\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 66, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/66\hfil Strong maximum principle]
{Remarks on the strong maximum principle for
 nonlocal operators}

\author[J. Coville \hfil EJDE-2008/66\hfilneg]
{J\'er\^ome Coville}

\address{J\'er\^ome Coville \newline
Max Planck Institute for mathematical science\\
Inselstrasse 22, D-04103  Leipzig, Germany}
\email{coville@mis.mpg.de}


\thanks{Submitted January 25, 2008. Published May 1, 2008.}
\thanks{Supported  by the Ceremade-Universit\'e Paris Dauphine,
 and CMM-Universidad de Chile  \hfill\break\indent
 through an Ecos-Conicyt project.}
\subjclass[2000]{35B50, 47G20, 35J60}
\keywords{Nonlocal diffusion operators; maximum principles;
\hfill\break\indent Geometric condition}

\begin{abstract}
 In this note, we study the  existence of a
 strong maximum principle for the nonlocal operator
 $$
 \mathcal{M}[u](x) :=\int_{G}J(g)u(x*g^{-1})d\mu(g) - u(x),
 $$
 where $G$ is a topological group  acting continuously on a
 Hausdorff space $X$ and $u \in C(X)$.
 First  we investigate the general situation and derive a pre-maximum
 principle. Then  we restrict our analysis to the case of homogeneous
 spaces (i.e., $ X=G /H$).  For such Hausdorff spaces,  depending
 on the topology, we give a condition on $J$ such that a strong
 maximum principle holds for $\mathcal{M}$. We also revisit the classical
 case of the convolution operator (i.e.
 $G=(\mathbb{R}^n,+), X=\mathbb{R}^n, d\mu =dy$).
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction and Main results}

This note is devoted to the study of the  strong maximum
principle satisfied by an operator
\begin{equation}\label{pajc-gen}
\mathcal{M}[u] := \int_{G}J(g)u(x*g^{-1})d\mu(g) -u(x),
\end{equation}
where $G,*,X,J,d\mu$ satisfy the following assumptions:
\begin{itemize}
\item[(H1)] $X$ is a Hausdorff space,
\item[(H2)] $G$ is a topological group acting continuously on $X$
   with the operation  $*$,
\item[(H3)] $d\mu$ is a Borel measure on $G$ such that for all nonempty
   open sets $A\subset G$ we have $d\mu(A)>0$,
\item[(H4)] $J\in C(G,\mathbb{R})$ is a non-negative function of unit mass
   with respect to  $d\mu$.
\end{itemize}

Such kind of operators have been recently introduced in various
models where long range interactions play an important role, see for
example  \cite{BFRW,CF,CD2, DOPT1, MMP, Sch}.
A first example of such models  is given by  the well known  nonlocal
reaction diffusion equation below,
\begin{equation}
 \frac{\partial u}{\partial t}= \int_{\mathbb{R}^n}J(x-y) u(y)\,dy - u + u(1-u)
 \quad\text{in } \mathbb{R}^+\times\mathbb{R}^n. \label{pajc.nl}
\end{equation}
The above equation  models the evolution of a population density
through a homogeneous environment with a constant  rate of  birth and death.
  In this case, we have $(G,*)=(\mathbb{R}^n,+)$,  $X=\mathbb{R}^n$, $J\in C(\mathbb{R}^n)$  and
$d\mu=dy$  is the Lebesgue measure.
Such an equation, with a different type of nonlinearity, appears also
in some Ising models and in ecology, see for example
\cite{BFRW,CD2,DGP,Sch} and their many references.

Other examples  are given by  the  following two discrete versions
 of \eqref{pajc.nl},
\begin{gather}
 \frac{\partial u}{\partial t}=\frac{1}{2}[u(x+1)+u(x-1)-2u(x)]+ f(u)
  \quad\text{in }\mathbb{R}^+\times\mathbb{R}, \label{pajc.nld1} \\
 \frac{\partial u}{\partial t}=\frac{1}{2}[u(p+1)+u(p-1)-2u(p)]+ f(u)
  \quad\text{in }\mathbb{R}^+\times\mathbb{Z}. \label{pajc.nld2}
\end{gather}
In both situations the discrete diffusion operator can be reformulated
in terms of a nonlocal operator $\mathcal{M}$ defined in \eqref{pajc-gen}.
Indeed,  in these two  cases, by taking $(G,*)=(\mathbb{Z},+)$,  $d\mu$
the counting measure and   $J\in C(\mathbb{Z},\mathbb{R})$  defined as follows:
$$
J(p):=\begin{cases}
 \frac{1}{2}&\text{if }  p=-1 \text{ or } p=1,\\
0 &\text{otherwise,}
\end{cases}
$$
it follows that for any $x$ in the Hausdorff space  $\mathbb{R}$ or $\mathbb{Z}$ we have
$$
\frac{1}{2}[u(x+1)+u(x-1)-2u(x)]=\int_{G}J(g)u(x*g^{-1})d\mu(g) -u(x).
$$
As for their continuous version \eqref{pajc.nl},
equations \eqref{pajc.nld1} and \eqref{pajc.nld2}  appear  in
discrete reaction diffusion models  describing  a wide variety of
 phenomenon, ranging from combustion to  nerve propagation and phase
transitions.  We point the interested reader to  \cite{CF,CG,HHZ}
and the many references cited therein.

