# A non-linear characterization of stochastic completeness of graphs

Marcel Schmidt  Mathematisches Institut, Universität Leipzig, 04109 Leipzig,
Germany<EMAIL_ADDRESS>and  Ian Zimmermann
Mathematisches Institut, Friedrich-Schiller-Universität Jena, 07743 Jena,
Germany<EMAIL_ADDRESS>
###### Abstract.

We study non-linear Schrödinger operators on graphs. We construct minimal
nonnegative solutions to corresponding semi-linear elliptic equations and use
them to introduce the notion of stochastic completeness at infinity in a non-
linear setting. We provide characterizations for this property in terms of a
semi-linear Liouville theorem. It is employed to establish a non-linear
characterization for stochastic completeness, which is a graph version of a
recent result on Riemannian manifolds.

##  1\. Introduction

A Riemannian manifold $M$ is called stochastically complete if the minimal
Brownian motion on it has infinite lifetime. In terms of the semigroup induced
by the Friedrichs extension of the Laplacian (which we simply denote by
$\Delta$), this is equivalent to the conservation property

$e^{t\Delta}1=1,\text{ all }t>0.$

By standard arguments it can also be expressed through the resolvent and then
reads $\alpha(\alpha-\Delta)^{-1}1=1$, all $\alpha>0$.

There is a huge amount of literature on geometric criteria involving volume
growth and curvature bounds that ensure stochastic completeness on manifolds
[Gaf59, Has60, Aze74, KL, Yau78, Ich82, Gri86, Hsu89, Tak89, Dav92, Mur94,
Stu94, IM01, Mur03, PRS03, RSV05, MV13] (this list is not comprehensive and we
refer to [Gri99] for further background). Beyond manifolds, in recent years
there has been an increased interest in such geometric criteria for stochastic
completeness of nonlocal operators e.g. for discrete graph Laplacians [Woj08,
Woj11, Hua11, Hua12, GHM12, Fol14, Hua14, HKS20] (we refer to the notes after
Chapter 14 in [KLW21] for a more detailed account of the subject).

The mentioned geometric criteria for stochastic completeness are based on
analytic and probabilistic characterizations and we refer to the survey
[Gri99] for a comprehensive overview (see also [KLW21] for the discrete case).
Here, we only mention two characterizations: Due to the parabolic maximum
principle the semigroup $(e^{t\Delta})_{t>0}$ induces minimal nonnegative
solutions to the heat equation

$\begin{cases}\partial_{t}u=\Delta u&\text{on }(0,\infty)\times M\\\
u(0,\cdot)=u_{0}&\text{on }M\end{cases}$

for given $u_{0}\geq 0$. Similarly, by the elliptic maximum principle the
resolvent $(\alpha-\Delta)^{-1}$, $\alpha>0$, yields minimal nonnegative
solutions to the Poisson equation $(\alpha-\Delta)f=h$ for given $h\geq 0$ and
$\alpha>0$. With these observations stochastic completeness (more precisely
the conservation property) can be seen to be equivalent to uniqueness of
bounded solutions to the heat equation/poisson equation.

In recent years these characterizations of stochastic completeness have been
expanded to more general equations in two different directions.

1\. Schrödinger operators: If one considers the Schrödinger operator
$\Delta-V$ with nonnegative potential $V\neq 0$, then the conservation
property of the semigroup does not hold. In this case, one always finds
$e^{t(\Delta-V)}1<1$ for all $t>0$, but the heat equation and the Poisson
equation with the operator $\Delta-V$ (instead of $\Delta$) might still have
unique bounded solutions. It was first observed in [KL12] for graphs and then
extended to Riemannian manifolds in [MS20] that uniqueness of bounded
solutions to the heat equation/Poisson equation with respect to $\Delta-V$ is
equivalent to a generalized conservation property, which is also referred to
as stochastic completeness at infinity.

2\. Non-linear equations: A different perspective is taken in [GIM20, GIMP23].
There, it is proven that stochastic completeness of $M$ is equivalent to:

  * •

For one strictly increasing and sufficiently regular
$\psi\colon[0,\infty)\to[0,\infty)$ and all $u_{0}\in L_{+}^{\infty}(M)$, the
non-linear filtration equation

$\begin{cases}\partial_{t}u=\Delta\psi(u)&\text{on }(0,\infty)\times M\\\
u(0,\cdot)=u_{0}&\text{on }M\end{cases}$

has unique bounded solutions.

  * •

For one strictly increasing continuous $\varphi\colon[0,\infty)\to[0,\infty)$
with $\varphi(0)=0$ the semi-linear elliptic equation $\Delta f=\varphi(f)$
only has trivial nonnegative bounded (sub)solutions.

The aim of this paper is to extend the elliptic part of both perspectives
(generalized conservation property, non-linear characterizations of stochastic
completeness) to non-linear Schrödinger operators on graphs. To discuss our
main results we start with a reformulation of the semi-linear elliptic
equation. If $\varphi\colon\mathbb{R}\to\mathbb{R}$ is strictly increasing
with $\varphi(0)=0$, the equation $\Delta f=\varphi(f)$ can be reformulated as
$\varphi^{-1}(\Delta f)-f=0$. We replace $f$ by $-g$, $-\Delta$ by the
weighted graph Laplacian ${\mathcal{L}}$, additionally introduce a nonnegative
potential $W$ with $\inf W>0$, and study the semi-linear equation

$\varphi^{-1}({\mathcal{L}}g)+Wg=h.$

For nonnegative $h$ we use a non-linear comparison principle and variational
methods to construct minimal (super)solutions, see Theorem 3.14, which is our
first main result. We employ these minimal (super)solutions to characterize
the triviality of nonpositive bounded supersolutions to the equation
$\varphi^{-1}({\mathcal{L}}g)+Wg=0$ via a non-linear conservation property,
see Theorem 4.2, which is our second main result. This extends the main result
of [KL12] to certain non-linear operators and gives rise to a definition of
stochastic completeness at infinity in a semi-linear setting, see Definition
4.3. We provide a geometric condition that ensures stochastic completeness at
infinity for big enough $W$, see Proposition 4.5

We also use minimal supersolutions to establish (and slightly generalize) the
main elliptic result of [GIMP23] for graphs. We show that stochastic
completeness is equivalent to the triviality of bounded nonnegative solutions
to $-{\mathcal{L}}f=\varphi(Wf)$ for one strictly increasing continuous
$\varphi\colon\mathbb{R}\to\mathbb{R}$ with $\varphi(0)=0$ and one bounded $W$
with $\inf W>0$ (or all $W$ with $\inf W>0$, [GIMP23] treats the case $W=1$),
see Theorem 5.1, which is our third main result. It actually goes beyond
[GIMP23] by also providing non-linear conservation properties that
characterize stochastic completeness.

We wrote this paper specifically for the graph case for two reasons. On the
one hand, as discussed above, stochastic completeness of graphs has gained
quite some attention in recent years. On the other hand, the graph Laplacian
is a non-local operator and certain tools (chain rules and local estimates)
are not available. We wanted to provide a proof of one of the main results of
[GIM20, GIMP23] that is less reliant on local PDE techniques but rather uses
variational methods (plus a comparison principle). This approach makes it
possible to extend the results to other operators, such as e.g. fractional
Laplacians or Laplacians on metric graphs.

A characterization of stochastic completeness of graphs in terms of uniqueness
of bounded solutions to non-linear filtration equations is beyond the scope of
this paper. The reason is the lack of a sufficiently general theory of
positive solutions to non-linear parabolic equations on graphs. A first step
towards such a theory has been recently taken in [BSW22], where
$\ell^{1}$-solutions for the porous medium equation on graphs are considered.
We note however, that a characterization of stochastic completeness via
uniqueness of bounded nonnegative subsolutions to filtration equation with
respect to the special nonlinearity $\psi(t)=\log(t)$ (which is somewhat skew
to the assumptions in [GIMP23], which require $\psi(0)=0$) is already treated
in [Ma22].

Acknowledgements. M.S. acknowledges financial support of the DFG within the
priority programme ’Geometry at Infinity’.

##  2\. Preliminaries

In this section we introduce the setting of operators on weighted discrete
graphs. We closely follow [KLW21] and refer to it for claims which are not
explained in detail.

###  2.1. Graphs and formal objects

Throughout we let $X\neq\emptyset$ be a countable set. We write $C(X)$ for the
real-valued functions on $X$ and $C_{c}(X)$ for the real-valued functions of
finite support. Every function $m\colon X\to(0,\infty)$ induces a measure on
all subsets of $X$, which we also denote by $m$, via

$m(A)=\sum_{x\in A}m(x),\quad A\subseteq X.$

We call such a pair $(X,m)$ discrete measure space. For $1\leq p<\infty$ we
let

$\ell^{p}(X,m)=\\{f\colon X\to\mathbb{R}\mid\sum_{x\in
X}|f(x)|^{p}m(x)<\infty\\}$

denote the real-valued discrete Lebesgue space and equip it with the norm

$\left\|f\right\|_{p}=\left(\sum_{x\in X}|f(x)|^{p}m(x)\right)^{1/p}.$

The space of bounded real-valued functions is denoted by $\ell^{\infty}(X)$
and equipped with the supremum norm $\left\|\cdot\right\|_{\infty}$.

A weighted graph over $X$ is a symmetric function $b\colon X\times
X\to[0,\infty)$ with $b(x,x)=0$, $x\in X$, such that at all $x\in X$ the
weighted vertex degree is finite, i.e.,

$\deg(x):=\sum_{y\in X}b(x,y)<\infty.$

The elements of $X$ are called vertices. Two vertices $x,y\in X$ are thought
to be connected by an edge if $b(x,y)>0$, in which case we write $x\sim y$. A
path is a finite of infinite sequence of vertices
$\gamma=(x_{1},x_{2},\ldots)$ such that $x_{j}\sim x_{j+1}$, for all
$j=1,2,\ldots$. We say that a subset $U\subseteq X$ is connected if for every
$x,y\in U$ there exists a path $\gamma$ in $U$ that contains $x$ and $y$.

