\leq\sum_{I}\left|\sum_{J}A_{IJ}(t)-A_{IJ}(s)\right|$
$\displaystyle\leq\varepsilon$

Hence,

$\begin{split}||\Gamma(t)u-\Gamma(s)u||_{L^{2}}^{2}&\leq\int_{K_{N}}|\gamma(t)(x)-\gamma(t)(x)|^{2}|u(y)|^{2}dx\\\
&\leq\varepsilon^{2}||u||_{L^{2}}^{2}\end{split}$

Finally, we have

$\displaystyle||\mathbb{L}(t)u-\mathbb{L}(s)u||_{L^{2}}$
$\displaystyle\leq||\mathcal{A}(t)u-\mathcal{A}(s)u||_{L^{2}}+||\Gamma(t)u-\Gamma(s)u||_{L^{2}}$
$\displaystyle\leq\varepsilon||u||_{L^{2}}\left(\operatorname{Vol}(K_{N})^{\frac{1}{2}}+1\right)$

This ends the proof. ∎

In particular the proof of Theorem 2.11 shows that $t\mapsto\mathbb{L}(t)$ is
continuous in the uniform operator topology.

For each $t>0$, $\mathbb{L}(t)$ is a generator of a contraction semigroup
$\\{e^{s\mathbb{L}(t)}\\}$, associated to the Cauchy problem of the heat
equation:

$\begin{cases}\frac{\partial u}{\partial s}(s,x)=\mathcal{L}(t)u(s,x)\\\
u(0,x)=u_{0}(x)\in L^{2}(K_{n},\mathbb{C})\end{cases}\,.$ (4)

with $t\geq 0$. Now we present the initial value problem called the evolution
system or the non autonomous abstract Cauchy problem

$\begin{cases}\frac{\partial u}{\partial t}(t,x)=\mathbb{L}(t)u(t,x)\\\
u(s,x)=u_{0}(x)\in L^{2}(K_{n},\mathbb{C})\end{cases}\,.$ (5)

In the next section we present some results of the theory of semigroups and
non autonomous differential equations.

###  2.5 Dirichlet and von Neumann eigenvalues

Assume that the kernel function $A(x,y,t)$ is symmetric in $(x,y)$, and here
also that it is an analytic function for $t\geq 0$. Define for fixed $t\geq 0$
the $p$-adic edge set of $\mathcal{G}(t)$ as:

$\Omega_{E}(t)={\left\\{(x,y)\in K_{N}^{2}\mid A(x,y,t)\neq 0\right\\}},$

as well as the $p$-adic vertex boundary:

$\delta_{N}S(t)={\left\\{x\in K_{N}\setminus S\mid\exists y\in S\colon
A(x,y,t)\neq 0\right\\}}$

as well as the $p$-adic edge boundary of $\mathcal{G}(t)$:

$\partial_{N}S(t)={\left\\{(x,y)\in\Omega_{E}(t)\mid x\in S\wedge y\notin
S\right\\}}$

which could also be called the vertex and edge boundaries of the operator
$\mathcal{A}(t)$ defined in Section 2.4. Further, define the compact open set

$\Omega_{S}^{*}(t)=\left(\Omega_{E}(t)\cap S\times
S\right)\cup\partial_{N}S(t)\subset\mathbb{Q}_{p}^{2}$

Now, let $S\subset K_{N}$ be a compact open subset. Then define:

$V(S):={\left\\{I\in V\mid B_{p^{-N}}(I)\cap S\neq\emptyset\right\\}}.$

The space of functions satisfying the _Dirichlet boundary condition_ for $S$
is

$D_{S}^{*}(t)={\left\\{f\in L^{2}(K_{N})\mid\text{$f(x)=0$ for all
$x\in\delta_{N}S(t)$}\right\\}},$

and the space of functions satisfying the _von Neumann boundary condition_ for
$S$ is

$\displaystyle N_{S}^{*}(t)=$ $\displaystyle{\left\\{f\in
L^{2}(K_{N})\mid\forall
x\in\delta_{N}S(t)\colon\int_{{\left\\{(x,y)\in\partial_{N}S(t)\right\\}}}A(x,y,t)(f(x)-f(y))\,dy=0\right\\}},$

again with fixed $t\geq 0$.

In view of [4], the von Neumann boundary condition can be formulated as

$D_{\partial S(t),+}f(x)=0$

where

$D_{\partial
S(t),+}u(x)=\int_{{\left\\{(x,y)\in\partial_{N}S(t)\right\\}}}A(x,y,t)(f(x)-f(y))\,dy$

is an advection operator defined similarly as $D_{+}^{\alpha}$ from [4, §3],
and imitating the normal derivative w.r.t. the boundary of $S$ in the
classical case of Riemann manifolds.

The _first Dirichlet eigenvalue_ of $\mathbb{L}(t)$ (with respect to $S$) is
defined as:

$\lambda_{1,\mathbb{L}}^{(D)}(t)=\inf\limits_{f\neq 0\atop f\in
D_{S}^{*}(t)}\frac{p^{N}\int_{\Omega_{S}^{*}(t)}A(x,y,t)\left\lvert
f(x)-f(y)\right\rvert^{2}\,dx\,dy}{\int_{S}\left\lvert
f(x)\right\rvert^{2}\gamma(x)(t)\,dx}$

The _von Neumann eigenvalue_ of $\mathbb{L}(t)$ with respect to $S$ is defined
as:

$\lambda_{S,\mathbb{L}}^{(N)}(t)=\inf\limits_{f\neq
0\atop\int_{S}f(x)\gamma(x)\,dx=0}\frac{p^{N}\int_{\Omega_{S}^{*}(t)}A(x,y,t)\left\lvert
f(x)-f(y)\right\rvert^{2}\,dx\,dy}{\int_{S}\left\lvert
f(x)\right\rvert^{2}\gamma(x)(t)\,dx}$

where it is assumed that $\delta_{N}S(t)\neq\emptyset$ for this instance of
$t\geq 0$.

The Dirichlet and von Neumann eigenvalues of $\mathcal{G}(t)$ w.r.t. $S$ will
be denoted as

$\lambda_{1,\mathcal{G}}^{(D)}(t)\quad\text{and}\quad\lambda_{S,\mathcal{G}}^{(N)}(t),$

respectively.

######  Lemma 2.12.

Let $\psi\colon S\to\mathbb{C}$ be a Kozyrev wavelet. Then $\psi\in
N_{S}^{*}(t)$, i.e. $\psi$ satisfies the von Neumann boundary condition.