Another example  comes from the following  size structured population model,
recently introduced  by Perthame \textit{et al.} in \cite{MMP,PR},
\begin{equation}
 \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}
 =\int_{0}^{+\infty}u(\frac{x}{y})b(y)dy - u(x)
\quad\text{in }\mathbb{R}^+\times\mathbb{R}^+. \label{pajc.fg}
\end{equation}
In such case, we have $(G,*)=(\mathbb{R}^{+}\setminus\{0\},\cdot)$,
$X=\mathbb{R}^{+}$
 and  $d\mu(y) = dy$ is the Lebesgue measure.

In all these examples, depending on the group and the measure considered,
the properties satisfied by the corresponding operator $\mathcal{M}$ show significant
differences. However, as for the classical Laplace operator ($\Delta$),
they all satisfy the following positive maximum principle.

\begin{definition}[Courr\`ege Positive maximum principle \cite{BCP}]
An operator $A\in   \mathcal{L}(C(X))$ is said  to satisfy the positive maximum
principle if  for all $f \in C(X)$ and $x\in X$ such that
 $f(x)=\sup (f)$    we have $A(f)(x)\le 0$.
\end{definition}

For the  Laplace operator ($\Delta$), in addition to the above property,
it is well known, see \cite{GT,PW}, that a sub-harmonic functions
satisfies a strong maximum principle:

\begin{theorem}[Elliptic Strong maximum principle] \label{thm1.2}
Let $u\in C^2(\mathbb{R}^n)$ be such that
$\Delta u\ge 0$  in $\mathbb{R}^n$.
Then $u$ cannot achieve a global maximum without being constant.
\end{theorem}


In this note, we investigate  the conditions on  $(G,*),X,J$ and $d\mu$
in order to achieve such strong maximum principles for $\mathcal{M}$.
More precisely , we are interested in finding simple conditions on
 $(G,*),X,J$ and $d\mu$ for the strong maximum principle to hold:

\begin{theorem}[Strong maximum principle] \label{thm1.3}
Let $u\in C(X,\mathbb{R})$ be such that
$\mathcal{M}[u]\ge 0$  in $X$.
Then $u$ cannot achieve a global maximum without being constant.
\end{theorem}

In the analysis of nonlinear elliptic equations, the strong maximum
principle plays a very important role in proving  key \textit{ a priori }
estimates.  It is expected that such a strong maximum property for $\mathcal{M}$
will play a similar role in the analysis of nonlinear  equations
involving nonlocal operators. It is therefore of great interest
to investigate the conditions on $G, X, d\mu$ and $J$ in order  that
a strong maximum principle  hold for $\mathcal{M}$.

In this direction, we first establish a generic result satisfied by all
operators $\mathcal{M}$. More precisely, we show the following result.

\begin{theorem}[Pre-maximum principle]  \label{pajc.theo.1}
Let  $(G,*,X,J,d\mu)$ be such that $(H1-H4)$ are satisfied and let
$u\in C(X,\mathbb{R})$ be such that
$$
\mathcal{M}[u] \ge 0\quad \quad ( \text{resp.} \le 0 ).
$$
 Assume that  $u$ achieves  a global maximum (resp. minimum) at some
point $x_0\in X$ and let $F_{x_0}$ denote  the smallest closed subset
of $X$ such that
\begin{itemize}
\item $x_0\in F_{x_0}$,
\item $F_{x_0}*\{g^{-1} \in G|J(g)>0\}\subset F_{x_0}$.
\end{itemize}
Then $u\equiv u(x_0)$ in $F_{x_0}$.
 \end{theorem}

Our next result is a characterization of the set  $F_{x_0}$ defined
in the above Theorem \ref{pajc.theo.1}.