The graph energy is the functional

${\mathcal{Q}}\colon
C(X)\to[0,\infty],\quad{\mathcal{Q}}(f)=\frac{1}{2}\sum_{x,y\in
X}b(x,y)(f(x)-f(y))^{2}$

and the corresponding space of functions of finite energy is given by

${\mathcal{D}}=\\{f\in C(X)\mid{\mathcal{Q}}(f)<\infty\\}.$

By polarization ${\mathcal{Q}}$ induces a bilinear form on ${\mathcal{D}}$,
which we also denote by ${\mathcal{Q}}$. In this sense, we have
${\mathcal{Q}}(f)={\mathcal{Q}}(f,f)$ for $f\in{\mathcal{D}}$.

By our assumption on the finiteness of the weighted vertex degree we have
$C_{c}(X)\subseteq{\mathcal{D}}$. We let ${\mathcal{D}}_{0}$ be the space of
those $f\in{\mathcal{D}}$ for which there exists a sequence $(f_{n})$ in
$C_{c}(X)$ with $f_{n}\to f$ pointwise, as $n\to\infty$, and
$\lim_{n\to\infty}{\mathcal{Q}}(f-f_{n})=0$. Moreover, we let

${\mathcal{Q}}_{0}\colon
C(X)\to[0,\infty],\quad{\mathcal{Q}}_{0}(f)=\begin{cases}{\mathcal{Q}}(f)&\text{if
}f\in{\mathcal{D}}_{0}\\\ \infty&\text{else}\end{cases}.$

######  Lemma 2.1.

The functionals ${\mathcal{Q}}$ and ${\mathcal{Q}}_{0}$ are lower
semicontinuous with respect to pointwise convergence. In particular, if
$(f_{n})$ in $C(X)$ converges pointwise to $f\in C(X)$ and
$({\mathcal{Q}}_{0}(f_{n}))$ is bounded, then $f\in{\mathcal{D}}_{0}$ and

${\mathcal{Q}}_{0}(f)\leq\liminf_{n\to\infty}{\mathcal{Q}}_{0}(f_{n}).$

###### Proof.

The lower semicontinuity of ${\mathcal{Q}}$ is a simple application of Fatou’s
lemma. For the lower semicontinuity of ${\mathcal{Q}}_{0}$ we refer to [MS23,
Proposition 3.5]. ∎

######  Lemma 2.2.

Let $f\in C(X)$ and let $C\colon\mathbb{R}\to\mathbb{R}$ be a normal
contraction. Then

${\mathcal{Q}}(C\circ f)\leq{\mathcal{Q}}(f)\text{ and
}{\mathcal{Q}}_{0}(C\circ f)\leq{\mathcal{Q}}_{0}(f).$

In particular, for $f,g\in C(X)$ we have

${\mathcal{Q}}(f\wedge g)\leq{\mathcal{Q}}(f)+{\mathcal{Q}}(g),$

where $f\wedge g=\min\\{f,g\\}$.

###### Proof.

For ${\mathcal{Q}}$ this is a direct consequence of the definition. For
${\mathcal{Q}}_{0}$ we have to take into account where it is finite and make
use of the lower semicontinuity. More precisely, for $f\in{\mathcal{D}}_{0}$
(for other $f$ there is nothing to show) we choose a sequence $(f_{n})$ in
$C_{c}(X)$ with $f_{n}\to f$ pointwise and ${\mathcal{Q}}(f-f_{n})\to 0$
(which also yields
${\mathcal{Q}}(f_{n})\to{\mathcal{Q}}(f)={\mathcal{Q}}_{0}(f)$). Then
$C(f_{n})\to C(f)$ pointwise and $C(f_{n})\in
C_{c}(X)\subseteq{\mathcal{D}}_{0}$. Using ${\mathcal{Q}}={\mathcal{Q}}_{0}$
on ${\mathcal{D}}_{0}$, the pointwise lower semicontinuity of
${\mathcal{Q}}_{0}$ and the inequality for ${\mathcal{Q}}$, we obtain

$\displaystyle{\mathcal{Q}}_{0}(C\circ f)$
$\displaystyle\leq\liminf_{n\to\infty}{\mathcal{Q}}_{0}(C\circ
f_{n})=\liminf_{n\to\infty}{\mathcal{Q}}(C\circ f_{n})$
$\displaystyle\leq\liminf_{n\to\infty}{\mathcal{Q}}(f_{n})={\mathcal{Q}}_{0}(f).$

’In particular’-part: Applying the already established inequality to the
absolute value yields ${\mathcal{Q}}(|f|)\leq{\mathcal{Q}}(f)$ for all $f\in
C(X)$. It is well-known that for quadratic forms this inequality leads to
${\mathcal{Q}}(f\wedge g)\leq{\mathcal{Q}}(f)+{\mathcal{Q}}(g)$, see e.g. the
proof of [LSW20, Lemma 1.22] for the required computations. ∎

As an application of both properties we show that nonnegative functions in
${\mathcal{D}}_{0}$ can be approximated by nonnegative functions in $C_{c}(X)$
satisfying a certain bound.

######  Lemma 2.3.

Let $0\leq f\in{\mathcal{D}}_{0}$. Then there exists a sequence $(f_{n})$ in
$C_{c}(X)$ with $0\leq f_{n}\leq f$, $f_{n}\to f$ pointwise and
${\mathcal{Q}}(f_{n}-f)\to 0$.

###### Proof.

We first choose a sequence $(g_{n})$ in $C_{c}(X)$ with $g_{n}\to f$ pointwise
and ${\mathcal{Q}}(f-g_{n})\to 0$. We then consider the sequence
$h_{n}:=(g_{n})_{+}\wedge f$. Then $0\leq h_{n}\leq f$ and $h_{n}\to f$
pointwise. From Lemma 2.2 we infer

${\mathcal{Q}}(h_{n})\leq{\mathcal{Q}}((g_{n})_{+})+{\mathcal{Q}}(f)\leq{\mathcal{Q}}(g_{n})+{\mathcal{Q}}(f).$

Since ${\mathcal{Q}}(g_{n}-f)\to 0$ and ${\mathcal{Q}}$ is a quadratic form,
the sequence $(h_{n})$ is ${\mathcal{Q}}$-bounded. By the Banach-Saks theorem
it has a subsequence $(h_{n_{j}})$ such that the sequence $(f_{N})$ defined by

$f_{N}=\frac{1}{N}\sum_{j=1}^{N}h_{n_{j}}$

is ${\mathcal{Q}}$-Cauchy. By construction it satisfies $0\leq f_{N}\leq f$
and $f_{N}\to f$ pointwise. Moreover, the pointwise lower semicontinuity of
${\mathcal{Q}}$ yields

${\mathcal{Q}}(f-f_{M})\leq\liminf_{N\to\infty}{\mathcal{Q}}(f_{N}-f_{M}).$

The right side of this inequality becomes small for large $M$ because
$(f_{N})$ is ${\mathcal{Q}}$-Cauchy. ∎

The formal Laplacian ${\mathcal{L}}={\mathcal{L}}_{b,m}$ of a graph $b$ over
the discrete measure space $(X,m)$ has the domain

${\mathcal{F}}={\mathcal{F}}_{b}=\\{f\in C(X)\mid\sum_{y\in
X}b(x,y)|f(y)|<\infty\text{ for all }x\in X\\},$

on which it acts by

${\mathcal{L}}f(x)=\frac{1}{m(x)}\sum_{y\in X}b(x,y)(f(x)-f(y)).$

Our assumption on the finiteness of the weighted vertex degree implies
$\ell^{\infty}(X)\subseteq{\mathcal{F}}$. The formal Laplacian is related to
the quadratic form ${\mathcal{Q}}$ through the following Green formula, see
e.g. [KLW21, Proposition 1.4].

######  Lemma 2.4 (Green’s formula).

${\mathcal{D}}\subseteq{\mathcal{F}}$ and for $f\in{\mathcal{D}}$ and
$\varphi\in C_{c}(X)$ we have

${\mathcal{Q}}(f,\varphi)=\sum_{x\in X}({\mathcal{L}}f)(x)\varphi(x)m(x).$

###  2.2. Dirichlet forms and stochastic completeness (at infinity)

In this subsection we briefly recall the concept of stochastic completeness
and stochastic completeness at infinity for graphs. Let $c\colon
X\to[0,\infty)$. We consider the quadratic form $Q=Q_{b,c}$ on $\ell^{2}(X,m)$
with domain

$D(Q)=\\{u\in{\mathcal{D}}_{0}\cap\ell^{2}(X,m)\mid\sum_{x\in
X}u(x)^{2}c(x)<\infty\\},$

on which it acts by

$Q(u)={\mathcal{Q}}(u)+\sum_{x\in X}u(x)^{2}c(x).$

It is a regular Dirichlet form and by Green’s formula the associated
nonnegative self-adjoint operator $L=L_{b,c,m}$ on $\ell^{2}(X,m)$ satisfies
$D(L)\subseteq{\mathcal{F}}$ with

$Lf(x)={\mathcal{L}}f(x)+\frac{c(x)}{m(x)}f(x)$

for $f\in D(L)$, $x\in X$.

######  Remark.

For more details see [KLW21, Chapter 1]. The operator $L$ can be thought of
being the self-adjoint realization of the Schrödinger operator
${\mathcal{L}}+c/m$ with abstract Dirichlet boundary conditions at infinity.

