# Positive harmonically bounded solutions
for semi-linear equations00footnotetext: MSC2020: Primary 31C05, 35J61, 35K58;
Secondary 60J45, 60J35, 45K05.
Keywords: semi-linear equation, balayage space, Green function, Hunt process.
The second named author was partially supported by NCN grant
2017/27/B/ST1/01339.

Krzysztof Bogdan and Wolfhard Hansen

###### Abstract

For open sets $U$ in some space $X$, we are interested in positive solutions
to semi-linear equations $Lu=\varphi(\cdot,u)\mu$ on $U$. Here $L$ may be an
elliptic or parabolic operator of second order (generator of a diffusion
process) or an integro-differential operator (generator of a jump process),
$\mu$ is a positive measure on $U$ and $\varphi$ is an arbitrary measurable
real function on $U\times\mathbbm{R}^{+}$ such that the functions
$t\mapsto\varphi(x,t)$, $x\in U$, are continuous, increasing and vanish at
$t=0$.

More precisely, given a measurable function $h\geq 0$ on $X$ which is
$L$-harmonic on $U$, that is, continuous real on $U$ with $Lh=0$ on $U$, we
give necessary and sufficient conditions for the existence of positive
solutions $u$ such that $u=h$ on $X\setminus U$ and $u$ has the same “boundary
behavior” as $h$ on $U$ (Problem 1) or, alternatively, $u\leq h$ on $U$, but
$u\not\equiv 0$ on $U$ (Problem 2).

We show that these problems are equivalent to problems of the existence of
positive solutions to certain integral equations $u+K\varphi(\cdot,u)=g$ on
$U$, $K$ being a potential kernel. We solve them in the general setting of
balayage spaces $(X,\mathcal{W})$ which, in probabilistic terms, corresponds
to the setting of transient Hunt processes with strong Feller resolvent.

##  1 Introduction

In this paper we are looking for positive solutions to semi-linear equations

(1.1) $Lu=\varphi(\cdot,u)\mu\quad\text{ on }U$

for linear operators $L$ which may be elliptic or parabolic differential
operators of second order (generators of diffusion processes) as well as
integro-differential operators (generators of jump processes). Here $U$ is a
(non-empty) open set in some space $X$, $\mu$ is a measure on $U$ and
$\varphi$ is a real function on $U\times\mathbbm{R}^{+}$ such that the
functions $t\mapsto\varphi(x,t)$, $x\in X$, are continuous, _increasing_ and
vanish at $0$.111In [10] we discuss real solutions $u$ for $\varphi\colon
U\times\mathbbm{R}\to\mathbbm{R}$ such that the functions
$t\mapsto\varphi(x,t)$ need _not_ be increasing.

Given a function $h\geq 0$ on $X$ which is ($L$-)harmonic on $U$, that is,
continuous real on $U$ with $Lh=0$ on $U$, may or may not vanish on
$X\setminus U$, but is _not_ identically $0$ on $U$, we are interested in the
following.

  * •

Problem 1: Is there a (unique) positive solution $u$ such that $h-u$ is a
potential on $U$ (with respect to $L$) and $u=h$ on $X\setminus U$?

  * •

Problem 2: Are there solutions $0\leq u\leq h$ with $u\not\equiv 0$ on $U$,
$u=h$ on $X\setminus U$?

In both cases, the solution $u$ is “harmonically bounded”, $u\leq h$. In
Example 2.4(1), we have $U=X$, $h$ is a minimal harmonic function, Problem 1
may fail to have a solution, but Problem 2 has solutions bounded by a small
multiple of $h$.

We shall illustrate our results with the following _standard examples_ :

  * (i)

Classical case: $L=\Delta:=\sum_{j=1}^{d}\partial^{2}/\partial_{x_{j}}^{2}$,
$X=\mathbbm{R}^{d}$ with $d\geq 3$,
or $X={\mathbbm{R}^{d}}\setminus A$ with $d\leq 2$ and closed non-polar set
$A$.

  * (ii)

Riesz potentials: $L=-(-\Delta)^{\alpha/2}$, $X={\mathbbm{R}^{d}}$, $d\geq 1$,
$0<\alpha<2\wedge d$.