\begin{proposition} \label{pajc.prop.descr}
Let $(G,*,X,J,d\mu)$ be such that  {\rm (H1)--(H4)} are satisfied and
let $F_{x_0}$ be the set defined in Theorem \ref{pajc.theo.1}.
Then
$$
F_{x_0}=\overline{\bigcup_{n\in \mathbb{N}}F_n},
$$
where the $F_n$ are defined by induction as follows
\[
F_0=\{x_0\},\quad\text{and}\quad
\forall n\ge 0 \quad F_{n+1}:= F_n*\{g^{-1} \in G|J(g)>0\}.
\]
\end{proposition}

In view of the above generic result,  in order to get a strong maximum
principle for $\mathcal{M}$, we need to find conditions on $(G,*),X,d\mu$ and
 $J$ which imply  that $F_{x_0}=X$. Note that, from the characterization
of the set $F_{x_0}$, the condition $F_{x_0}=X$ implies that
$X=F_{x_0}\subset \overline{orb(x_0)}:=\overline{\{x*g^{-1}|g\in G\}}\subset X$,
 which means that $orb(x_0)$ is a dense set in $X$.

Observe that for the discrete diffusion operator considered in
\eqref{pajc.nld1}, the set $orb(x)$ is never dense in $\mathbb{R}$.
Therefore, we cannot expect to have a strong maximum principle in
such situation. On the contrary,  for the same diffusion operator
considered in \eqref{pajc.nld2}, the set $orb(x)$ is always dense in $\mathbb{Z}$.
Moreover we can easily see that in this situation the discrete operator
satisfies a  strong maximum principle.

Considering the above remarks, in what follows we restrict our attention
to the case of Hausdorff homogeneous spaces $X$
(i.e. $X:=G/H$, where $H$ is a closed subgroup of $G$).
For such Hausdorff spaces, the set $orb(x)$ is always dense in  $X$
and sufficient conditions  on $(G,*),X,J$ and $d\mu$ for the strong
maximum principle to hold reduce to find some simple conditions on $J$.
In this direction, we first give a sufficient condition on $J$ to ensure
 that $\mathcal{M}$ satisfies the strong maximum principle. Namely, we have
the following result.

\begin{theorem} \label{pajc.theo.2}
Let $X$ be a connected homogeneous space and let $(G,*),J,d\mu$ be as
in Theorem  \ref{pajc.theo.1}. Let $e$ be the unit element of $G$
and assume that $J(e)>0$.  Then $\mathcal{M}$ satisfies the strong maximum principle.
\end{theorem}

When $X$ is a compact connected homogeneous spaces, we can generalize
the previous statement to the following result.

\begin{theorem} \label{pajc.theo.3}
Let $X$ be a connected compact homogeneous space and $(G,*),J,d\mu$ as in
Theorem  \ref{pajc.theo.1}. Then $\mathcal{M}$ satisfies the strong maximum principle.
\end{theorem}

 Next, we state  optimal condition on $J$ in two special cases.
Namely,  we first retrieve the  Markov necessary and sufficient
condition for the convolution operator (i.e.
$(G,*)=(\mathbb{R}^n,+), X=\mathbb{R}^n, d\mu=dy$), which is well known  among experts
 in stochastic processes.

\begin{theorem}[Markov condition]\label{pajc.theo.4}
Assume that $(G,*)=(\mathbb{R}^n,+), X=\mathbb{R}^n$ and $d\mu=dy$. Then $\mathcal{M}$ satisfies
the strong maximum principle iff the convex hull of
$\{y\in \mathbb{R}^n|\, J(y)>0\}$ contains $0$.
\end{theorem}

As a consequence of the above Markov condition, we  derive  the following
optimal condition when  $(G,*)=(\mathbb{R}^+\setminus \{0\},\bullet), X=\mathbb{R}^+$ and
 $d\mu=dy$:

\begin{corollary}\label{pajc.theo.5}
Assume that  $(G,*)=(\mathbb{R}^+\setminus \{0\},\bullet), X=\mathbb{R}^+$ and $d\mu=dy$.
Then $\mathcal{M}$ satisfies the strong maximum principle iff there exists $2$
points $x_1$ and $x_2$ such that $J(x_i)>0$ and   $0< x_1\le 1\le x_2)$.
\end{corollary}

\subsection{General comments}
We first note that, provided an extra assumption on the sign  of the
maximum (minimum) is made, we can easily extend the above results
to operators  $\mathcal{M}[u] +c(x)u$ with non-positive zero order term
(i.e. $c(x)\le 0$).
As for $\mathcal{M}$, the operator $\mathcal{M} +c(x)$  satisfies a Courr\`eges
 positive maximum principle \cite{BCP}, which in this case state
the following definition.