The subsequent discussion follows [KLW21, Chapter 7]. Since $Q$ is a Dirichlet
form, the induced semigroup $(e^{-tL})_{t>0}$ and resolvent
$((L+\beta)^{-1})_{\beta>0}$ are positivity preserving. Hence, they can be
extended to linear maps on the cone $C(X)_{+}=\\{f\in C(X)\mid f\geq 0\\}$.
More precisely, for $x\in X$ we let

$e^{-tL}f(x):=\sup\\{e^{-tL}g(x)\mid g\in\ell^{2}(X,m)\text{ with }0\leq g\leq
f\\}$

and

$(L+\beta)^{-1}f(x):=\sup\\{(L+\beta)^{-1}g(x)\mid g\in\ell^{2}(X,m)\text{
with }0\leq g\leq f\\}.$

These extensions may attain the value $\infty$. Since both resolvent and
semigroup are continuous on $\ell^{2}(X,m)$, it suffices to take the $\sup$
over $g\in C_{c}(X)$ with $0\leq g\leq f$.

Since $Q$ is a Dirichlet form, the semigroup and resolvent are not only
positivity preserving but also Markovian. This means that for $0\leq f\leq 1$
we have

$0\leq e^{-tL}f\leq 1\text{ and }0\leq\beta(L+\beta)^{-1}f\leq 1.$

In particular, for nonnegative bounded functions the extended resolvent and
semigroup only attain finite values.

######  Definition 2.5 (Stochastic completeness).

Assume $c=0$. We say that $b$ over $(X,m)$ is stochastically complete if for
all $t>0$ the semigroup $(e^{-tL})_{t>0}$ satisfies the conservation property
$1=e^{-tL}1.$

If $c\neq 0$, then the semigroup $(e^{-tL})_{t>0}$ never satisfies the
conservation property. This is why the following concept of stochastic
completeness at infinity was introduced in [KL12].

######  Definition 2.6 (Stochastic completeness at infinity).

We say that $(b,c)$ over $(X,m)$ is stochastically complete at infinity if for
all $t>0$ the semigroup $(e^{-tL})_{t>0})$ satisfies the generalized
conservation property

$1=e^{-tL}1+\int_{0}^{t}e^{-sL}\frac{c}{m}ds.$

######  Remark.

The conservation property is equivalent to $1=\beta(L+\beta)^{-1}1$ for all
$\beta>0$ and the generalized conservation property is equivalent to
$1=\beta(L+\beta)^{-1}1+(L+\beta)^{-1}\frac{c}{m}$ for all $\beta>0$.

Below we need the following characterizations of stochastic completeness (at
infinity), which can be found in [KLW21, Theorem 7.2].

######  Proposition 2.7 (Some characterizations of stochastic completeness at
infinity).

The following assertions are equivalent.

  1. (i)

$(b,c)$ over $(X,m)$ is stochastically complete at infinity.

  2. (ii)

For one/all $\beta>0$ every $0\leq g\in\ell^{\infty}(X)$ with
${\mathcal{L}}g+\frac{c}{m}g+\beta g\leq 0$ satisfies $g=0$.

  3. (iii)

If $f\in{\mathcal{F}}$ satisfies $f^{*}:=\sup f\in(0,\infty)$ and
$\delta\in(0,f^{*})$, then

$\sup_{x\in
X_{\delta}}\left({\mathcal{L}}f(x)+\frac{c(x)}{m(x)}f(x)\right)\geq 0,$

where $X_{\delta}=\\{x\in X\mid f(x)>f^{*}-\delta\\}$ (’Omori-Yau maximum
principle’).

##  3\. A resolvent of non-linear Schrödinger operators with bounded below
potential

In this section we introduce minimal resolvents of the non-linear Schrödinger
operator $\varphi^{-1}\circ\mathcal{L}+W$. Here and throughout this section we
make the following assumptions.

Assumptions:

  * •

$\varphi\colon\mathbb{R}\to\mathbb{R}$ is continuous and strictly increasing
with $\varphi(0)=0$.

  * •

$W\colon X\to(0,\infty)$ with $W_{0}:=\inf_{x\in X}W(x)>0$.

Since $\varphi$ is strictly increasing and continuous, its range is a possibly
unbounded open interval and $\varphi$ is a bijection onto it. Hence,
$\varphi^{-1}({\mathcal{L}}f(x))$ is well-defined as long as
${\mathcal{L}}f(x)\in{\rm ran}\,\varphi$. We let

${\mathcal{F}}_{\varphi}=\\{f\in{\mathcal{F}}\mid{\mathcal{L}}f(x)\in{\rm
ran}\,\varphi\text{ for all }x\in X\\}$

be the domain of the formal non-linear Schrödinger operator
$\varphi^{-1}\circ\mathcal{L}+W$. For $f\in{\mathcal{F}}_{\varphi}$ and $x\in
X$ the operator acts by

$(\varphi^{-1}\circ\mathcal{L}+W)(f)(x)=\varphi^{-1}({\mathcal{L}}f(x))+W(x)f(x).$

Note that ${\mathcal{F}}_{\varphi}$ is convex.

For nonnegative $f\in C(X)$ we construct a minimal nonnegative function
$g\in{\mathcal{F}}_{\varphi}$ that solves the equation

$\varphi^{-1}\left(\mathcal{L}g\right)+Wg=f$

(if such nonnegative solutions exist). Using certain convex functionals we
first show existence for finitely supported $f$ and extend it by using
monotonicity properties. We then use these resolvents to characterize when all
nonnegative bounded solutions to the semi-linear equation

$-{\mathcal{L}}g=\varphi(Wg)$

vanish.

###  3.1. Dirichlet resolvent of finitely supported functions

In this subsection we use variational methods to construct the minimal
solution to $\varphi^{-1}\left(\mathcal{L}g\right)+Wg=f$ for compactly
supported $f$, which we call Dirichlet resolvent.

We denote by $\Phi$ the antiderivative of $2\varphi$ with $\Phi(0)=0$, i.e.,

$\Phi\colon\mathbb{R}\to\mathbb{R},\quad\Phi(s)=\int_{0}^{s}2\varphi(t)dt.$

Then $\Phi$ is nonnegative and since $\varphi$ is strictly increasing, $\Phi$
is strictly convex.

Let $f\in C_{c}(X)$ and $\kappa_{f}=\kappa_{f}^{\varphi,W}\colon
C(X)\to[0,\infty]$,

$\kappa_{f}(u)=\sum_{x\in X}\Phi(f(x)-W(x)u(x))\frac{m(x)}{W(x)}.$

We define the functional ${\mathcal{E}}_{f}={\mathcal{E}}_{f}^{\varphi,W}$ by
$D({\mathcal{E}}_{f})=\\{u\in{\mathcal{D}}_{0}\mid\kappa_{f}(u)<\infty\\}$ and

${\mathcal{E}}_{f}\colon
D({\mathcal{E}}_{f})\to[0,\infty),\quad{\mathcal{E}}_{f}(u)={\mathcal{Q}}(u)+\kappa_{f}(u).$

######  Remark.

Since $f$ has finite support and $\Phi(0)=0$, we have

$\kappa_{f}(g)=\sum_{x\in X}\Phi(f(x)-W(x)g(x))\frac{m(x)}{W(x)}<\infty$

for all $g\in C_{c}(X)$, which implies $C_{c}(X)\subseteq
D({\mathcal{E}}_{f})$. Moreover, if $u\in D({\mathcal{E}}_{f})$ and $g\in
C_{c}(X)$, then $u+g\in D({\mathcal{E}}_{f})$.

######  Lemma 3.1 (Pointwise lower semicontinuity).

If $(u_{n})$ is a sequence in $D({\mathcal{E}}_{f})$ with $u_{n}\to u$
pointwise and $\liminf_{n\to\infty}{\mathcal{E}}_{f}(u_{n})<\infty$, then
$u\in D({\mathcal{E}}_{f})$ and

${\mathcal{E}}_{f}(u)\leq\liminf_{n\to\infty}{\mathcal{E}}_{f}(u_{n}).$

###### Proof.

The continuity of $\Phi$ and Fatou’s lemma imply

$\sum_{x\in
X}\Phi(f(x)-W(x)u(x))\frac{m(x)}{W(x)}\leq\liminf_{n\to\infty}\sum_{x\in
X}\Phi(f(x)-W(x)u_{n}(x))\frac{m(x)}{W(x)}.$

This observation combined with Lemma 2.1 yields the claim. ∎

######  Lemma 3.2 (Approximation in $C_{c}(X)$).

For any $0\leq u\in D({\mathcal{E}}_{f})$ there exists a sequence $(g_{n})$ in
$C_{c}(X)$ with $0\leq g_{n}\leq u$ such that

${\mathcal{E}}_{f}(u)=\lim_{n\to\infty}{\mathcal{E}}_{f}(g_{n}).$

###### Proof.

According to Lemma 2.3 there exists a sequence $(g_{n})$ in $C_{c}(X)$ with
$0\leq g_{n}\leq u$, $g_{n}\to u$ pointwise and ${\mathcal{Q}}(g_{n}-u)\to 0$.
Since ${\mathcal{Q}}$ is a quadratic form, we also have
${\mathcal{Q}}(g_{n})\to{\mathcal{Q}}(u)$. The inequality $0\leq g_{n}\leq u$
yields

$f-Wu\leq f-Wg_{n}\leq f,$

which by the definition of $\Phi$ implies

$\Phi(f-Wg_{n})\leq\Phi(f)+\Phi(f-Wu).$

Since $f\in C_{c}(X)$ and $\kappa_{f}(u)<\infty$, the right side of this
inequality is summable with respect to the weight $m/W$. Hence, the pointwise
convergence $\Phi(f-Wg_{n})\to\Phi(f-Wu)$ and Lebesgue’s dominated convergence
theorem yield $\kappa_{f}(g_{n})\to\kappa_{f}(u)$. Together with the already
established ${\mathcal{Q}}(g_{n})\to{\mathcal{Q}}(u)$ this proves the claim. ∎

######  Proposition 3.3 (Existence of the Dirichlet resolvent).

There exists a unique function $Rf=R^{\varphi,W}f\in D({\mathcal{E}}_{f})$
such that

${\mathcal{E}}_{f}(Rf)=\inf\\{{\mathcal{E}}_{f}(u)\mid u\in
D({\mathcal{E}}_{f})\\}=\inf\\{{\mathcal{E}}_{f}(g)\mid g\in C_{c}(X)\\}.$

It satisfies $W_{0}\left\|Rf\right\|_{\infty}\leq\left\|f\right\|_{\infty}$
and $Rf\in{\mathcal{F}}_{\varphi}$, and solves the equation

$\varphi^{-1}\left(\mathcal{L}Rf\right)+WRf=f.$

###### Proof.

The strict convexity of $\Phi$ and the convexity of ${\mathcal{Q}}$ imply the
strict convexity of ${\mathcal{E}}_{f}$. Hence, if a minimizer $Rf$ exists, it
is unique. That taking the infimum over $D({\mathcal{E}}_{f})$ and over
$C_{c}(X)$ yields the same result follows from Lemma 3.2.