  * (iii)

Heat equation: $L=\Delta-\partial/\partial_{x_{n+1}}$, $X=\mathbbm{R}^{d+1}$,
$d\geq 1$.

Under some additional assumptions, the first property of the solution $u$ in
Problem 1 is equivalent to the convergence $h-u\to 0$ along regular sequences
in $U$ which tend to the boundary $\partial U$ (see Proposition 2.6). So it
can be considered as a boundary condition for $u$. The condition $u=h$ on
$X\setminus U$ can be ignored if $U=X$ or $L$ is a _local_ operator, as in the
standard examples (i) and (iii). Let us also note that Problem 2 is a
relaxation of Problem 1, certainly when $1_{U}h$ is not a potential on $U$.
This is also true if Proposition 2.1 below holds, as in the standard examples.

Our results for the standard examples are given below in Theorems 1.1, 2.3 and
2.5. Note that in the setting of the standard examples we have a Green
function $G_{U}$ of $U$ and an associated Hunt process on $U$ (the Brownian
motion, the isotropic $\alpha$-stable Lévy process or the space-time Brownian
motion, all killed when exiting $U$). These can be used to present the
following answer to Problem 1 in the special case when $h$ is a constant (see
Theorems 2.3 and 2.5 for the general case). Below, we let
$\tau_{V}:=\inf\\{t\geq 0\colon X_{t}\notin V\\}$ be the first exit time of
open set $V\subset X$ and we denote by ${\mathcal{U}}(U)$ the set of all open
$V$ with compact closure in $U$.

######  THEOREM 1.1.

Let $c\in(0,\infty)$. Assuming that $\mu\geq 0$ is a Radon measure on $U$ such
that, for every compact $A$ in $U$, the potential
$G_{U}(1_{A}\mu):=\int_{A}G_{U}(\cdot,y)\,d\mu(y)$ is continuous real and
$G_{U}(1_{A}\varphi(\cdot,c)\mu)<\infty$, the following statements are
equivalent:

  * (1)

There exists a (unique) Borel function $0\leq u\leq c$ on $X$ such that $u=c$
on $U^{c}$,

$Lu=\varphi(\cdot,u)\mu\quad\mbox{ on }U,$

and $\mathbb{E}^{x}u(X_{\tau_{V}})\to c$, $x\in U$, as $V\in{\mathcal{U}}(U)$
increases to $U$.

  * (2)

For every $\eta\in(0,1)$, there is a Borel set $B$ in $U$ with
$G_{U}(1_{B^{c}}\varphi(\cdot,\eta c)\mu)<\infty$ and such that, starting at
$x\in U$, the probability of $(X_{t})$ hitting $B$ after exiting from
$V\in{\mathcal{U}}(U)$ tends to $0$ as $V\uparrow U$.

The function $u$ in $(1)$ is continuous if the functions
$G_{U}(1_{A}\varphi(\cdot,c)\mu)$, $A$ compact in $U$, are continuous.
Moreover, $(2)$ holds if, for every $\eta\in(0,1)$, there is a compact $B$ in
$U$ such that $G_{U}(1_{B^{c}}\varphi(\cdot,\eta c)\mu)<\infty$.

Let us stress that, without further restriction on $\varphi$, the condition
(2) is optimal (see Examples 2.4).

In the _general_ framework of balayage spaces we answer Problems 1 and 2 by
Theorems 6.14, 6.16 and 7.1 below. For Problem 1 they extend and simplify
results that were obtained in [19, Theorem 7.19] for harmonic spaces (in
particular, for differential operators $L$ on manifolds) in the linear case
$\varphi(x,t)=\rho(x)t$ and in [14] under rather restrictive assumptions on
$\varphi$. Let us note that Problem 2 has been discussed in [3] for the
fractional Laplacian $L$, Lebesgue measure on $U={\mathbbm{R}^{d}}$ and
$\varphi(x,t)=\rho(x)f(t)$, however the equivalence asserted in [3, Theorem 1]
only holds if $\alpha>1$, see Remark 2.7.

The literature of the semilinear equation is extremely rich, with much
emphasis on special cases. We mention the seminal contributions [31], [26],
[25] and [12], the monograph [29] on elliptic (local) operators, the paper
[13] with early probabilistic results, the contribution [19] on Riemannian
manifolds (more generally, harmonic spaces), which inspired a large part of
our development, and recent results on the fractional Laplacian [1, 9], [2],
[14], [15], [3]. For general functions $\varphi$ the corresponding Dirichlet
problem has been studied in Baalal-Hansen [2].