###### Proof.

Assume that the Kozyrev wavelet $\psi$ is supported in a ball $B\subset
B_{p^{-N}(I)}$ for some vertex $I\in V(S)$. Then

$\displaystyle\int_{(x,y)\in\partial_{N}S(t)}$ $\displaystyle
A(x,y,t)(\psi(x)-\psi(y))\,dy$
$\displaystyle=\int_{(x,y)\in\partial_{N}S(t)}A(x,y,t)\,dy\,\psi(x)$
$\displaystyle-\int_{(x,y)\in\partial_{N}S(t)}A(x,y,t)\psi(y)\,dy$
$\displaystyle=0-0=0$

where the left term vanishes, because $x\in\delta_{N}S(t)$, and the right
integral vanishes, because $\psi$ is a Kozyrev wavelet, and from Kozyrev’s
theorem that

$\int_{K_{N}}\psi(x)\,dx=0,$

cf. e.g. [10, Thm. 3.29] or [1, Thm. 9.4.2]. ∎

######  Theorem 2.13.

Let $S\subseteq K_{N}$ be compact open, and assume that $\delta S(t)$ is non-
empty for $t\in[0,a)$ with $a>0$. Then the following statements hold true:

  1. 1.

$\lambda_{1,\mathbb{L}}^{(D)}(t)\leq\min{\left\\{\lambda_{1,\mathcal{G}}^{(D)}(t),\hat{\gamma}_{S}(t)\right\\}}\leq
1,$

  2. 2.

$\lambda_{S,{\mathbb{L}}}^{(N)}(t)\leq\min{\left\\{\lambda_{S,\mathcal{G}}^{(N)}(t),\hat{\gamma}_{S}(t)\right\\}}$,

where

$\hat{\gamma}_{S}(t)=\min\limits_{I\in V(S)}\frac{\deg_{\partial
V(S)(t)}(I)}{\int_{S}\gamma(t)(x)\,dx}$

and $\deg_{\partial V(S)(t)}(I)$ is the degree of vertex $I$ in the edge
boundary $\partial V(S)(t)$, viewed as a subgraph of $\mathcal{G}(t)$, for
$t\in[0,a)$.

###### Proof.

The inequalities
$\lambda_{1,\mathbb{L}}^{(D)}(t)\leq\lambda_{1,\mathcal{G}}^{(D)}(t)$ and
$\lambda_{S,\mathbb{L}}^{(N)}(t)\leq\lambda_{S,\mathcal{G}}^{(N)}(t)$ are
obvious. Now, let $\psi$ be a Kozyrev wavelet supported in
$B_{p^{-r}}(I)\subset B_{p^{-N}}(I)\cap S$ for some vertex $I\in V(S)$. It
clearly satisfies the Dirichlet boundary condition, and, by Lemma 2.12, also
the von Neumann boundary condition. Hence, the Dirichlet and the von Neumann
eigenvalues are both less than or equal to:

$\frac{\sum\limits_{J\in\delta
V(S)(t)}A_{IJ}}{\int_{S}\gamma(x)(t)\,dx}=\frac{\deg_{\partial
V(S)(t)}(I)}{\int_{S}\gamma(x)(t)\,dx}$

where $\delta V(S)(t)$ is the vertex boundary of the set $V(S)\subset V$ in
$\mathcal{G}(t)$. This now shows the first inequality in _1._ , and that of
_2._ The remaining inequality in _1_. is given in [5, §8.4]. This proves the
theorem. ∎

The following examples show that the inequalities can be strict:

######  Example 2.14.

Let $\mathcal{G}(t)$ be the complete graph on three vertices $a,b,c$, and let
$S=B_{p^{-N}}(a)\sqcup B_{p^{-N}}(b)$. Then, according to [5, §8.4],
$\lambda_{1,\mathcal{G}}^{(D)}(t)$ is an eigenvalue of the submatrix of the
graph-Laplacian of $\mathcal{G}(t)$ spanned by the vertices $a,b$. This is the
matrix

$L_{S}=\begin{pmatrix}1&-\frac{1}{2}\\\ -\frac{1}{2}&1\end{pmatrix}$

with eigenvalues $\frac{3}{2}$ and $\frac{1}{2}$. Hence,
$\lambda_{1,\mathcal{G}}^{(D)}(t)=\frac{1}{2}$. But since the degree of each
vertex of $\mathcal{G}(t)$ equals $2$, it follows that

$\lambda_{1,\mathbb{L}}^{(D)}(t)\leq\hat{\gamma}_{S}(t)=\frac{1}{4}<\frac{1}{2}=\lambda_{1,\mathcal{G}}^{(D)}(t)$

for given $t\geq 0$.

######  Example 2.15.

Let $\mathcal{G}(t)$ be the complete graph on two vertices $a,b$, and let
$S=B_{p^{-N}}(a)$. Then any non-trivial function $f$ constant on the vertex
balls $B_{p^{-N}}(a)$ and $B_{p^{-N}}(b)$ satisfying the von Neumann boundary
condition, must satisfy $f(a)=0$ and thus

$\int_{S}\left\lvert f(x)\right\rvert^{2}\gamma(x)(t)\,dx=0$

This means that

$\lambda_{S,\mathcal{G}(t)}^{(N)}(t)=\infty$

and consequently

$\lambda_{S,\mathbb{L}}^{(N)}(t)<\lambda_{S,\mathcal{G}}^{(N)}(t)$

because $\hat{\gamma}_{S}(t)<\infty$.

##  3 Non Autonomous Cauchy Problem

Consider the $p$-adic non autonomous Cauchy problem for a finite graph. Most
of the underlying theory for this section is taken from [6, 7].

###  3.1 Review of the abstract Cauchy problem

Let $A(t)$ be a family of operators with $t\in\mathbb{R}$ on a Banach space
$X$. The following problem is addressed:

$\begin{cases}\frac{\partial u}{\partial t}(t)=A(t)u(t)\\\ u(s)=x\in
X\end{cases}$ (6)

with $t,s\in\mathbb{R}$ and $t\geq s$. The system $(\ref{nACP_abstract})$ is
called (nACP).

######  Definition 3.1.

A continuous function $u:[s,\infty)\rightarrow X$ is called a (strict)
solution of (6), if $u\in C^{1}([s,\infty),X)$, $u(t)\in D(A(t))$ for all
$t\geq s$, $u(s)=x$, and $\frac{\partial u}{\partial t}=A(t)u$ for $t\geq 0$.

The solution of non-autonomous Cauchy problem on some Banach space $X$ can be
given by a so called evolution family which can be defined as follows, cf.
[15]:

######  Definition 3.2.

A family $(U(t,s))_{t\geq s}$ of linear operators on a Banach space $X$ is
called an evolution family for (6), if

  1. 1.

$U(t,r)U(r,s)=U(t,s)$, $U(t,t)=id$, $t\geq s\in\mathbb{R}$.

  2. 2.

The mapping $(s,t)\mapsto U(t,s)$ is strongly continuous.

  3. 3.

$||U(t,s)||\leq Me^{\omega(t-s)}$ for some $M\geq 1$, $\omega\in\mathbb{R}$,
$t\geq s\in\mathbb{R}$.

  4. 4.

For $s\in\mathbb{R}$ the regularity space

$Y_{s}:=\\{y\in X:[s,\infty)\ni t\rightarrow U(t,s)y\ solves\ (nACP)\\}$

is dense in X, where $U(t,s)y$ is the unique solution of (nACP).