To prove existence we first show that functions in a minimizing sequence can
be chosen to satisfy a uniform pointwise bound. Let
$K=\left\|f\right\|_{\infty}/W_{0}$ and consider the normal contraction
$C\colon\mathbb{R}\to\mathbb{R},C(t)=(t\wedge K)\vee(-K)$. The definition of
$\Phi$ implies $\Phi(s)\leq\Phi(t)$ if $|s|\leq|t|$. Moreover, if
$|t|\leq\left\|f\right\|_{\infty}$, then for all $x\in X$ we have

$\displaystyle|t-W(x)C(s)|$
$\displaystyle=W(x)|t/W(x)-C(s)|=W(x)|C(t/W(x))-C(s)|$
$\displaystyle\leq|t-W(x)s|.$

Combining both inequalities we obtain

$\Phi(f(x)-W(x)C(u(x)))\leq\Phi(f(x)-W(x)u(x))$

for $u\in C(X)$ and $x\in X$.

Since $\mathcal{Q}$ on $\mathcal{D}_{0}$ is compatible with normal
contractions, see Lemma 2.2, we arrive at $C\circ u\in D({\mathcal{E}}_{f})$
and ${\mathcal{E}}_{f}(C\circ u)\leq{\mathcal{E}}_{f}(u)$ for any $u\in
D({\mathcal{E}}_{f})$.

Let $(u_{n})$ be a minimizing sequence for ${\mathcal{E}}_{f}$. By the
contraction property we proved above we can assume without loss of generality
$u_{n}=C\circ u_{n}$, i.e., $|u_{n}|\leq K=\left\|f\right\|_{\infty}/W_{0}$.
With this at hand we can assume without loss of generality that there exists
$Rf\in C(X)$ with $u_{n}\to Rf$ pointwise. Then
$\left\|Rf\right\|_{\infty}\leq\left\|f\right\|_{\infty}/W_{0}$ and the lower
semicontinuity of ${\mathcal{E}}_{f}$ with respect to pointwise convergence
yields $Rf\in D({\mathcal{E}}_{f})$ and

${\mathcal{E}}_{f}(Rf)\leq\liminf_{n\to\infty}{\mathcal{E}}_{f}(u_{n})=\inf\\{{\mathcal{E}}_{f}(u)\mid
u\in D({\mathcal{E}}_{f})\\}.$

To prove that the minimizer satisfies the claimed equation we recall $Rf+g\in
D({\mathcal{E}}_{f})$ for any $g\in C_{c}(X)$. For $h\in\mathbb{R}$ and $x\in
X$ this implies

${\mathcal{E}}_{f}(Rf+h\delta_{x})\geq{\mathcal{E}}_{f}(Rf).$

Here, $\delta_{x}(x)=1$ and $\delta_{x}(y)=0$ if $x\neq y$. Comparing the
expressions on both sides of this inequality and using that $\mathcal{Q}$ is a
quadratic form we obtain

$\displaystyle 2h{\mathcal{Q}}(Rf,\delta_{x})+h^{2}\mathcal{Q}(\delta_{x})$
$\displaystyle\geq\frac{m(x)}{W(x)}\left(\Phi(f(x)-W(x)Rf(x))-\Phi(f(x)-W(x)(Rf(x)+h)\right)$

Dividing by $2h$, letting $h\to 0\pm$ and using $\Phi^{\prime}=2\varphi$
yields

${\mathcal{Q}}(Rf,\delta_{x})=m(x)\varphi(f(x)-W(x)Rf(x)).$

Dividing by $m(x)$ and using Green’s formula Lemma 2.4, we infer
${\mathcal{L}}Rf(x)=\varphi(f(x)-W(x)Rf(x))$, which shows the claim. ∎

In order to establish further properties of $Rf$ we approximate it via
resolvents on finite sets, where we can employ a comparison principle (see
below). To this end, for $U\subseteq X$ and $f\in C_{c}(X)$ we define the
convex functional ${\mathcal{E}}_{f,U}={\mathcal{E}}_{f,U}^{\varphi,W}$ as the
restriction of ${\mathcal{E}}_{f}$ to $D({\mathcal{E}}_{f,U})=\\{u\in
D({\mathcal{E}}_{f})\mid{\rm supp}\,u\subseteq U\\}.$ The same arguments as in
the proof of Proposition 3.3 show that it has a unique minimizer
$R_{U}f=R_{U}^{\varphi,W}f\in D({\mathcal{E}}_{f,U})$, which satisfies
$W_{0}\left\|R_{U}f\right\|_{\infty}\leq\left\|f\right\|_{\infty}$ and
$R_{U}f\in{\mathcal{F}}_{\varphi}$, and solves the non-local Dirichlet problem

$\begin{cases}\varphi^{-1}\left(\mathcal{L}R_{U}f\right)+WR_{U}f=f&\text{on
}U\\\ R_{U}f=0&\text{on }X\setminus U\end{cases}.$

######  Definition 3.4 (Dirichlet resolvent).

We call $Rf$ (respectively $R_{U}f$) the Dirichlet resolvent (respectively the
Dirichlet resolvent on $U$) of $f$ with respect to the operator
$\varphi^{-1}\circ\mathcal{L}+W$.

######  Remark (Neumann resolvent).

If we only wanted to construct solutions $u$ to the non-linear equation
$\varphi^{-1}(\mathcal{L}u)+Wu=f$, instead of ${\mathcal{E}}_{f}$, which is
defined on a subset of ${\mathcal{D}}_{0}$, we could have considered the
functional

$\widetilde{{\mathcal{E}}}_{f}(u)={\mathcal{Q}}(u)+\kappa_{f}(u),$

on the domain
$D(\widetilde{{\mathcal{E}}}_{f})=\\{u\in{\mathcal{D}}\mid\kappa_{f}(u)<\infty\\}$.
It also has a unique minimizer $\widetilde{R}f$ that solves the mentioned
equation. The function $\widetilde{R}f$ is the Neumann resolvent of $f$ with
respect to the operator $\varphi^{-1}\circ\mathcal{L}+W$. The names Dirichlet
resolvent and Neumann resolvent come from the observation that they correspond
to solutions with abstract ’Dirichlet boundary conditions’ and ’Neumann
boundary conditions’, respectively. This analogy can be best seen for the
Dirichlet resolvent on a subset $U$.

We use the Dirichlet resolvent because it can be approximated by resolvents on
finite subsets and turns out to be the minimal solution. This is discussed
next.

The main tool for studying properties of the resolvent is the following
comparison principle.

######  Lemma 3.5 (Comparison principle - I).

Let $u,v\in{\mathcal{F}}_{\varphi}$ and assume that for some $U\subseteq X$
they satisfy the following conditions:

  1. (a)

$\varphi^{-1}({\mathcal{L}}u)+Wu\geq\varphi^{-1}({\mathcal{L}}v)+Wv$ on $U$.

  2. (b)

$u-v$ attains a minimum on $U$.

  3. (c)

$u\geq v$ on $X\setminus U$.

Then $u\geq v$ on $X$.

###### Proof.

Let $g:=u-v$ attain its minimum on $U$ at $x\in U$. If $g(x)<0$, then
assumption (a) and $W(x)>0$ imply

$\varphi^{-1}({\mathcal{L}}u(x))-\varphi^{-1}({\mathcal{L}}v(x))\geq-W(x)g(x)>0.$

The strict monotonicity of $\varphi$ yields

$\frac{1}{m(x)}\sum_{y\in
X}b(x,y)(g(x)-g(y))={\mathcal{L}}g(x)={\mathcal{L}}u(x)-{\mathcal{L}}v(x)>0.$

This implies the existence of $y\sim x$ such that $g(y)<g(x)<0$. Since $g$
attains its minimum on $U$ in $x$, we infer $y\in X\setminus U$. But this
implies $g(y)\geq 0$ by assumption (c), which is a contradiction. Hence,
$u-v=g\geq 0$ on $U$. ∎

As a corollary we also note the following version of the comparison principle,
which compares different nonlinearities and potentials, but only holds under a
further nonnegativity assumption.