\begin{definition}[Positive maximum principle] \rm
An operator $A\in   \mathcal{L}(C(X))$ is said to satisfy the positive maximum
principle if  for all $f \in C(X)$ and $x\in X$ such that $f(x)\ge 0$
 and $f(x)=\sup (f)$   we have $A(f)(x)\le 0$.
\end{definition}

In our investigation of homogeneous spaces, we also observe that to obtain
a strong maximum principle for $\mathcal{M}$, we only need the inequality
$\mathcal{M}[u]\ge  0$ at points where the function $u$ achieves its global
maximum.  As a consequence, in the two  situation investigated above
(Theorems \ref{pajc.theo.2} and \ref{pajc.theo.3} ), we have
the following characterization:

\begin{proposition} \label{prop1.11}
Let $(G,*), X,  d\mu$ and $J$ be as in Theorem  \ref{pajc.theo.2}
or \ref{pajc.theo.3}. Then  for all $u\in C(X)$ and $x\in X$ such
 that  $u(x)=\sup (u)$ we have the following alternatives: Either
\begin{itemize}
\item  there exists $y\in X$  such that $u(y)=u(x)$ and
$\mathcal{M}[u](y)<0$, or
 \item $u$ is a constant.
\end{itemize}
\end{proposition}

We also want to point out that, although  the Markov condition is well
known among  experts in stochastic analysis, we present here a simple
analytical proof, which we believe is new.  Using such a point of
view allows us  to relate a simple recovering problem with the
conditions for the strong maximum principle.

The outline of this note is the following.
 In the two first Sections (Sections \ref{pajc.s.pre} and
\ref{pajc.s.premp}), we recall some basic topological results and prove
 the pre-maximum principle and the characterization of $F_x$
(Theorems  \ref{pajc.theo.1} and Proposition  \ref{pajc.prop.descr}).
 Then in Section \ref{pajc.s.smp}, we establish the strong maximum
principle (Theorems  \ref{pajc.theo.2} and  \ref{pajc.theo.3}).
Finally, in the last section, we prove the optimal conditions
(Theorems  \ref{pajc.theo.4} and  \ref{pajc.theo.5}).
\medskip

\section{Preliminaries\label{pajc.s.pre}}

In this section, we  first present some definitions and notation that we
will use in this paper. Then  we establish a useful proposition.
Let us first define some notations:
\begin{itemize}
\item $\Sigma:=\{g^{-1}\in G|J(g)>0\}$.
\item For a function $u$, we define $\Gamma_{y}:=\{x\in X|u(y)=u(x)\}$.
\end{itemize}
Let us now  introduce the following  two definitions:

\begin{definition} \label{def2.1} \rm
Let $A\subset X$ and $B\subset G$ be two sets, then we define $A*B\subset X$ as follows
$$
A*B:=\{a*b\;|\; a\in A \text{ and } b\in B\}.
$$
\end{definition}

\begin{definition} \label{def2.2} \rm
Let $A\subset X$ and $B\subset G$ be two sets, then
we say that  $A$ is $B*$ stable if $$A*B\subset A.$$
\end{definition}

Next,  let us  recall  the following  basic property of $*$ stable sets.

\begin{proposition} \label{pajc.prop.stabi}
Let $A\subset X$ and $B\subset G$ be two sets. If  $A$ is $B*$ stable,
then $\bar A$ is  $B*$ stable, where $\bar A$ denotes the closure of $A$.
 \end{proposition}

\begin{proof}
Let  $y\in \bar A*B$ and $V(y)$ be an open neighbourhood of $y$.
By definition, we have $y:=x_1*b_1$ for some $x_1 \in \bar A$ and $b_1\in B$.
Since the operation $*$ is  continuous,
the following map $T$ is continuous:
\begin{align*}
 T:X &\to X\\
 z&\mapsto z*b_1.
\end{align*}
Therefore, $T^{-1}(V(y))$ is a open neighbourhood of $x_1$.
Since $\bar A$ is a closed set and  $x_1\in \bar A$, we have
$T^{-1}(V(y))\cap  A\neq\emptyset$.
By definition of $T^{-1}(V(y))$, using the stability of  $A$,
it follows that for all $z\in T^{-1}(V(y))\cap  A$, $z*b_1\in A$.
Therefore,
$$
z*b_1\in V(y)\cap  A\quad \text{for all }
 z \in T^{-1}(V(y))\cap  A,
$$
and yields
$ V(y)\cap  A\neq \emptyset$.