If there exists a solving evolution family, then (6) is called well-posed. We
have the following Theorem.

######  Theorem 3.3.

Let $X$ a Banach space and for every $t>0$ let $A(t)$ be a bounded linear
operator on $X$. If the function $t\mapsto A(t)$ is strongly continuous for
$0\leq t<T$ (that is, $t\mapsto A(t)u$ is continuous for each $u\in X$), then
for every $x\in X$ the problem (6) is well-posed.

###### Proof.

[7, Thm. 7.1.1]. ∎

We can apply the above Theorem to the problem (5) for time-changing graphs.

######  Proposition 3.4.

The system (5) is well-posed.

Proof. This is a direct consequence of Theorem 2.11. ∎

######  Remark 3.5.

If we let $U(t,s)x=u(t)$, where $u(t)$ is the solution of (5), then the
following holds true:

  1. 1.

$||U(t,s)||\leq e^{\int_{s}^{t}||\mathbb{L}(\tau)||d\tau}$.

  2. 2.

$(t,s)\mapsto U(t,s)$ is continuous in the uniform operator topology for
$s\leq t$.

  3. 3.

$\partial U(t,s)/\partial t=\mathcal{L}(t)U(t,s)$.

  4. 4.

$\partial U(t,s)/\partial s=-U(t,s)\mathcal{L}(s)$.

This is due to the fact that $t\mapsto\mathbb{L}(t)$ is continuous in the
uniform norm topology, cf. [14, Theorem 5.2].

In general, the solution does not have a closed form, but in special cases it
does, as the following example shows.

######  Example 3.6.

Under the hypothesis of Theorem 3.3, assume that $A(t)$ and
$\int_{s}^{t}A(\tau)d\tau$ commute for each $s,t$. Then

$U(t,s)=e^{\int_{s}^{t}A(\tau)d\tau}$

with the integral $\int A(\tau)d\tau$ being interpreted ”elementwise”.

###### Proof.

[7, Example 7.1.9]. ∎

###  3.2 Reviewing the autonomous case for graphs

First, we review how to solve the heat equation in the autonomous case:

$\begin{cases}\frac{\partial u}{\partial t}(x,t)=\mathbb{L}u(x,t)\\\
u(0,x)=u_{0}(x)\in L^{2}(K_{n},\mathbb{C})\end{cases}\,,$ (7)

where $\mathbb{L}$ is interpreted as $\mathbb{L}=\mathbb{L}(t_{0})$ for some
$t_{0}\geq 0$. The following Theorem allows us to apply the Fourier Method (or
the separation of variables method) to solve the system (7).

######  Theorem 3.7 (Zúñiga-Galindo, 2020).

The elements of the set:

$\operatorname{Spec}(L)\bigsqcup\\{-\gamma_{I}:I\in V(\mathcal{G})\\}$

are the eigenvalues of of $\mathbb{L}$. The corresponding eigenfunctions are
given by the following infinite set

$\left\\{\frac{\varphi_{\mu}}{||\varphi_{\mu}||_{2}}:\mu\in\operatorname{Spec}(L)\right\\}\bigsqcup\left\\{\Psi_{j,I,r}:I\in
V(\mathcal{G}),j\in\\{1,...,p-1\\},r\in\mathbb{Z},r\geq N\right\\},$

where the functions $\varphi_{\mu}(x)$ are defined by:

$\varphi_{\mu}(x)=\sum_{J\in
V(\mathcal{G})}\varphi_{\mu}(J)\Omega(p^{-N}|x-J|_{p})$ (8)

and the vector $(\varphi_{\mu}(J))_{J\in V(\mathcal{G})}$ is an eigenvector of
the (negative semi-definite) Laplacian matrix $L$ associated with adjacency
matrix $A$, corresponding to $\mu\in\operatorname{Spec}(L)$, and
$\Psi_{r,I,j}$ are the Kozyrev functions of the form (1). Furthermore we have
the following orthogonal decomposition

$L^{2}(K_{N},\mathbb{C})=X_{N}\oplus\mathcal{L}_{0}^{2}(K_{N}).$ (9)

###### Proof.

[19, Thm. 10.1]. ∎

According to [19, Thm. 4.2], the Cauchy problem of the heat equation (7), with
$t\geq 0$ has a solution which is unique. The solution is given by the
semigroup $\\{e^{t\mathbb{L}}\\}$ generated by $\mathbb{L}$. By the
decomposition (9), we propose a solution of the form

$u(x,t)=\sum_{\mu\in\operatorname{Spec}(L)}c_{\mu}(t)\varphi_{\mu}(x)+\sum_{\operatorname{supp}(\Psi_{j,I,r})\subset
K_{N}}c_{r,I,j}(t)\psi_{j,I,r}(x),$ (10)

where the infinite set
$\\{\Psi_{j,I,r}|\operatorname{supp}(\Psi_{j,I,r})\subset K_{N}\\}$ is
described in Theorem 3.7. Hence, the solution is an infinite sum, and the
convergence takes place in $L^{2}(K_{N})$. After inserting this into (7) we
obtain

$\begin{cases}\frac{\partial c_{\mu}}{\partial t}=\mu c_{\mu}\\\
\frac{\partial c_{r,I,j}}{\partial t}=-\gamma_{I}c_{j,I,r}\end{cases}\,.$

where $\mu$ and $\gamma_{I}$ are the eigenvalues associated to $\varphi_{\mu}$
and $\psi_{j,I,r}$, respectively.

This proves that

$e^{t\mathbb{L}}u_{0}(x)=u(x,t)=\sum_{\mu\in\operatorname{Spec}(L)}C_{\mu}e^{\mu
t}\varphi_{\mu}(x)+\sum_{\operatorname{supp}(\Psi_{j,I,r})\subset
K_{N}}C_{j,I,r}e^{-\gamma_{I}t}\psi_{j,I,r}(x).$ (11)

with $C_{I}$ and $C_{j,I,r}$ is uniquely determined by the initial conditions

$u_{0}(x)=u(x,0)=\sum_{\mu\in\operatorname{Spec}(L)}C_{\mu}\varphi_{\mu}(x)+\sum_{\operatorname{supp}(\Psi_{j,I,r})\subset
K_{N}}C_{j,I,r}\psi_{j,I,r}(x),$ (12)

and solution (11) is an infinite convergent sum.

###  3.3 Approximating the non-autonomous case for graphs

We now want to use the Trotter product formula to approximate the operator
$U(t,s)$, i.e., the solutions of (5). As approximating operators we choose the
product

$\prod_{k=1}^{n}e^{t/n\mathcal{L}(s+kt/n)}=e^{t/n\mathcal{L}(s+t)}\cdot
e^{t/n\mathcal{L}(s+t-t/n)}\dots e^{t/n\mathcal{L}(s+t/n)}.$

We have the following proposition:

######  Proposition 3.8.

Let $\\{A(t)\\}_{t\in\mathbb{R}}$ be a family of bounded and strongly
continuous operators on a Banach space $X$, and let $U(t,s)$ be its respective
evolution family. Then

$\lim_{n\rightarrow\infty}\prod_{k=1}^{n}e^{t/nA(s+kt/n)}x=U(s+t,s)x$ (13)

for all $x\in X$ and uniformly for $s$ and $t$ in compact intervals of
$\mathbb{R}$ and $\mathbb{R}_{+}$, respectively.