######  Corollary 3.6 (Comparison principle - II).

Let $\psi\colon\mathbb{R}\to\mathbb{R}$ be strictly increasing, continuous
with $\psi(0)=0$ and $\psi(t)\leq\varphi(t)$ for $t\geq 0$. Let $V\colon
X\to(0,\infty)$ with $V\geq W$. Let $u\in{\mathcal{F}}_{\varphi}$ and
$v\in{\mathcal{F}}_{\psi}$ and assume that for some $U\subseteq X$ they
satisfy the following conditions:

  1. (a)

$\varphi^{-1}({\mathcal{L}}u)+Wu\geq\psi^{-1}({\mathcal{L}}v)+Vv$ on $U$.

  2. (b)

$u-v$ attains a minimum on $U$.

  3. (c)

$u\geq v$ on $X\setminus U$.

  4. (d)

${\mathcal{L}}v\geq 0$ on $U$ and $v\geq 0$ on $U$.

Then $u\geq v$ on $X$.

###### Proof.

Since $\psi,\varphi$ are continuous and strictly increasing, their ranges are
open intervals. Hence, the inequality $\psi(t)\leq\varphi(t)$, $t\geq 0$,
yields

$\operatorname{ran}\psi\cap[0,\infty)\subseteq\operatorname{ran}\varphi\cap[0,\infty)$

and $\psi^{-1}(s)\geq\varphi^{-1}(s)$ for $s\geq 0$ with
$s\in\operatorname{ran}\psi$. Using these observations and the assumptions
${\mathcal{L}}v\geq 0$ and $v\in{\mathcal{F}}_{\psi}$, shows
$v\in{\mathcal{F}}_{\varphi}$. Moreover, using (a) and ${\mathcal{L}}v\geq 0$,
$v\geq 0$, $V\geq W$, we obtain

$\varphi^{-1}({\mathcal{L}}u)+Wu\geq\psi^{-1}({\mathcal{L}}v)+Vv\geq\varphi^{-1}({\mathcal{L}}v)+Wv.$

With this at hand the statement follows with the assumptions (b) and (c) and
the previous lemma. ∎

######  Lemma 3.7 (Domain monotonicity).

Let $U,V\subseteq X$ be finite with $U\subseteq V$. Furthermore, let $f,g\in
C_{c}(X)$ with $0\leq f\leq g$. Then

$R_{U}f\leq R_{V}g.$

In particular, $R_{V}g\geq 0$.

###### Proof.

We first show $R_{V}g\geq 0$. By definition $R_{V}g=0$ on $X\setminus V$ and
$\varphi^{-1}\left(\mathcal{L}R_{V}g\right)+WR_{V}g=g\geq 0$ on $V$. Moreover,
$R_{V}g$ attains its minimum on the finite set $V$. Hence, the comparison
principle Lemma 3.5 applied to $u=R_{V}g$ and $v=0$ yields $R_{V}g\geq 0$.

The other inequality also follows from the comparison principle. More
precisely, on $U$ we have

$\varphi^{-1}\left(\mathcal{L}R_{V}g\right)+WR_{V}g=g\geq
f=\varphi^{-1}\left(\mathcal{L}R_{U}f\right)+WR_{U}f$

and on $X\setminus U$ we have

$R_{V}g\geq 0=R_{U}f.$

Since $R_{V}g-R_{U}f$ attains its minimum on the finite set $U$, we infer
$R_{U}f\leq R_{V}g$ from the comparison principle Lemma 3.5. ∎

######  Lemma 3.8 (Approximation via finite subsets - I).

Let $0\leq f\in C_{c}(X)$ and let $(K_{n})$ be finite subsets in $X$ with
$K_{n}\nearrow X$. Then $R_{K_{n}}f\nearrow Rf$ pointwise.

###### Proof.

The domain monotonicity implies that $R_{K_{n}}f$ is increasing. Since
$W_{0}\left\|R_{K_{n}}f\right\|_{\infty}\leq\left\|f\right\|_{\infty}$, the
limit

$F:=\lim_{n\to\infty}R_{K_{n}}f$

exists pointwise. We show that it is the minimizer of ${\mathcal{E}}_{f}$.

The domains $D({\mathcal{E}}_{f,K_{n}})$ are increasing. Therefore,
${\mathcal{E}}_{f}(R_{K_{n}}f)={\mathcal{E}}_{f,K_{n}}(R_{K_{n}}f)$ is
decreasing and bounded by ${\mathcal{E}}_{f,K_{1}}(R_{K_{1}}f)$. The pointwise
lower semicontinuity of ${\mathcal{E}}_{f}$, Lemma 3.1, yields $F\in
D({\mathcal{E}}_{f})$ and

${\mathcal{E}}_{f}(F)\leq\liminf_{n\to\infty}{\mathcal{E}}_{f}(R_{K_{n}}f)=\liminf_{n\to\infty}{\mathcal{E}}_{f,K_{n}}(R_{K_{n}}f).$

For $g\in C_{c}(X)$ there exists $N\in\mathbb{N}$ such that ${\rm
supp}\,g\subseteq K_{n}$ for each $n\geq N$. Since $R_{K_{n}}f$ is the
minimizer of ${\mathcal{E}}_{f,K_{n}}$, this implies

${\mathcal{E}}_{f,K_{n}}(R_{K_{n}}f)\leq{\mathcal{E}}_{f,K_{n}}(g)={\mathcal{E}}_{f}(g)$

for $n\geq N$. Combined with the previous inequality, we infer
${\mathcal{E}}_{f}(F)\leq{\mathcal{E}}_{f}(g)$ for all $g\in C_{c}(X)$.
According to Proposition 3.3, this implies $F=Rf$. ∎

######  Corollary 3.9 (Monotonicity of the resolvent).

Let $f,g\in C_{c}(X)$ with $0\leq f\leq g$. Then $0\leq Rf\leq Rg$.

###### Proof.

This follows from combining the previous two lemmas. ∎

We established the monotonicity of the resolvent as a consequence of the
comparison principle Lemma 3.5 and the approximation via finite sets. Using
the same argument with the slightly more general comparison principle
Corollary 3.6, we obtain the following result on domination of the resolvents.

######  Proposition 3.10 (Domination).

Let $\psi\colon\mathbb{R}\to\mathbb{R}$ be strictly increasing, continuous
with $\psi(0)=0$ and $\psi(t)\leq\varphi(t)$ for $t\geq 0$. Let $V\colon
X\to(0,\infty)$ with $V\geq W$. For all $0\leq f\in C_{c}(X)$ we have

$0\leq R^{\psi,V}f\leq R^{\varphi,W}f.$

###  3.2. The extended Dirichlet resolvent

The monotonicity of the Dirichlet resolvent on finitely supported functions
allows us to extend it to all positive functions. In this subsection we show
that this extension yields minimal solutions to the equation
$\varphi^{-1}({\mathcal{L}}g)+Wg=f$.

######  Definition 3.11 (Extended Dirichlet resolvent).

For $0\leq f\in C(X)$ we define the extended Dirichlet resolvent by

$Rf\colon X\to[0,\infty],\quad Rf(x)=\sup\\{Rg(x)\mid g\in C_{c}(X)\text{ with
}0\leq g\leq f\\}.$

As a first consequence of this definition we note that the domination we
observed at the end of the last section extends to the extended resolvent.

######  Proposition 3.12 (Domination).

Let $\psi\colon\mathbb{R}\to\mathbb{R}$ be strictly increasing, continuous
with $\psi(0)=0$ and $\psi(t)\leq\varphi(t)$ for $t\geq 0$. Let $V\colon
X\to(0,\infty)$ with $V\geq W$. For all $0\leq f\in C(X)$ we have

$0\leq R^{\psi,V}f\leq R^{\varphi,W}f.$

###### Proof.

This follows directly from Proposition 3.10 and the definition of the extended
resolvent. ∎

######  Lemma 3.13 (Approximation via finite subsets - II).

Let $0\leq f\in C(X)$ and let $(K_{n})$ be finite subsets in $X$ with
$K_{n}\nearrow X$. Then $R_{K_{n}}(f1_{K_{n}})\nearrow Rf$ pointwise.

###### Proof.

The approximation via finite subsets of the resolvent on finitely supported
functions, Lemma 3.8, and the definition of the extended resolvent yield

$R_{K_{n}}(f1_{K_{n}})\leq R(f1_{K_{n}})\leq Rf.$

Moreover, domain monotonicity implies that $R_{K_{n}}(f1_{K_{n}})$ is
increasing in $n$.

Let $x\in X$. Case 1: $Rf(x)<\infty$. Let $\varepsilon>0$ and choose $w\in
C_{c}(X)$ with $0\leq w\leq f$ and $Rf(x)\leq Rw(x)+\varepsilon$. Using Lemma
3.8 and $w\leq f1_{K_{n}}$ for large enough $n$, we infer

$Rf(x)\leq
Rw(x)+\varepsilon=\lim_{n\to\infty}R_{K_{n}}w(x)+\varepsilon\leq\lim_{n\to\infty}R_{K_{n}}(f1_{K_{n}})(x)+\varepsilon.$

Case 2: $Rf(x)=\infty$: Let $C\geq 0$ and choose $0\leq w\leq f$ with
$Rw(x)\geq C$. For $n$ large enough we have $f1_{K_{n}}\geq w$ and infer

$C\leq
Rw(x)=\lim_{n\to\infty}R_{K_{n}}w(x)\leq\lim_{n\to\infty}R_{K_{n}}(f1_{K_{n}})(x),$

where we use Lemma 3.8 for the equality. ∎

######  Theorem 3.14 (Minimality of the extended Dirichlet resolvent).

Let $0\leq f\in C(X)$.

  1. (a)

$Rf(x)<\infty$ for all $x\in X$ implies $Rf\in{\mathcal{F}}_{\varphi}$ and

$\varphi^{-1}({\mathcal{L}}Rf)+WRf=f.$

  2. (b)

$Rf(x)<\infty$ for all $x\in X$ if and only if there exists $0\leq
g\in{\mathcal{F}}_{\varphi}$ with

$\varphi^{-1}({\mathcal{L}}g)+Wg\geq f.$

In this case, $g\geq Rf$.