The above argumentation, being independent of the choice of $V(y)$, shows that $y\in \bar A$.
Now, since $y$ is  arbitrary, we end up with
$ \bar A*B \subset \bar A$.
\end{proof}


\section{Pre-maximum principle and Characterizations of $F_x$
 \label{pajc.s.premp}}


In this Section we prove  Theorem \ref{pajc.theo.1} and Proposition
\ref{pajc.prop.descr}.
Let us first start with the  proof of the  pre-maximum principle.

\begin{proof}[Proof of Theorem \ref{pajc.theo.1}]
The proof is rather simple.
 Let us first recall the definition of $\Gamma_{x_0}$:
\begin{equation}
 \Gamma_{x_0}:=\{x\in X|u(x)=u(x_0)\}.\label{defG}
\end{equation}
Since $u$ is continuous, $\Gamma_{x_0}$ is a closed subset of $X$.
Now observe that $\Gamma_{x_0}$ is $\Sigma*$ stable
(i.e. $\Gamma_{x_0}*\Sigma\subset \Gamma_{x_0}$).
Indeed, choose any $\bar x\in \Gamma_{x_0}$. At $\bar x, u$
satisfies
 \begin{equation*}
0\le \mathcal{M}[u](\bar x)
=\int_{G}J(g)u(\bar x*g^{-1})\,d\mu-u(\bar x)
=\int_{G}J(g)[u(\bar x*g^{-1})-u(\bar x)]\,d\mu \le 0.
 \end{equation*}
Therefore,
\begin{equation}\label{pajc.eq.zero}
 \int_{G}J(g)[u(\bar x*g^{-1})-u(\bar x)]\,d\mu=0.
\end{equation}
 Using that $J\ge 0$  and that for all
$g\in G, [u(\bar x*g)-u(\bar x)]\le 0$,  \eqref{pajc.eq.zero} yields
$$
u(\bar x*g^{-1})=u(\bar x)\quad\text{ for all }\quad g \in \Sigma.
$$
Thus,  we have
 \begin{align*}
 &u(y)=u(x_0) \quad \text{ for all }\quad y\in \{\bar x\}*\Sigma.
 \end{align*}
Hence,
$\{\bar x\}*\Sigma\subset\Gamma_{x_0}$.
Since this computation holds for any element $\bar x$ of $\Gamma_{x_0}$,
we have $\Gamma_{x_0}*\Sigma\subset\Gamma_{x_0}$.

Recall now that $F_{x_0}$ is the smallest closed subset of $X$ such that
\begin{itemize}
\item $x_0\in F_{x_0}$,
\item $F_{0}*\Sigma\subset F_{x_0}$.
\end{itemize}
Since $\Gamma_{x_0}$ satisfies the above conditions, we then have
$F_{x_0}\subset\Gamma_{x_0}$.
\end{proof}


Note that $\Gamma_{x_0}$ is independent of the choice of the point where $u$
takes its global maximum. Indeed, we easily see that $\Gamma_{x_0}=\Gamma_{y}$
for any $y\in \Gamma_{x_0}$. On the contrary, the set $F_{x_0}$ strongly
depends on $x_0$ and there is no reason to always have $F_{x_0}=F_{y}$.
Indeed, for $X=G=\mathbb{R}$, if $\Sigma=\mathbb{R}^+$ then for $x_0<y$,
$F_y\subset_{\neq} F_{x_0}$.

Now, we give a  characterization of the set $F_{x_0}$ defined in
Theorem \ref{pajc.theo.1} and prove Proposition \ref{pajc.prop.descr}.
For the sake of clarity, let us first recall
Proposition \ref{pajc.prop.descr}.

\begin{proposition} \label{prop3.1}
Let $F_{x_0}$ be the set defined in Theorem \ref{pajc.theo.1}, then
$$
F_{x_0}=\overline{\bigcup_{n\in \mathbb{N}}F_n},
$$
where the $F_n$ are defined by induction as follows:
$F_0=\{x_0\}$ and for $n\ge 0$, $F_{n+1}:= F_n*\Sigma$.
\end{proposition}

\begin{proof}
Let us define the set
$$
F_\infty:=\bigcup_{n\in \mathbb{N}}F_n.
$$
Using the definition of $F_\infty$, we easily see that $F_\infty$ is  $\Sigma*$
stable. From Proposition \ref{pajc.prop.stabi}, it follows that
$\bar F_{\infty}$ is  $\Sigma*$ stable. Therefore, by definition
of $F_{x_0}$, we have  $F\subset\bar F_{\infty}$.