###### Proof.

[6, Ch. III.5.9, Prop.]. ∎

Now we outline how to apply Proposition 3.8 to approximate the solution of
(5). For $0<b$, and $\varepsilon>0$, we have

$||e^{t/n\mathbb{L}(t)}\cdot e^{t/n\mathbb{L}(t-t/n)}\dots
e^{t/n\mathbb{L}(t/n)}u_{0}(x)-U(t,0)u_{0}(x)||_{L^{2}}<\varepsilon$

with $0\leq t<b$, and some $n\in\mathbb{N}$. Take $0\leq t_{0}<b$ fixed,
hence, in order to compute the approximate (respect to the $L^{2}$ norm)
solution we have to solve (7) by iterating the initial conditions, or in other
words we have to compute $e^{t_{0}/n\mathbb{L}(t_{0})}\cdot
e^{t_{0}/n\mathbb{L}(t_{0}-t_{0}/n)}\dots
e^{t_{0}/n\mathbb{L}(t_{0}/n)}u_{0}(x)$. We can view this approximation like
snapshots of the system at $n\geq 0$ discrete time points. We call this the
_temporal discretisation_ , and we fix some notation:

  1. 1.

Let $\mu(t_{0})$ be an eigenvalue of the matrix $L(t_{0})$ and
$\varphi_{\mu}(x,t_{0})$ the associated eigenfunction.

  2. 2.

Let $\\{e^{t\mathbb{L}(t_{0})}\\}_{t\geq 0}$ be the semigroup generated by the
operator $\mathbb{L}(t_{0})$ for some $t_{0}\geq 0$.

  3. 3.

Let $\gamma_{I}(t_{0})=\sum_{J}A_{IJ}(t_{0})$

Therefore, for a fixed $t_{0}$, the solution of the Cauchy problem in the
autonomous case (7) associated to the operator $\mathbb{L}(t_{0})$, with
initial condition $u(x,0)=u_{0}(x)$ as in (12), is given by

$\displaystyle e^{t\mathbb{L}(t_{0})}$ $\displaystyle u_{0}(x)=u(x,t)$
$\displaystyle=\sum_{\mu=\mu(t_{0})\in\operatorname{Spec}(L(t_{0}))}C_{\mu}(t_{0})e^{\mu(t_{0})t}\varphi_{\mu}(x,t_{0})$
$\displaystyle+\sum_{r,I,j}C_{j,I,r}(t_{0})e^{-\gamma_{I}(t_{0})t}\psi_{j,I,r}(x).$

Now it is clear that

$\displaystyle e^{t_{0}/n\mathbb{L}(t_{0}/n)}u_{0}(x)$
$\displaystyle=\sum_{\mu=\mu(t_{0})\in\operatorname{Spec}(L(t_{0}))}C_{\mu}(t_{0}/n)e^{\mu(t_{0}/n)t_{0}/n}\varphi_{\mu}(x,t_{0}/n)$
$\displaystyle+\sum_{j,I,r}C_{r,I,j}(t_{0}/n)e^{-\gamma_{I}(t_{0}/n)t_{0}/n}\psi_{j,I,r}(x).$
(14)

The next step is to compute
$e^{t_{0}/n\mathbb{L}(2t_{0}/n)}e^{t_{0}/n\mathbb{L}(t_{0}/n)}u_{0}(x)$. For
this we need to find the solution to the Cauchy problem (7) associated to the
operator $\mathbb{L}(2t_{0}/n)$ with initial condition
$u_{1}(x)=e^{t_{0}/n\mathbb{L}(t_{0}/n)}u_{0}(x)$, and evaluate the solution
$t\mapsto e^{t\mathbb{L}(2t_{0}/n)}e^{t_{0}/n\mathbb{L}(t_{0}/n)}u_{0}(x)$ in
$t=t_{0}/n$.

Define for a given adjacency matrix $A(t)$ the corresponding weighted graph
$G_{A(t)}$ whose adjacency matrix is $A(t)$. The corresponding $L^{2}$-space
will be denoted as $L^{2}(G_{A(t)},\mathbb{C})$. It is isomorphic to
$\mathbb{C}^{\left\lvert V(\mathcal{G})\right\rvert}$.

To find the solution of the form (10) we need to express the initial condition
$u_{1}(x)=e^{t_{0}/n\mathbb{L}(t_{0}/n)}u_{0}(x)$ as an element of
$L^{2}(G_{A(2t_{0}/n)},\mathbb{C})\oplus\mathcal{L}_{0}^{2}(K_{N})$, that is

$e^{t_{0}/n\mathbb{L}(t_{0}/n)}u_{0}(x)=\sum_{\mu\in\operatorname{Spec}(L(t_{0}))}B_{\mu}\varphi_{\mu}(x,2t_{0}/n)+\sum_{j,I,r}B_{j,I,r}\psi_{r,I,j}(x).$

Then, after computing the coefficients $B_{\mu}$ and $B_{j,I,r}$ the desired
solution will be given by

$\begin{split}e^{t\mathbb{L}(2t_{0}/n)}&e^{t_{0}/n\mathbb{L}(t_{0}/n)}u_{0}(x)=u_{1}(x,t)\\\
&=\sum_{\mu\in\operatorname{Spec}(L(t_{0}))}B_{\mu}e^{\mu(2t_{0}/n)t}\varphi_{\mu}(x,2t_{0}/n)+\sum_{j,I,r}B_{j,I,r}e^{-\gamma_{I}(2t_{0}/n)t}\psi_{j,I,r}(x).\end{split}$
(15)

Comparing (14) and (LABEL:expansion4), it is clear that
$B_{j,I,r}=C_{j,I,r}e^{-\gamma_{I}(t_{0}/n)t_{0}/n}$. This is due to the fact
that in decomposition (9) the space $\mathcal{L}_{0}^{2}(K_{N})$ remains
constant in time. Now we need to compute the constants $B_{\mu}$. For this,
note that

$\omega:=\sum_{\mu\in\operatorname{Spec}(L(t_{0}))}B_{\mu}\varphi_{\mu}(x,2t_{0}/n)=\sum_{\mu\in\operatorname{Spec}(L(t_{0}))}C_{\mu}e^{\mu(t_{0}/n)t_{0}/n}\varphi_{\mu}(x,t_{0}/n)$

Note that for each $s\geq 0$, the set

${\left\\{\varphi_{\mu}(x,s)=\varphi_{\mu(s)}(x)\mid\mu(s)\in\operatorname{Spec}(L(s))\right\\}}$

is a basis for the finite dimensional space $X_{N}$. Hence, the coefficients
$B_{\mu}$ are given by the basis change matrix from the basis
$\\{\varphi_{\mu}(x,t_{0}/n)\\}$ to the basis
$\\{\varphi_{\mu}(x,2t_{0}/n)\\}$. The basis change matrix
$M=[M_{\lambda,\mu}]$ is given by the following equalities:

$\varphi_{\mu}(x,2t_{0}/n)=\sum_{\lambda\in
V(\mathcal{G})}M_{\lambda,\mu}\varphi_{\lambda}(x,t_{0}/n).$

In general, the basis change matrix from the basis $\\{\varphi_{\mu}(x,s)\\}$
to the basis $\\{\varphi_{\mu}(x,t)\\}$ $M$ will be given by

$M=M(s)^{-1}M(t),$

where the matrices $M(\tau)$ are the modal matrices of the Laplacian matrix,
that is, the $\mu$-column of $M(\tau)$ is given by the eigenvector
$[\varphi_{\mu}(K,\tau)]_{K\in V(\mathcal{G})}$. Then, it is clear that the
coefficients $B_{\mu}$ are given by

$[B_{\mu}]_{\mu}=M^{-1}[C_{\mu}e^{\mu(t_{0}/n)t_{0}/n}]_{\mu}.$

Thus,

$[B_{\mu}]_{\mu}=M(2t_{0}/n)^{-1}M(t_{0}/n)\operatorname{diag}(e^{\mu(t_{0}/n)t_{0}/n})[C_{\mu}]_{\mu}$

and

$B_{j,I,r}=C_{j,I,r}e^{-\gamma_{I}(t_{0}/n)t_{0}/n}.$

The process continues by solving the Cauchy problem for the operator
$\mathbb{L}(3t_{0}/n)$ with initial condition
$e^{t_{0}/n\mathbb{L}(2t_{0}/n)}e^{t/n\mathbb{L}(t_{0}/n)}u_{0}(x)$. In order
to do so, we repeat the process described above. That is we take as initial
condition

$\displaystyle u_{1}(x,t_{0}/n)$
$\displaystyle=\sum_{\mu\in\operatorname{Spec}(L(t_{0}))}B_{\mu}e^{\mu(2t_{0}/n)t_{0}/n}\varphi_{\mu}(x,2t_{0}/n)$
$\displaystyle+\sum_{j,I,r}C_{j,I,r}e^{-\gamma_{I}(t_{0}/n)t_{0}/n-\gamma_{I}(2t_{0}/n)t}\psi_{j,I,r}(x)$

and express it as an element of
$L^{2}(G_{A(3t_{0}/n)},\mathbb{C})\oplus\mathcal{L}_{0}^{2}(K_{N})$. Now,

$u_{1}(x,t_{0}/n)=\sum_{\mu\in\operatorname{Spec}(L(t_{0}))}D_{\mu}\varphi_{\mu}(x,3t_{0}/n)+\sum_{j,I,r}D_{j,I,r}\psi_{r,I,j}(x).$

Again, we obtain the coefficients $D_{\mu}$ and $D_{j,I,r}$ by:

$[D_{\mu}]_{\mu}=M(3t_{0}/n)^{-1}M(2t_{0}/n)\operatorname{diag}(e^{\mu(2t_{0}/n)t_{0}/n})[B_{\mu}]_{\mu}$

and

$D_{j,I,r}=B_{j,I,r}e^{-\gamma_{I}(2t_{0}/n)t_{0}/n}.$

We can simplify these expressions by putting the value of $B_{\mu}$ and
$B_{j,I,r}$ in the above equalities, and we get

$[D_{\mu}]_{\mu}=M(3t_{0}/n)^{-1}M(t_{0}/n)\operatorname{diag}(e^{\mu(t_{0}/n)+\mu_{I}(2t_{0}/n)})[C_{\mu}]_{\mu}$

and

$D_{j,I,r}=C_{j,I,r}e^{-\gamma_{I}(t_{0}/n)t_{0}/n-\gamma_{I}(2t_{0}/n)t}.$

In the end, we obtain the function $e^{t_{0}/n\mathbb{L}(t_{0})}\dots
e^{t_{0}/n\mathbb{L}(t_{0}/n)}u_{0}(x)\in L^{2}(K_{N})$, of the form

$\begin{split}e^{t_{0}/n\mathbb{L}(t_{0})}&\cdot
e^{t_{0}/n\mathbb{L}(t_{0}-t_{0}/n)}\dots
e^{t_{0}/n\mathbb{L}(t_{0}/n)}u_{0}(x)=\sum_{\mu\in\operatorname{Spec}(L(t_{0}))}Q_{\mu}(n,t_{0})\varphi_{\mu}(x,t_{0})\\\
&+\sum_{\operatorname{supp}(\Psi_{j,I,r})\subset
K_{N}}C_{r,I,j}(t_{0}/n)e^{-\frac{t_{0}}{n}\sum_{k=1}^{n}\gamma_{I}(k\frac{t_{0}}{n})}\Psi_{j,I,r}(x),\end{split}$

where the coefficients $Q_{\mu}(n,t_{0})$ are given by the equation

$[Q_{\mu}(n,t_{0})]_{\mu}=M(t_{0})^{-1}M(t_{0}/n)\operatorname{diag}(e^{\frac{t_{0}}{n}\sum_{k=1}^{n}\mu(k\frac{t_{0}}{n})})[C_{\mu}(t_{0}/n)]_{\mu}$

For simplicity, we rename the approximation above:

$\displaystyle U_{n}(t,0)u_{0}(x)$
$\displaystyle=\sum_{\mu\in\operatorname{Spec}(L(t_{0}))}Q_{\mu}(n,t)\varphi_{\mu}(x,t)$
$\displaystyle+\sum_{\operatorname{supp}(\Psi_{j,I,r})\subset
K_{N}}C_{r,I,j}(t/n)e^{-\frac{t}{n}\sum_{k=1}^{n}\gamma_{I}(k\frac{t}{n})}\Psi_{j,I,r}(x)$

Then, for a general initial condition $s$, we obtain:

$\displaystyle U_{n}(t,s)u_{0}(x)$
$\displaystyle=\sum_{\mu\in\operatorname{Spec}(L(t_{0}))}Q_{\mu}(n,t,s)\varphi_{\mu}(x,t_{0})$
$\displaystyle+\sum_{\operatorname{supp}(\Psi_{j,I,r})\subset
K_{N}}C_{r,I,j}(s+\frac{t-s}{n})e^{-\frac{t-s}{n}\sum_{k=1}^{n}\gamma_{I}(s+k\frac{t-s}{n})}\Psi_{j,I,r}(x),$

where the coefficients $Q_{\mu}(n,t,s)$ are computed by the following
equation:

$[Q_{\mu}(n,t,s)]=M(t)^{-1}M\left(s+\frac{t-s}{n}\right)\operatorname{diag}\left(e^{\frac{t-s}{n}\sum_{k=1}^{n}\mu(s+k\frac{t-s}{n})}\right)[C_{\mu}(s+\frac{t-s}{n})]_{\mu}$

where the numbers $C_{\mu}(s+\frac{t-s}{n})$ are the Fourier coefficients of
the initial condition at time $s+\frac{t-s}{n}$. Note that in $L^{2}(K_{N})$
we have the following convergence:

$\lim_{n\rightarrow\infty}U_{n}(t,s)u_{0}(x)=U(t,s)u_{0}(x),$

and we can expand the function $U(t,s)u_{0}(x)\in L^{2}(K_{N})$ by its Fourier
series using the decomposition
$L^{2}(G_{A(t_{0})},\mathbb{C})\oplus\mathcal{L}_{0}^{2}(K_{N})$:

$U(t,s)u_{0}(x)=\sum_{\mu\in\operatorname{Spec}(L(t_{0}))}Q_{\mu}(t,s)\varphi_{\mu}(x,t_{0})+\sum_{\operatorname{supp}(\Psi_{j,I,r})\subset
K_{N}}Q_{r,I,j}\Psi_{j,I,r}(x).$

Hence, by Parseval’s equality we obtain:

$\lim_{n\rightarrow\infty}Q_{\mu}(n,t,s)=Q_{\mu}(t,s)$

and

$\lim_{n\rightarrow\infty}C_{r,I,j}\left(s+\frac{t-s}{n}\right)e^{-\frac{t-s}{n}\sum_{k=1}^{n}\gamma_{I}(s+k\frac{t-s}{n})}=C_{r,I,j}(s)e^{-\int_{s}^{t}\gamma_{I}(\tau)d\tau},$

where the last equality occurs by the continuity of $t\mapsto\gamma_{I}(t)$.
On the other hand, the first limit is more complicated, because it depends on
the continuity of the matrix function $t\mapsto M(t)$, more specifically, on
the continuity of the eigenvectors: $t\mapsto\varphi_{\mu}(t)$, where the
vector $\varphi_{\mu}=[\varphi_{\mu}(K,t)]_{K}$ is the eigenvector of the
Laplacian matrix $L(t)$ associated to $\mu(t)\in\operatorname{Spec}(L(t))$.

Let $t\mapsto B(t)$ be a matrix function. If $B(t)$ is real symmetric, then we
can find an orthonormal basis $\\{\varphi_{\mu}(t)\\}_{\mu}$ of
$\mathbb{C}^{N}$ formed by eigenvectors of $B(t)$. It can be shown that if the
matrix $B(t)$ depends analytically on $t$ inside a real interval, then we can
chose an orthonormal basis $\\{\varphi_{\mu}(t)\\}_{\mu}$ that depends
analytically on $t$ [9, Ch. II.4], and in particular, continuously on $t$. If
the analyticity assumption is deleted, the eigenvectors may not depend
continuously on $t$. Therefore, if we suppose that the adjacency matrix $A(t)$
depends analytically on $t$, then we have the following:

$\begin{split}\lim_{n\rightarrow\infty}[Q_{\mu}(n,t,s)]_{\mu}&=\lim_{n\rightarrow\infty}M(t)^{-1}M\left(s+\frac{t-s}{n}\right)\\\
&\cdot\operatorname{diag}\left(e^{\frac{t-s}{n}\sum_{k=1}^{n}\mu(s+k\frac{t-s}{n})}\right)\left[C_{I}\left(s+\frac{t-s}{n}\right)\right]_{I}\\\
&=M(t)^{-1}M(s)\operatorname{diag}\left(e^{\int_{s}^{t}\mu(\tau)d\tau}\right)[C_{\mu}(s)]_{\mu}=[Q_{\mu}(t,s)]_{\mu}.\end{split}$

But, in order to have the above limit, it is sufficient to suppose that the
eigenvectors $\varphi_{\mu}(t)$ are right continuous at the initial condition
$s\geq t_{0}$.

This proves the following:

######  Theorem 3.9.

If the eigenvectors of the Laplacian matrices $L(t)$ are right continuous at
some $s\geq 0$, then the evolution family of the Cauchy problem (5) (with
initial condition at $s$) is given by

$\displaystyle U(t,s)u_{0}(x)$
$\displaystyle=\sum_{\mu\in\operatorname{Spec}(L(t_{0}))}Q_{\mu}(t,s)\varphi_{\mu}(x,t)$
$\displaystyle+\sum_{\operatorname{supp}(\Psi_{j,I,r})\subset
K_{N}}C_{r,I,j}(s)e^{-\int_{s}^{t}\gamma_{I}(\tau)d\tau}\Psi_{j,I,r}(x),$

where $t\geq s$,

$[Q_{\mu}(t,s)]=M(t)^{-1}M(s)\operatorname{diag}\left(e^{\int_{s}^{t}\mu(\tau)d\tau}\right)[C_{\mu}(s)]_{\mu}\,,$

and the coefficients $C_{\mu}(s)$ and $C_{r,I,j}(s)$ are uniquely determined
by the initial condition

$u_{0}(x)=\sum_{\mu\in\operatorname{Spec}(L(t_{0}))}C_{\mu}(s)\varphi_{\mu}(x,s)+\sum_{\operatorname{supp}(\Psi_{j,I,r})\subset
K_{N}}C_{r,I,j}(s)\Psi_{j,I,r}(x).$

In particular, this result holds for all $s\geq 0$ if the adjacency matrix
$A(t)$ depends analytically on $t$.

######  Remark 3.10.

Note that this is consistent with the autonomous problem, that is, when
$L(t)=L$ is constant. In this case, the matrices $M(t)$ and the functions
$t\mapsto\mu(t)$, $t\mapsto\varphi_{\mu}(t)$ are constant, thus the solution
is given by

$\displaystyle U(t,s)u_{0}(x)$
$\displaystyle=\sum_{\mu\in\operatorname{Spec}(L)}C_{\mu}(s)e^{(t-s)\mu}$
$\displaystyle+\sum_{\operatorname{supp}(\Psi_{j,I,r})\subset
K_{N}}C_{r,I,j}(s)e^{-(t-s)\gamma_{I}}\Psi_{j,I,r}(x)$

for $t\geq s\geq 0$.

######  Corollary 3.11.

Assuming analyticity of $L(t)$, the first-order approximation error in the
temporal discretisation is $O\left(\frac{1}{n}e^{\frac{1}{n}}\right)$.

###### Proof.

For the temporal discretisation, observe that by replacing each analytic
function with its first order approximation leads to an error of $O(1/n)$ in
each entry of the eigenvectors and eigenvalues of $L(t)$, and an error of
$O(e^{1/n})$ in each exponential integral, because the temporal discretisation
is in this case a Riemann sum. Hence, the error in each coefficient of the
solution is given by the product error, and this is in first order
$O\left(\frac{1}{n}e^{\frac{1}{n}}\right)$, which proves the assertion. ∎

##  4 Inhomogeneous Markov processes and Feller evolution.

###  4.1 Reviewing inhomogeneous Markov processes

In this section we review some definitions and results necessary to prove that
the non-autonomous abstract Cauchy problem (5) has attached to it an
inhomogeneous Markov process.