###### Proof.

(a): Choose a sequence of finite sets $(K_{n})$ with $K_{n}\nearrow X$ and let
$g_{n}:=R_{K_{n}}(f1_{K_{n}})$. The definition of $R_{K_{n}}$ yields

${\mathcal{L}}g_{n}=\varphi(f-Wg_{n})\text{ on }K_{n}.$

Spelled out, for each $x\in K_{n}$ this reads

$\frac{\deg(x)}{m(x)}g_{n}(x)-\frac{1}{m(x)}\sum_{y\in
X}b(x,y)g_{n}(y)=\varphi(f(x)-W(x)g_{n}(x)).$

Lemma 3.13 implies $g_{n}\nearrow Rf$. The monotone convergence theorem yields
that the left side of the previous equation converges and the right side of
the equation converges because $\varphi$ is continuous. Hence, we obtain

$\frac{\deg(x)}{m(x)}Rf(x)-\frac{1}{m(x)}\sum_{y\in
X}b(x,y)Rf(y)=\varphi(f(x)-W(x)Rf(x))$

for all $x\in X$. In particular, this shows $\sum_{y\in X}b(x,y)Rf(y)<\infty$
for all $x\in X$ (i.e. $Rf\in{\mathcal{F}}$) and
${\mathcal{L}}Rf=\varphi(f-WRf)$.

(b): If $Rf(x)<\infty$ for all $x\in X$, then using (a) we can choose $g=Rf$.

Now suppose $0\leq g\in{\mathcal{F}}_{\varphi}$ with
$\varphi^{-1}({\mathcal{L}}g)+Wg\geq f.$ Let $g_{n}=R_{K_{n}}(f1_{K_{n}})$ be
as in the proof of (a). Then

$\varphi^{-1}({\mathcal{L}}g)+Wg\geq
f=\varphi^{-1}({\mathcal{L}}g_{n})+Wg_{n}\text{ on }K_{n}.$

Moreover, $g\geq 0=g_{n}$ on $X\setminus K_{n}$ and $g-g_{n}$ attains a
minimum on the finite set $K_{n}$. Hence, the comparison principle Lemma 3.5
yields $g\geq g_{n}$. The approximation of Lemma 3.13 yields $g\geq Rf$. ∎

######  Corollary 3.15.

For any $\alpha\geq 0$ we have $R(\alpha W)\leq\alpha$.

###### Proof.

The constant function $f=\alpha 1_{X}$ satisfies

$\varphi^{-1}({\mathcal{L}}f)+Wf=\alpha W.$

With this at hand the previous theorem shows $R(\alpha W)\leq f=\alpha 1_{X}$.
∎

######  Remark.

  1. (a)

Recall the situation of Subsection 2.2. In this case, if we let $\varphi={\rm
id}_{\mathbb{R}}$ and $W=\beta+\frac{c}{m}$, then for $0\leq f\in C(X)$ we
have $Rf=(L+\beta)^{-1}f$. Here, $(L+\beta)^{-1}f$ is the extended resolvent
discussed in Subsection 2.2. In particular, we obtain

$R(\beta+\frac{c}{m})=\beta(L+\beta)^{-1}1+(L+\beta)^{-1}\frac{c}{m}.$

This shows that $R(\alpha W)$ is the non-linear analogue of the quantity
appearing in the generalized conservation property. In particular, the
previous theorem and its corollary are non-linear versions of [KL12, Theorem
11 and Proposition 6.1].

  2. (b)

For general $\varphi$ the resolvent $R$ is not homogenous. It is therefore
quite remarkable that the inequality $R(\alpha W)\leq\alpha$ holds for all
$\alpha\geq 0$.

##  4\. Stochastic completeness at infinity and uniqueness of bounded
(super)harmonic functions

In this section we characterize triviality of bounded solutions (nonpositive
bounded supersolutions) to the equation

$\varphi^{-1}({\mathcal{L}}g)+Wg=0$

with the help of the non-linear resolvent introduced in the previous sections.
The most important observation is that resolvents yield minimal bounded below
(super)solutions to it, which we discuss in the following lemma.

######  Lemma 4.1 (Minimal (super)solution).

Let $\alpha\geq 0$. The function $w=R(\alpha W)-\alpha$ satisfies $-\alpha\leq
w\leq 0$, $w\in{\mathcal{F}}_{\varphi}$ and
$\varphi^{-1}({\mathcal{L}}w)+Ww=0$. If $h\in{\mathcal{F}}_{\varphi}$
satisfies $\varphi^{-1}({\mathcal{L}}h)+Wh\geq 0$ and $h\geq-\alpha$, then
$w\leq h$.

###### Proof.

By Corollary 3.15 $w$ is nonpositive and by Corollary 3.9 it is bounded from
below by $-\alpha$. Moreover, Theorem 3.14 shows $w\in{\mathcal{F}}_{\varphi}$
and

$\varphi^{-1}({\mathcal{L}}w)+Ww=\varphi^{-1}({\mathcal{L}}R(\alpha
W))+WR(\alpha W)-\alpha W=\alpha W-\alpha W=0.$

Let $g=\alpha+h\geq 0$. Then $g\in{\mathcal{F}}_{\varphi}$ with

$\varphi^{-1}({\mathcal{L}}g)+Wg=\varphi^{-1}({\mathcal{L}}h)+Wh+\alpha
W\geq\alpha W.$

With this at hand Theorem 3.14 yields $R(\alpha W)\leq g=\alpha+h$. ∎

With this at hand we obtain the following characterization of an
$\ell^{\infty}$-Liouville type theorem for the operator
$\varphi^{-1}\circ{\mathcal{L}}+W$.

######  Theorem 4.2 (Uniqueness of bounded supersolutions).

The following assertions are equivalent.

  1. (i)

$R(\alpha W)=\alpha$ for all $\alpha\geq 0$.

  2. (ii)

Any $0\geq g\in\ell^{\infty}(X)\cap{\mathcal{F}}_{\varphi}$ with
$\varphi^{-1}({\mathcal{L}}g)+Wg=0$ satisfies $g=0$.

  3. (iii)

Any $0\geq g\in\ell^{\infty}(X)\cap{\mathcal{F}}_{\varphi}$ with
$\varphi^{-1}({\mathcal{L}}g)+Wg\geq 0$ satisfies $g=0$.

  4. (iv)

Any $g\in\ell^{\infty}(X)\cap{\mathcal{F}}_{\varphi}$ with
$\varphi^{-1}({\mathcal{L}}g)+Wg\geq 0$ satisfies $g\geq 0$.

  5. (v)

Any $0\leq f\in\ell^{\infty}(X)$ with $-{\mathcal{L}}f=\varphi(Wf)$ satisfies
$f=0$.

###### Proof.

(i) $\Rightarrow$ (iv): Let $\alpha=\left\|g\right\|_{\infty}$. Then
$g\geq-\alpha$ and Lemma 4.1 yields $R(\alpha W)-\alpha\leq g$. Since by
assumption $R(\alpha W)=\alpha$, this leads to $g\geq 0$.

(iv) $\Rightarrow$ (iii) $\Rightarrow$ (ii): This is trivial.

(ii) $\Rightarrow$ (i): Let $\alpha\geq 0$ and consider the function
$w=R(\alpha W)-\alpha$. As seen in Lemma 4.1 it satisfies $0\geq
w\in\ell^{\infty}(X)\cap{\mathcal{F}}_{\varphi}$ and
$\varphi^{-1}({\mathcal{L}}w)+Ww=0$. Hence, (ii) implies $w=0$.

(ii) $\Leftrightarrow$ (v): It is readily verified that
$g\in{\mathcal{F}}_{\varphi}$ and $\varphi^{-1}({\mathcal{L}}g)+Wg=0$ if and
only if $g\in{\mathcal{F}}$ and ${\mathcal{L}}g=\varphi(-Wg)$. Since also
$\ell^{\infty}(X)\subseteq{\mathcal{F}}$, the equivalence of (ii) and (v)
follows immediately. ∎

######  Remark.

  1. (a)

It is quite remarkable that the conditions (ii), (iii) and (v) in the previous
theorem only depend on $\varphi|_{[0,\infty)}$ (and hence also the equivalent
conditions (i) and (iv)). The reason is that $R(\alpha W)$, which yields
minimal supersolutions, does only depend on $\varphi|_{[0,\infty)}$ (this can
be seen by carefully following the construction of the resolvent). Note
however, that for general $f\geq 0$ the resolvent $Rf$ may depend on
$\varphi|_{(-\infty,0)}$.

  2. (b)

For $W=1$ the equation $-{\mathcal{L}}f=\varphi(f)$ in (v) is a discrete
version of the semi-linear elliptic equation considered in [GIM20, GIMP23]. We
discuss its connection to stochastic completeness in the next section.

  3. (c)

Assume that $\varphi\colon\mathbb{R}\to\mathbb{R}$ is surjective (i.e.
$\lim_{t\to\pm\infty}\varphi(t)=\pm\infty$). Then
${\mathcal{F}}_{\varphi}={\mathcal{F}}$. Since also
$\ell^{\infty}(X)\subseteq{\mathcal{F}}$, in this case, the previous theorem
simply deals with bounded (super)solutions of
$\varphi^{-1}({\mathcal{L}}g)+Wg=0$.