Now, since $x_0\in F_{x_0}$ and $F_{x_0}$ is $\Sigma *$ stable, by induction
we easily see that  $\forall n\in \mathbb{N}, F_n \subset F_{x_0}$.
Thus, $F_\infty\subset F_{x_0}$ and yields
$F_{x_0}\subset \bar F_\infty\subset F_{x_0}$.
\end{proof}


\begin{remark} \label{rmk3.2} \rm
As already mentioned in the introduction,  to obtain a strong maximum
principle for $\mathcal{M}$, we only need to find conditions on $X,d\mu$ and $J$
such that $F_{x_0}=\Gamma_{x_0}=X$.
\end{remark}

\section{Strong maximum principle when $X$  is an homogeneous space
\label{pajc.s.smp}}

In this Section, we treat the case of connected homogeneous space $X$
and prove sufficient conditions on $J$ (Theorems \ref{pajc.theo.2}
and \ref{pajc.theo.3}) in order to have a  strong maximum principle f
or $\mathcal{M}$. Let us start with the proof of  Theorem \ref{pajc.theo.2}.

\begin{proof}[Proof of Theorem \ref{pajc.theo.2}]
Again the proof is rather simple. We must check that for any $u\in C(X,\mathbb{R})$
such that
 $$
\mathcal{M}[u] \ge 0\quad \quad ( \text{resp.} \le 0 )
$$
 then $u$ cannot achieve  a global maximum (resp. minimum) in $X$ without
being constant.
So consider $u\in C(X,\mathbb{R}) $ such that $u$ achieves a maximum at $x_0$
and satisfies
 $\mathcal{M}[u] \ge 0\quad \quad ( \text{resp.} \le 0 )$.
 By definition of $\Gamma_{x}$, we only need to show that $\Gamma_{x_0}=X$.
 To this end, we will prove that  $\Gamma_{x_0}$ is a closed and open set.
 By definition of $\Gamma_{x_0}$, $\Gamma_{x_0}$ is a closed set of $X$.
Now, let us show that $\Gamma_{x_0}$ is  open.
 Choose any  $y\in \Gamma_{x_0}$. Then at this point
 \begin{equation*}
 0\le \mathcal{M}[u](y)=\int_{G}J(g)u(y*g^{-1})\,d\mu-u(y)
=\int_{G}J(g)[u(y*g^{-1})-u(y)]\,d\mu(g)\le 0.
 \end{equation*}
 Arguing as in the proof of Theorem  \ref{pajc.theo.1},  we have
$u(y*g^{-1})=u(y)=u(x_0)$ for all $g \in \Sigma$.
Since $e\in \Sigma$,   we have for some open neighbourhood $B(e)$ of $e$
$$
u(y*g^{-1})=u(x_0) \quad \text{for all } g^{-1}\in B(e).
$$
Using that $G$ is a topological group, $y*B(e)$ is then an open
neighbourhood of $y$. Thus,
$$
B(y):=y*B(e)\subset \Gamma_{x_0}.
$$
Therefore $\Gamma_{x_0}$ is an open set.
Hence, $X=\Gamma_{x_0}$ since $X$ is connected.
\end{proof}


Let  us now turn our attention to the case of compact homogeneous space
and prove Theorem \ref{pajc.theo.3}.
First, let us prove the following  technical  Lemma.

\begin{lemma}  \label{pajc.lem.poincare}
For any $g \in X$ there exists a sequence  of integers
$(n_k)_{_{k\in\mathbb{N}}}$ with $n_k\ge 1$ and $g^{n_k} \to e$ as $k\to +\infty$,
where $e$ is the unit element of $G$.
\end{lemma}

\begin{proof}
Take $g\in X$ and let us consider the following sequence $(g^{m})_{m\in\mathbb{N}}$.
Since $X$ is compact, $(g_m)_{m\in\mathbb{N}}$ has a convergent sub-sequence
$(g_{m_k})_{k\in \mathbb{N}}$.
Without any restriction, we can assume that $m_{k+1}\ge m_k+1$.
Consider now the following sequence, $w_k:=g^{m_{k+1}-m_k}$.
By construction, $w_k\to e$ and $m_{k+1}-m_k\in\mathbb{N}^*$.
Hence, with $n_k:=m_{k+1}-m_k$, $g^{n_k}\to e$.
\end{proof}

We are now in a position to prove Theorem \ref{pajc.theo.3}.