In the sequel $E$ denotes a separable complete metrizable topological
Hausdorff space. In other words $E$ is a Polish space. The space $C_{b}(E)$ is
the space of all complex valued bounded continuous functions. The space
$C_{b}(E)$ is equipped with the uniform norm: $||\cdot||_{\infty}$, and the
strict topology $\mathcal{T}_{\beta}$. It is considered as a subspace of the
bounded Borel measurable functions $L^{\infty}(E)$, also endowed with the
supremum norm.

Let $(\Omega,\mathcal{F})$ be a measurable space. A filtration
$\mathcal{F}_{t}$, $t\in[0,T]$, is a family of sub-$\sigma$-algebras of
$\mathcal{F}$ such that if $0\leq t_{1}\leq t_{2}\leq T$, then
$\mathcal{F}_{t_{1}}\subset\mathcal{F}_{t_{2}}$. A two-parameter filtration
$\mathcal{F}_{t}^{\tau}$, $0\leq\tau\leq t\leq T$, is a family of
sub-$\sigma$-algebras of $\mathcal{F}$ that is increasing in $t$ and
decreasing in $\tau$. Let $X_{s}$, $s\in[0,T]$ be a stochastic process, with
state space $(E,\mathcal{E})$. The $\sigma$-algebra $\sigma(X_{s}:0\leq s\leq
t)$ is the smallest $\sigma$-algebra such that all the state variables $X_{s}$
with $0\leq s\leq t$ are measurable. The process $X_{s}$, $s\in[0,T]$,
generates the following two-parameter filtration:

$\mathcal{F}_{t}^{\tau}=\sigma(X_{s}:\tau\leq s\leq t)$

with $0\leq\tau\leq t\leq T$.

We recall some definitions:

######  Definition 4.1.

A transition probability funcition $P(r,x;s,A)$, where $0\leq r<s\leq T$,
$x\in E$, and $A\in\mathcal{E}$, is a nonnegative function for which the
following condition hold:

  * •

For fixed $r,s$ and $A$, $P$ is a nonnegative Borel measurable function on
$E$.

  * •

For fixed $r,s$ and $x$, $P$ is a Borel measure on $\mathcal{E}$.

  * •

$P(r,x;s,E)=1$ for all $r,s$ and $x$.

  * •

$P(r,x;s,A)=\int_{E}P(r,x;y,dy)P(u,y;s,A)$ for all $r<u<s$, and $A$.

The number $P(r,x;s,A)$ can be interpreted as the probability of the following
event: The random system located at $x\in E$ at time $r$ hits the target
$A\subset E$ at time $s$.

######  Definition 4.2.

The process

$\\{(\Omega,\mathcal{F}_{T}^{\tau},\mathbb{P}_{\tau,x}),(X(t):T\geq t\geq
0),(E,\mathcal{E})\\}$

is called a time-inhomogeneous Markov process if

$\mathbb{E}_{\tau,x}\left[f(X(t))|\mathcal{F}_{s}^{\tau}\right]=\mathbb{E}_{s,X(s)}[f(X(t))]$

$\mathbb{P}_{\tau,x}$-almost surely. Here $f$ is a bounded Borel measurable
function defined on the state space $E$ and $\tau\leq s\leq t\leq T$.

Let $P$ be a transition probability function. It is possible to construct a
filtered measurable space $(\Omega,\mathcal{F},\mathcal{F}_{t}^{\tau})$, a
family of probability measures $\mathbb{P}_{\tau,x}$, $x\in E$,
$\tau\in[0,T]$, on the space $(\Omega,\mathcal{F}_{T}^{\tau})$, and a Markov
process $X_{t}$, $t\in[0,T]$ such that

$\mathcal{F}_{t}^{\tau}=\sigma(X_{s}:\tau\leq s\leq t)$

with $0\leq\tau\leq t\leq T$ and

$\mathbb{P}_{\tau,x}(X_{t}\in A)=P(\tau,x;t,A)$

with $0\leq\tau\leq t\leq T,$ $A\in\mathcal{E}$.

######  Definition 4.3.

A family $\\{P(s,t):0\leq s\leq t\leq T\\}$ of operators defined on
$L^{\infty}(E)$ is called a Feller Evolution or Feller propagator on
$C_{b}(E)$ if it possesses the following properties:

  1. 1.

It leaves $C_{b}(E)$ invariant: $P(s,t)C_{b}(E)\subset C_{b}(E)$ for $0\leq
s\leq t\leq T$;

  2. 2.

It is an evolution: $P(\tau,t)=P(\tau,s)P(s,\tau)$ for all $\tau,s,t$ for
which $0\leq\tau\leq s\leq t$ and $P(t,t)=I$, $t\in[0,T]$;

  3. 3.

If $0\leq f\leq 1$, $f\in C_{b}(E)$, then $0\leq P(s,t)f\leq 1$, for $0\leq
s\leq t\leq T$.

  4. 4.

If the function $(s,t,x)\mapsto P(s,t)f(x)$ is continuous on the space
$\Lambda:=\\{(s,t,x)\in[0,T]\times[0,T]\times E:s\leq t\\}$.

In the following theorem we see that with a Feller evolution a strong Markov
process can be associated in such a way that the one-dimensional distributions
or marginals are determined by the operators $f\mapsto P(\tau,t)f$, $f\in
C_{b}(E)$. In fact every operator $P(\tau,t)$ can be written as

$P(\tau,t)f(x)=\int P(\tau,x;t,dy)f(y)$

for $f\in C_{b}(E)$. Where the mapping

$(\tau,x,t,B)\mapsto P(\tau,x;t,B)$

with $(\tau,x,t,B)\in[0,T]\times E\times[0,T]\times\mathcal{E}$, $\tau\leq t$,
is a sub-probability transition function.

######  Theorem 4.4.

Let $\\{P(\tau,t):\tau\leq t\leq T\\}$ be a Feller evolution in $C_{b}(E)$.
Then there exists a strong Markov process (in fact a Hunt Process) such that
$[P(\tau,t)f](X)=\mathbb{E}_{\tau,x}[f(X(t))]$, $f\in C_{b}(E)$, $t\geq 0$.
Moreover, this Markov process possesses the properties in [17, Thm. 2.9.].

###### Proof.

[17, Thm. 2.9]. ∎

###  4.2 Stochastic process attached to a non-autonomous equation

In this section we prove that the system

$\begin{cases}\frac{\partial u}{\partial t}(x,t)=\mathbb{L}(t)u(x,t)\\\
u(x,s)=u_{s}(x)\in C(K_{n},\mathbb{C})\end{cases}$ (16)

generates a time-inhomogeneous Markov process, where the operators
$\mathbb{L}(t)$ satisfies the hypothesis of Theorem 2.11. In more generality,
we study the following Cauchy problem

$\begin{cases}\frac{\partial u}{\partial t}(x,t)=A(t)u(x,t)\\\
u(x,s)=u_{s}(x)\in C(E)\,,\end{cases}$ (17)

where $E$ is a compact Polish space. Thus, $C_{b}(E)=C(E)$. Also, we suppose
that $t\mapsto A(t)$ is continuous in the uniform norm topology (In particular
$A(t)$ is bounded for all $t\geq 0$). Moreover, we suppose that for every
$\tau\geq 0$, $A(\tau)$ generates a strongly continuous, positive, contraction
semigroup $\\{e^{tA(\tau)}\\}_{t\geq 0}$ on $C(E)$. By Theorem 3.3, the system
(17) is well-posed for every initial condition $u_{s}(x)\in C(E)$. Therefore,
there exists an evolution family $U(t,s)$, such that $U(t,s)u_{s}(x)$ is the
unique solution of the Cauchy problem (17). Moreover, $U(t,s)$ satisfies the
properties in Remark 3.5.