  4. (d)

As already discussed in the remark following Theorem 3.14, if we let
$\varphi={\rm id}_{\mathbb{R}}$ and $W=\beta+\frac{c}{m}$, then

$R(\beta+\frac{c}{m})=\beta(L+\beta)^{-1}1+(L+\beta)^{-1}\frac{c}{m}.$

Hence, for fixed $\beta>0$ the identity $R(\beta+\frac{c}{m})=1$ is equivalent
to $\beta(L+\beta)^{-1}1+(L+\beta)^{-1}\frac{c}{m}=1$. By the resolvent
identity this in turn is equivalent to
$\beta^{\prime}(L+\beta^{\prime})^{-1}1+(L+\beta^{\prime})^{-1}\frac{c}{m}=1$
for all $\beta^{\prime}>0$, which is the generalized conservation property
discussed in Subsection 2.2.

These observations show that (iv) is a non-linear version of the generalized
conservation property and hence of stochastic completeness at infinity. For
general $\varphi$ and $W$ the resolvent is non-linear. Hence, it is not
sufficient to require $R(W)=1$ (as the previous computations in the linear
case suggest) but we need $R(\alpha W)=\alpha$ for all $\alpha\geq 0$.

Based on these observations we adopt the definition of stochastic completeness
at infinity to the non-linear setting.

######  Definition 4.3 (Stochastic completeness at infinity - non-linear
version).

We say that a graph $b$ over $(X,m)$ with $W$ and $\varphi$ as above is
stochastically complete at infinity if it satisfies the non-linear generalized
conservation property

$R^{\varphi,W}(\alpha W)=\alpha,\text{ all }\alpha>0.$

For later purposes we note a stability property of stochastic completeness at
infinity.

######  Proposition 4.4 (Stability of stochastic completeness at infinity).

Let $\psi\colon\mathbb{R}\to\mathbb{R}$ be strictly increasing, continuous
with $\psi(0)=0$ and $\psi(t)\leq\varphi(t)$ for $t\geq 0$. Let $V\colon
X\to(0,\infty)$ with $\inf_{x\in X}V(x)>0$ and $W\geq V$. If $b$ over $(X,m)$
with $V$ and $\psi$ is stochastically complete at infinity, then it is
stochastically complete at infinity with $W$ and $\varphi$.

###### Proof.

We show the statement for the case $V=W$ and for the case $\varphi=\psi$. The
general statement follows after applying one case first and then the other.

Case 1: $W=V$. For $\alpha>0$ stochastic completeness at infinity with respect
to $\psi,W$ and domination, see Proposition 3.12, yields

$\alpha=R^{\psi,W}(\alpha W)\leq R^{\varphi,W}(\alpha W)\leq\alpha.$

This implies $R^{\varphi,W}(\alpha W)=\alpha$, which is one of our
characterizations for stochastic completeness at infinity for $\varphi$,$W$.

Case 2: $\varphi=\psi$. Let $0\geq
g\in\ell^{\infty}(X)\cap{\mathcal{F}}_{\varphi}$ with
$\varphi^{-1}({\mathcal{L}}g)+Wg\geq 0$. Since $V\leq W$ and $g\leq 0$, we
obtain

$\varphi^{-1}({\mathcal{L}}g)+Vg\geq\varphi^{-1}({\mathcal{L}}g)+Wg\geq 0.$

Stochastic completeness at infinity with respect to $\varphi,V$ yields $g=0$,
and we obtain the statement. ∎

We finish this section by giving one geometric criterion for this property.

######  Proposition 4.5.

Assume that for all $0<\alpha\leq 1$ and all infinite paths $(x_{n})$ in $X$
we have

$\sum_{n=1}^{\infty}\frac{m(x_{n})\varphi(\alpha
W(x_{n}))}{\deg(x_{n})}=\infty.$

Then $b$ over $(X,m)$ with $W$ and $\varphi$ is stochastically complete at
infinity. In particular, this holds if $\deg/m$ is bounded.

###### Proof.

Let $0\geq g\in\ell^{\infty}(X)\cap{\mathcal{F}}_{\varphi}$ with
$\varphi^{-1}({\mathcal{L}}g)+Wg\geq 0$ be given and assume $g\neq 0$.
Moreover, we let $(\rho_{n})$ be a sequence with $\rho_{n}>1$,
$n\in\mathbb{N}$, with

$\beta:=\prod_{n=1}^{\infty}\frac{1}{\rho_{n}}>0.$

By our assumption we find $x_{0}\in X$ with $g(x_{0})<0$. We let $f=-g$ and
use the boundedness of $f$ to inductively choose a sequence $(x_{n})$ in $X$
with $x_{n+1}\sim x_{n}$ and

$\rho_{n+1}f(x_{n+1})\geq\sup\\{f(y)\mid y\sim x_{n}\\}.$

The inequality $\varphi^{-1}({\mathcal{L}}g)+Wg\geq 0$ leads to
${\mathcal{L}}f+\varphi(Wf)\leq 0$. Using the definition of ${\mathcal{L}}$,
for each $n\geq 0$ we infer

$\displaystyle\deg(x_{n})f(x_{n})+m(x_{n})\varphi(W(x_{n})f(x_{n}))$
$\displaystyle\leq\sum_{y\in X}b(x_{n},y)f(y)$
$\displaystyle\leq\deg(x_{n})\rho_{n+1}f(x_{n+1}).$

This leads to

$f(x_{n+1})\geq\frac{1}{\rho_{n+1}}\left(f(x_{n})+\frac{m(x_{n})\varphi(W(x_{n})f(x_{n}))}{\deg(x_{n})}\right),$

which immediately yields $f(x_{n})\geq\beta f(x_{0})>0$ for all
$n\in\mathbb{N}$.

Iterating the inequality and using $f(x_{n})\geq\beta f(x_{0})$, we infer

$f(x_{n+1})\geq\beta\sum_{k=0}^{n}\frac{m(x_{k})\varphi(W(x_{k})\beta
f(x_{0}))}{\deg(x_{k})}.$

Our assumptions imply that the sum on the right side diverges, as $n\to\infty$
(use that $\varphi(W(x_{k})\beta f(x_{0}))\geq\varphi(W(x_{k}))$ if $\beta
f(x_{0})>1$). This is a contradiction to $f$ being bounded.

The ’In particular’-part: If for some $C>0$ we have $\deg/m\leq C$, then

$\frac{m(x)\varphi(\alpha W(x))}{\deg(x)}\geq\frac{\varphi(\alpha
W_{0})}{C}>0,$

where $W_{0}=\inf_{x\in X}W(x)>0$. Hence, the summability criterion
immediately yields the statement. ∎

######  Remark.

In the linear case $\varphi={\rm id}_{\mathbb{R}}$ a similar criterion is used
to show that given a weighted graph $b$ over $(X,m)$ one can always find $W$
(respectively $c$ in the notation of Subsection 2.2) making it stochastically
complete at infinity, see [KL12, Theorem 2]. For example, $W=\deg/m+1$ does
the job. In the semi-linear case of general $\varphi$ however, this is no
longer possible. For example, if $\varphi$ is bounded, the sum in the
proposition may just always be finite. In contrast, under some mild growth
conditions on $\varphi$, we can establish the existence of $W$ making the
graph with $\varphi$ stochastically complete at infinity. This is discussed in
the next corollary.

######  Corollary 4.6.

Assume that a weighted graph $b$ over $(X,m)$ is given. Assume further
$\lim_{t\to\infty}\varphi(t)=\infty$ and that for each $0<\alpha\leq 1$ there
exists $C_{\alpha}>0$ such that $\varphi(\alpha t)\geq C_{\alpha}\varphi(t)$
for all $t\geq 0$. Then there exists $W\colon X\to(0,\infty)$ with $\inf_{x\in
X}W(x)>0$ such that $b$ over $(X,m)$ with $\varphi$ and $W$ is stochastically
complete at infinity.

###### Proof.

Using that $\varphi$ tends to $\infty$, as $t\to\infty$, we may choose $W$
such that $m(\varphi\circ W)\geq\deg$ and $W\geq 1$. For example,
$W=\varphi^{-1}\circ(\deg/m)+1$ does the job. Using our assumptions on
$\varphi$ we infer

$\frac{m(x)\varphi(\alpha W(x))}{\deg(x)}\geq
C_{\alpha}\frac{m(x)\varphi(W(x))}{\deg(x)}\geq C_{\alpha}.$

Hence, the previous proposition yields the claim. ∎

######  Remark.

The assumption of the corollary is satisfied if $\varphi|_{[0,\infty)}$ is
concave. For example, it can be applied if $\varphi(t)=\log(1+t)$ for $t\geq
0$. It is also satisfied for power functions $\varphi(t)=t^{p}$, $t\geq 0$,
where $0<p<\infty$.

##  5\. Stochastic completeness and semi-linear Liouville properties

In this section we apply the theory developed in the previous section to
characterize stochastic completeness by semi-linear Liouville properties. We
start with some notation. We let $\mathcal{I}$ be the set of all strictly
increasing continuous functions $\varphi\colon\mathbb{R}\to\mathbb{R}$ with
$\varphi(0)=0$. Moreover, we let $\mathcal{A}=\\{W\colon
X\to[0,\infty)\mid\inf_{x\in X}W(x)>0\\}$ and recall that
${\mathcal{F}}_{\varphi}=\\{f\in{\mathcal{F}}\mid{\mathcal{L}}f(x)\in{\rm
ran}\,\varphi\text{ for all }x\in X\\}$.

######  Theorem 5.1 (Characterization of stochastic completeness).

The following assertions are equivalent.

  1. (i)

The graph $b$ over $(X,m)$ is stochastically complete.