\begin{proof}[Proof of Theorem \ref{pajc.theo.3}]
 As for Theorem \ref{pajc.theo.2} we have to check that
 for any $u\in C(X,\mathbb{R}) $ such that
 $$
\mathcal{M}[u] \ge 0\quad \quad ( \text{resp.} \le 0 )
$$
 then $u$ cannot achieve  a global maximum (resp. minimum) in $X$ without
being constant.
 So consider $u\in C(X,\mathbb{R}) $ such that $u$ achieves a maximum at $x_0$
and satisfies
 $\mathcal{M}[u] \ge 0\quad \quad ( \text{resp.} \le 0 )$.
 By definition of $\Gamma_{x}$, we only need  to show that $\Gamma_{x_0}=X$.
 Again, as in the proof of Theorem \ref{pajc.theo.2}, we prove that
$\Gamma_{x_0}$ is an open and closed set and therefore $X=\Gamma_{x_0}$ since
$X$ is connected.
 By definition $\Gamma_{x_0}$ is closed. Now let us show that $\Gamma_{x_0}$ is open.
 Let $y\in \Gamma_{x_0}$ and $F_{y}$ be the set defined in
Theorem \ref{pajc.theo.1} with $y$ instead of $x_0$.
 Using now the characterization of $F_{y}$ given in
Proposition \ref{pajc.prop.descr}
 we have
 \begin{equation}
 F_{y}:=\overline{\bigcup_{n\in\mathbb{N}} F_n}\subset \Gamma_{x_0},
 \end{equation}
 where $F_{n}:=\{y\}*\Sigma^{n}$.

 Choose now $g\in \Sigma$. According to Lemma \ref{pajc.lem.poincare} there
exists  a sequence $(n_k)_{k\in\mathbb{N}}$ such that $g^{n_k}\to e$.
 By assumption, $\Sigma$ is an open subset of $G$. Therefore $\Sigma^{n_k}$
is a sequence of open subset of $G$.
 Since $g^{n_k}\to e$, $\Sigma^{n_k}$ is a open neighbourhood of $e$ for $k$
sufficiently large.
Therefore,
$$
\{y\}*\Sigma^{n_k}\subset F_y \subset \Gamma_{x_0}
$$
 Since $\Sigma^{n_k}$ is a open neighbourhood of $e$ for $k$ sufficiently large,
$\{y\}*\Sigma^{n_k}$ is then an open neighbourhood of $y$.
 Thus, $\Gamma_{x_0}$ contains an open neighbourhood of $y$ for any $y$ in
$\Gamma_{x_0}$. Hence, $\Gamma_{x_0}$ is open.
 \end{proof}

\section{Some optimal conditions \label{pajc.s.optc}}

In this section we  prove the optimal Markov condition for the convolution
operator (Theorem  \ref{pajc.theo.4}) and prove
Theorem  \ref{pajc.theo.5} .

 \subsection*{The classical convolution case $(X=G=\mathbb{R}^n)$ and $d\mu=dy$:}

 When $(X=G=\mathbb{R}^n)$ the operator $\mathcal{M}$ takes the form of the usual convolution;
i.e.,
$$
\mathcal{M}[u]:=\int_{\mathbb{R}^n}J(y)u(x-y)\,dy -u.
$$
 For such a convolution operator, the  optimal condition on $J$  in
order that $\mathcal{M}$ satisfy a strong maximum principle
is the following. This condition is known as the Markov condition.

\begin{theorem} \label{thm5.1}
$\mathcal{M}$ satisfies a strong maximum principle if and only if
 the convex hull of $\{y\in\mathbb{R}^n| J(y)>0\}$ contains $0$.
\end{theorem}


\begin{proof}
Let us start with the necessary condition.
 Assume that the Markov condition fails. We will show
that $\mathcal{M}$ does not satisfy the strong maximum principle.
To this end, we construct a non constant function $u$ that achieves
a global maximum and satisfies $\mathcal{M}[u] \ge 0$.

Let us denote $\mathop{\rm conv}(\{y\in\mathbb{R}^n| J(y)>0\} )$ the convex hull of
$\{y\in\mathbb{R}^n| J(y)>0\}$. By assumption,
$0\not\in \mathop{\rm conv}(\{y\in\mathbb{R}^n| J(y)>0\} )$.
Using the Hahn-Banach Theorem, there exists a hyperplane $H$  such that
$\mathop{\rm conv}(\{y\in\mathbb{R}^n| J(y)>0\})\subset H^+$, where
$H^+:=\{x\in\mathbb{R}^n|x_n\ge0\}$ in an orthonormal basis $(e_1; e_2;\ldots; e_n)$.
Let $v$ be a non-increasing function that is constant in $\mathbb{R}^-$,
and let us compute
$\mathcal{M}[u]$ with $u(x):=v(x_n)$.
Since the Lebesgue measure is invariant under  rotation and
$\mathop{\rm supp}(J)\subset H^+$ we have
\begin{align*}
\mathcal{M}[u]&=\int_{\mathbb{R}^{n-1}}\int_{\mathbb{R}}J(t,x_n-y_n) [v(y_n)-v(x_n)]\,dx_n \,dt\\
      &=\int_{\mathbb{R}^{n-1}}\int_{-\infty}^{x_n}J(t,x_n-y_n) [v(y_n)-v(x_n)]\,dx_n
\,dt.
\end{align*}
Therefore, since $v$ is non increasing we end up with
$\mathcal{M}[u]\ge 0$.
Since  $u$ achieves a global maximum without being constant,
$u$ is our desired function.