######  Lemma 4.5.

The evolution $U(t,s)$ described above is a Feller evolution.

Proof. We verify the properties of Definition 4.3. The verification of _1_.
and _2_. is straightforward. We now prove condition _3_. Let $f\in C(E)$ such
that $0\leq f\leq 1$. By Proposition 3.8, we have

$\lim_{n\rightarrow\infty}\prod_{k=1}^{n}e^{\frac{(t-s)}{n}A(s+k\frac{(t-s)}{n})}f(x)=U(t,s)f(x)$

in the supremum norm, and with $0\leq s\leq t\leq T$. Therefore, we also have
point-wise convergence. Since every semigroup $\\{e^{tA(\tau)}\\}_{t\geq 0}$
is a strongly continuous, positive, contraction semigroup we have

$0\leq\prod_{k=1}^{n}e^{\frac{(t-s)}{n}A(s+k\frac{(t-s)}{n})}f(x)\leq 1.$

We obtain the desire result by taking $n\rightarrow\infty$. For condition
_4_., note that, since the space $E$ is Polish, we can prove the continuity in
terms of sequences. Let $(t,s,x)\in\Lambda$, with $s\leq t$ and $x\in E$. And
let $(t_{n},s_{n},x_{n})$ a convergent sequence, with $s_{n}\leq t_{n}$ and
$x_{n}\in E$, such that $(t_{n},s_{n},x_{n})\rightarrow(t,s,x)$ as
$n\rightarrow\infty$. We have

$\begin{split}|U(t_{n},s_{n})f(x_{n})-U(t,s)f(x)|&\leq|U(t_{n},s_{n})f(x_{n})-U(t,s)f(x_{n})|\\\
&+|U(t,s)f(x_{n})-U(t,s)f(x)|\\\
\leq&||U(t_{n},s_{n})-U(t,s)||_{\infty}||f||_{\infty}\\\
&+|U(t,s)f(x_{n})-U(t,s)f(x)|\end{split}$

Since $(t,s)\mapsto U(t,s)$ is continuous in the uniform norm topology (Remark
3.5._2_.), and the function $x\mapsto U(t,s)f(x)$ is continuous in $E$, by
taking $n\rightarrow\infty$ the right hand side of the above inequality tends
to zero. This now proves that $U(t,s)$ is a Feller evolution. ∎

######  Theorem 4.6.

Let $\\{A(t)\\}_{t\geq 0}$ be a set of bounded operators on $C(E)$. Suppose
that $t\mapsto A(t)$ is continuous in the uniform norm topology and that each
$A(t)$ generates a strongly continuous, positive, contraction semi-group. Then
the Cauchy problem (17) is well-posed and its evolution family $U(t,s)$
generates a strong Markov process with the properties listed in Theorem 4.4.

Proof. This is a consequence of Lemma 4.5 and Theorem 4.4. ∎

######  Remark 4.7.

The literature does not seem to contain a result proving that a non-autonomous
Cauchy problem has attached to it a Markov process. That is why the above
result was obtained using the techniques learned above in order to find an
aproximate solution (Proposition 3.8). In particular, it is important to note
that evolution families are the two-parameter generalizations of semigroups,
and analogously, Feller evolution families are the two parameter
generalizations of Feller semigroups.

######  Theorem 4.8.

There exists a probabilty transition function $P(t,x;s,\cdot)$, where
$(t,x,s)\in[0,T]\times K_{N}\times[0,T]$, and $s\leq t$, on the Borel
$\sigma$-algebra of $K_{N}$, such that the Cauchy problem (16) has a unique
solution of the form

$u(x,t)=\int_{K_{N}}u_{s}(x)P(t,x;s,dy).$

In addition, $P(t,x;s,\cdot)$ is the transition function of a Markov process
with the properties listed in Theorem 4.4.

Proof. We have to proove that $t\mapsto\mathbb{L}(t)$ satisfies the hypothesis
in Theorem 4.6. By [19, Lem. 4.1], we know that $\mathbb{L}(t)$ generates a
strongly continuous, positive, contraction semigroup on $C(K_{N})$ for each
$t\geq 0$. hence, we only have to show that $t\mapsto\mathbb{L}(t)$ is
continuous in the uniform norm topology. Recall from the proof of Theorem 2.11
that

$||\mathbb{L}(t)u-\mathbb{L}(s)u||_{\infty}\leq||\mathcal{A}(t)u-\mathcal{A}(s)u||_{\infty}+||\Gamma(t)u-\Gamma(s)u||_{\infty}.$

Let $\varepsilon>0$, and $\delta>0$ such that

$|t-s|<\delta\implies|A(x,y,t)-A(x,y,s)|<\varepsilon\ and\
|\gamma(t)(x)-\gamma(s)(x)|<\varepsilon$

We have that

$\begin{split}||\mathcal{A}(t)u-\mathcal{A}(s)u||_{\infty}&=\sup_{x\in
K_{N}}\left|\int_{K_{N}}(A(x,y,t)-A(x,y,s))u(y)dy\right|\\\
&\leq\varepsilon||u||_{\infty}\operatorname{Vol}(K_{N})\end{split}$ (18)

and

$||\Gamma(t)u-\Gamma(s)u||_{\infty}=\sup_{x\in
K_{N}}|(\gamma(t)(x)-\gamma(s)(x))||u(x)|\leq\varepsilon||u||_{\infty}\,.$

If $||u||_{\infty}\leq 1$, then

$||\mathbb{L}(t)u-\mathbb{L}(s)u||_{\infty}\leq\varepsilon(\operatorname{Vol}(K_{N})+1).$

This inequality implies the continuity in the uniform norm topology. We now
apply Theorem 4.4 and Theorem 4.6 to obtain the result. ∎

## Acknowledgements

The first named author is indebted to Norbert Paul for coming up with
questions on how to study topological questions in the setting of finite
spaces a long time ago, which gave a long-lasting impulse to eventually
combine these with ultrametric analysis and mathematical physics. The second
named author is indebted to Wilson Zúñiga-Galindo for sharing valuable
insights into $p$-adic analysis when he was under his supervision as an
undergraduate student. Martin Breunig, Markus Jahn and David Weisbart are
warmly thanked for fruitful discussions. This work is supported by the
Deutsche Forschungsgemeinschaft under project number 469999674.

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