  2. (ii)

For one $W\in\mathcal{A}\cap\ell^{\infty}(X)$ (any $W\in\mathcal{A}$) and one
$\varphi\in\mathcal{I}$ (any $\varphi\in\mathcal{I}$) we have

$R^{\varphi,W}(\beta W)=\beta,\quad\text{ all }\beta>0.$

  3. (iii)

For one $W\in\mathcal{A}\cap\ell^{\infty}(X)$ (any $W\in\mathcal{A}$) and one
$\varphi\in\mathcal{\mathcal{missing}}I$ (any $\varphi\in\mathcal{I}$) every
$0\geq g\in\ell^{\infty}(X)\cap{\mathcal{F}}_{\varphi}$ with
$\varphi^{-1}({\mathcal{L}}g)+Wg=0$ satisfies $g=0$.

  4. (iv)

For one $W\in\mathcal{A}\cap\ell^{\infty}(X)$ (any $W\in\mathcal{A}$) and one
$\varphi\in\mathcal{I}$ (any $\varphi\in\mathcal{I}$) every
$g\in\ell^{\infty}(X)\cap{\mathcal{F}}_{\varphi}$ with
$\varphi^{-1}({\mathcal{L}}g)+Wg\geq 0$ satisfies $g\geq 0$.

  5. (v)

For one $W\in\mathcal{A}\cap\ell^{\infty}(X)$ (any $W\in\mathcal{A}$) and one
$\varphi\in\mathcal{I}$ (any $\varphi\in\mathcal{I}$) every $0\geq
g\in\ell^{\infty}(X)\cap{\mathcal{F}}_{\varphi}$ with
$\varphi^{-1}({\mathcal{L}}g)+Wg\geq 0$ satisfies $g=0$.

  6. (vi)

For one $W\in\mathcal{A}\cap\ell^{\infty}(X)$ (any $W\in\mathcal{A}$) and one
$\varphi\in\mathcal{I}$ (any $\varphi\in\mathcal{I}$) every $0\leq
f\in\ell^{\infty}(X)$ with $-{\mathcal{L}}f=\varphi(Wf)$ satisfies $f=0$.

###### Proof.

In this proof we use the following notation: If there are ’for
one/any’-statements in the assertions, we attach ’one’ or ’any’ to the number
of the assertion to specify them in the order of appearance. For example ’(v)
one any’ stands for the assertion ’For one
$W\in\mathcal{A}\cap\ell^{\infty}(X)$ and any $\varphi\in\mathcal{I}$ every
$0\geq g\in\ell^{\infty}(X)\cap{\mathcal{F}}_{\varphi}$ with
$\varphi^{-1}({\mathcal{L}}g)+Wg\geq 0$ satisfies $g=0$’.

Obviously the ’for any’-statements imply the ’for one’-statements and so we
only have to prove the opposite implications.

For fixed $A,B\in\\{\text{one,any}\\}$, Theorem 4.2 shows the equivalence of
(ii)$AB$, (iii) $AB$, (iv)$AB$, (v)$AB$ and (vi)$AB$. Moreover, it shows that
’(iv) any any implies (iii) one one’ and ’(iii) one one implies (iv) one one’.

Hence, in order to establish the claimed equivalences it suffices to show ’(i)
implies (v) any any’ and ’(v) one one implies (i)’.

(i) implies (v) any any: Let $W\in\mathcal{A}$, $\varphi\in\mathcal{I}$ and
let $0\geq g\in\ell^{\infty}(X)\cap{\mathcal{F}}_{\varphi}$ with
$\varphi^{-1}({\mathcal{L}}g)+Wg\geq 0$. Below we apply the Omori-Yau maximum
principle of Proposition 2.7 to $f=-g$ in order to conclude $g=0$.

Let $\alpha=\inf_{x\in X}W(x)>0$. Since $f$ is nonnegative, we have $\alpha
f\leq Wf$, which yields (after rearranging the inequality for $g$)

${\mathcal{L}}f+\varphi(\alpha f)\leq{\mathcal{L}}f+\varphi(Wf)\leq 0.$

Assume that $f^{*}=\sup_{x\in X}f(x)>0$. We let $\varepsilon>0$ and choose
$\delta>0$ such that $|t-\alpha f^{*}|<\delta$ implies
$|\varphi(t)-\varphi(\alpha f^{*})|<\varepsilon$. The Omori-Yau maximum
principle implies the existence of $\xi\in X$ with
$f(\xi)>f^{*}-\delta/\alpha$ and $\mathcal{{\mathcal{L}}}f(\xi)>-\varepsilon$.
Then $|\alpha f(\xi)-\alpha f^{*}|<\delta$ and our choices of the parameters
yield

$\varphi(\alpha f^{*})\leq\varphi(\alpha
f(\xi))+\varepsilon\leq-{\mathcal{L}}f(\xi)+\varepsilon<2\varepsilon.$

Since $\varepsilon>0$ was arbitrary, we infer $\varphi(\alpha f^{*})=0$. Using
that $\varphi$ is strictly increasing with $\varphi(0)=0$, we obtain
$f^{*}=0$, a contradiction to our assumption $f^{*}>0$. Hence, $0=f=-g$.

(v) one one implies (i): Let $W\in\mathcal{A}\cap\ell^{\infty}(X)$ and
$\varphi\in\mathcal{I}$ for which (v) is satisfied. The stability result
Proposition 4.4 yields that (v) is satisfied for any
$\varphi^{\prime}\in\mathcal{I}$ with $\varphi^{\prime}(t)\geq\varphi(t)$,
$t\geq 0$. Hence, we can assume without loss of generality, that
$\lim_{t\to\infty}\varphi(t)=\infty$.

Assume $b$ over $(X,m)$ is not stochastically complete. Our unboundedness
assumption on $\varphi$ implies that if $f\in{\mathcal{F}}$ satisfies
${\mathcal{L}}f\geq 0$, then $f\in{\mathcal{F}}_{\varphi}$. Below we construct
a nontrivial $0\leq h\in\ell^{\infty}(X)$ with ${\mathcal{L}}h+\varphi(Wh)\leq
0$. For $g=-h$ this implies ${\mathcal{L}}g\geq\varphi(-Wg)\geq 0$, which
shows $g\in{\mathcal{F}}_{\varphi}$ and $\varphi^{-1}({\mathcal{L}}g)+Wg\geq
0$, a contradiction to (v).

We let $\beta=\sup_{x\in X}W(x)>0$ and use the continuity of $\varphi$ to
choose $\alpha>0$ such that $\varphi(t)<\alpha/2$ for all $0\leq t\leq\beta$.

According to Proposition 2.7 (applied with $c=0$) there exists $0\leq
f\in\ell^{\infty}(X)$ with $\left\|f\right\|_{\infty}=1$ and
${\mathcal{L}}f\leq-\alpha f$. We consider the function $h=(f-1/2)_{+}$. Then
$h\neq 0$, $0\leq h\leq 1/2$ and a direct computation shows

${\mathcal{L}}h(x)\leq\begin{cases}{\mathcal{L}}f(x)&\text{if }f(x)>1/2\\\
0&\text{if }f(x)\leq 1/2\end{cases}.$

Moreover, if $f(x)\leq 1/2$, we have $\varphi(W(x)h(x))=0$ and if $f(x)>1/2$,
we obtain

$\varphi(W(x)h(x))\leq\varphi(\beta)<\alpha/2\leq\alpha
f(x)\leq-{\mathcal{L}}f(x).$

Put together, this implies ${\mathcal{L}}h+\varphi(Wh)\leq 0$, a contradiction
to (v). ∎

######  Remark.

  1. (a)

Our characterization shows in particular that if $b$ is stochastically
complete, then for all $W\in\mathcal{A}$ and all $\varphi\in\mathcal{I}$ the
graph $b$ with $W$ and $\varphi$ is stochastically complete at infinity. In
order to deduce stochastic completeness from stochastic completeness at
infinity for one particular $W\in\mathcal{A}$ and $\varphi\in I$, we have to
additionally assume that $W$ is bounded. Indeed, under mild assumptions on
$\varphi$ Corollary 4.6 shows that even if $b$ is not stochastically complete,
one can find a large $W$ making $b$ with $W$ and $\varphi$ stochastically
complete at infinity.

  2. (b)

In the case when $W=\alpha$ and $\varphi={\rm id}_{\mathbb{R}}$, assertion
(ii) is also known as the $L^{\infty}$-positivity preserving property, a
notion that was introduced in [Gü16]. That it is related to stochastic
completeness was observed in [BM23] for Riemannian manifolds. In the linear
situation on Riemannian manifolds the most delicate part is the question about
regularity of bounded functions satisfying $(-\Delta+\alpha)u\geq 0$ in the
sense of distributions, a problem which disappears in the discrete case.

The one/one statements yield particularly strong non-uniqueness results in the
stochastically incomplete case. We only mention one.

######  Corollary 5.2.

The following assertions are equivalent.

  1. (i)

The graph $b$ over $(X,m)$ is not stochastically complete.

  2. (ii)

For all $W\in\mathcal{A}\cap\ell^{\infty}(X)$ and all $\varphi\in\mathcal{I}$
there exists a nontrivial $0\leq f\in\ell^{\infty}(X)$ such that
$-{\mathcal{L}}f=\varphi(Wf)$.

In this case, for $\alpha>0$, $W\in\mathcal{A}\cap\ell^{\infty}(X)$ and
$\varphi\in\mathcal{I}$ the function $w_{\alpha}=\alpha-R(\alpha W)$ is the
maximal function $0\leq w_{\alpha}\leq\alpha$ satisfying
$-{\mathcal{L}}w_{\alpha}=\varphi(Ww_{\alpha})$.

######  Remark.

The previous corollary is the precise analogue of the elliptic
characterization of stochastic completeness in [GIMP23, Theorem 1.1] (if
$W=1$). Note however, that our considerations go beyond, as we also have a
characterization via supersolutions and a theory of maximal solutions.

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