Let us now turn our attention to the sufficient condition. Assume that
$0\in \mathop{\rm conv}(\{y\in\mathbb{R}^n| J(y)>0\})$, then there exists a simplex
$S(p_i)$  formed by $n+1$ points of $\mathbb{R}^n$ such that $0\in S$ and $J(p_i)>0$.

By continuity, we can always assume that $(p_1,\dots, p_n)$ is a
basis of $\mathbb{R}^n$. Let us now rewrite  $x_0$ in the basis $(p_1,\dots,p_n)$:
$$
x_0=-a_1p_1\dots-a_np_n \quad\text{with } a_i\ge 0.
$$
Observe now that for  $\mathbb{R}^n$ equipped with the  sup norm  associated to
the  base $(p_1,\dots,p_n)$, there exists $r>0$ so that
$B(x_0,r)\subset\{J>0\}$.
Now for all integer $m>0$, set $ y_m=mp_0+[ma_1]p_1+\dots+[ma_n]p_n$, where
$[\cdot]$ denotes the integer part.
Now let $u$ be a continuous function satisfying
$\mathcal{M}[u]\ge 0$ and that achieves a global maximum at some point $z\in \mathbb{R}$.
Without loss of generality, we may always assume that $z=0$.
Indeed, if $z\neq 0$,  we consider the function $u_{z}(x):=u(x-z)$,
instead of $u$.
We easily see that $u_z$ achieves a global maximum at $0$ and satisfies
$\mathcal{M}[u_z]\ge 0$.
Using now Theorem \ref{pajc.theo.1}, we see that for all $m\in \mathbb{N}$,
$$
\|y_m\|<1 \quad \text{and}\quad B(y_m;mr)\subset \Gamma_0.
$$
Therefore,
$$
\bigcup_{m\in\mathbb{N}} B(y_m;mr) \subset \Gamma_0.
$$
Hence, $\mathbb{R}^n\subset \Gamma_0$.
\end{proof}

The above necessary and sufficient condition for the convolution operator
can be weakened depending on the underlying topological structure of
the space. In particular, we have in mind the following setting.
Since $\mathcal{M}$ is translation invariant, $\mathcal{M}$ is also an operator on the
set of  periodic functions. On this set of functions, the  strong
maximum principle always holds.
This condition is not so surprising since the additional periodic
structure will in some sense compactify the homogeneous space $\mathbb{R}^n$.


\subsection*{Another special case:
$X=\mathbb{R}^+,(G,*)=(\mathbb{R}^{+}\setminus\{0\},\cdot)$
 and $d\mu=dy$}
In this situation,
$$
\mathcal{M}[u]:=\int_{\mathbb{R}^+}J(y)u\big(\frac{x}{y}\big)\,dy -u,
$$
and  the above operator has essentially the same property as the usual
convolution operator. Indeed,
let us make the following change of variables $x:=e^t$. Then we have
$$
\mathcal{M}[u](e^t)=\int_{\mathbb{R}}\tilde J(t-s)u(e^s)\,ds-u(e^t),
$$
where $\tilde J(t):=J(e^t)e^t$.
Therefore, letting $v(t)=u(e^t)$, we end up with
$$
\mathcal{R}[v](t)=\tilde J \star v(t) -v(t)\quad  \text{in } \mathbb{R},
$$
with $\int_{\mathbb{R}}\tilde J(t)dt=1$.
Hence, the optimal condition to achieve  a strong maximum principle for
such a kind of operator will be of the same type as the one used for
the convolution operator.

Namely, there exists two points $a<1<b$ such that $J(a)>0$ and $J(b)>0$.
This condition  corresponds to the one given for the convolution operator
which is the existence of two points  $a'<0< b'$ such that $\tilde J(a')>0$
and $\tilde J(b')>0$.
The above observation  proves  Corollary  \ref{pajc.theo.5}.

\subsection*{Acknowledgments}
The author would like to warmly thanks Dr. Pascal Autissier for
 suggesting me this problem and is constant support.

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\end{document}