The content of the paper is as follows. In Section 2 we illustrate our
findings by applying them to the Laplacian, fractional Laplacian, and the heat
equation. In Section 3 we present the general setting of _balayage spaces_
$(X,\mathcal{W})$, $X$ being a locally compact space with countable base and
$\mathcal{W}$ being a convex cone of positive (hyperharmonic) functions on
$X$. Up to Doob’s conditioning, the generic example for $\mathcal{W}$ is the
set ${\mathcal{E}}_{\mathbbm{P}}$ of excessive functions for a sub-Markov
semigroup $\mathbbm{P}$ on $X$ such that $\mathcal{E}_{\mathbbm{P}}$ is
linearly separating, every function in $\mathcal{E}_{\mathbbm{P}}$ is the
supremum of its continuous real minorants in $\mathcal{E}_{\mathbbm{P}}$ (for
example, by $\mathbbm{P}$ having a strong Feller resolvent), and there exist
continuous two real functions $u,v\in\mathcal{E}_{\mathbbm{P}}$ such that the
ratio $u/v$ vanishes at infinity. We also discuss the existence of
corresponding Hunt processes and a characterization of balayage spaces by
families of harmonic kernels. In Section 4, inspired by considerations in [2],
we present the domination principle and develop its consequences, in
particular uniqueness and comparison of solutions for (1.1).

Let us digress that in the setting of Section 2, solving (1.1) with given
boundary/exterior conditions is equivalent to solving the integral equation

(1.2) $u+\int G_{U}(\cdot,y)\varphi(y,u(y))\,d\mu(y)=h,$

where $G_{U}$ is the Green function for $L$ on $U$, $h\geq 0$ is a function on
$X$, continuous on $U$, with $Lh=0$ on $U$; see also Section 8, in particular
Proposition 8.6, for general Green functions. Hence in the general setting of
a balayage space $(X,\mathcal{W})$ we focus on equations

(1.3) $u+K\varphi(\cdot,u)=h,$

where $K$ is a _potential kernel_ on $U$.

Dealing with positive functions $u$ and increasing functions
$t\mapsto\varphi(\cdot,t)$, already a very simple application of Schauder’s
fixed point theorem, given in Section 5, is sufficient to deal with (1.3).
This is worked out in Section 6, leading to complete answers for Problems 1
and 2. The special case, where $1_{U}h$ is not harmonic on $U$ (relevant only
if $L$ is non-local, as in the second standard example) is investigated in
depth in Section 7. As mentioned, in Section 8 we discuss general balayage
spaces admitting a Green function.

In the Appendix 9 we first provide proofs of auxiliary results used in the
paper and not easily found in the literature. In particular, we detail
properties of products of sub-Markov semigroups and the existence of
corresponding Hunt processes. Moreover, we elaborate on the construction of
the Green function and space-time Green function for a given transition
density.

In a second paper [10] we address the more general problem of finding real-
valued solutions to equations $Lu=\varphi(\cdot,u)\mu+\nu$, where $\nu$ is a
signed measure on $U$ and $\varphi$ is a real function on $U\times\mathbbm{R}$
such that the functions $t\mapsto\varphi(x,t)$, $x\in X$, vanish for $t=0$,
are continuous, but need not satisfy $\varphi(\cdot,s)\leq\varphi(\cdot,t)$
for $s\leq t$.

Throughout the paper (and in [10]) we shall use the following notation. Given
a locally compact space $X$ with countable base of open sets, let
$\mathcal{B}(X)$ denote the set of all Borel measurable numerical functions on
$X$. Here and below numerical means having values in $[-\infty,\infty]$, real
means having values in $(-\infty,\infty)$, positive means $\geq 0$, strictly
positive means $>0$ (similarly for increasing and strictly increasing). Given
any set $\mathcal{F}$ of numerical functions, we add the superscript $+$ or
the subscript $r,b$, respectively, to denote the set of all the functions in
$\mathcal{F}$ which are positive or real, bounded, respectively. For instance,
$\mathcal{B}_{r}(X)$ are Borel real-valued functions on $X$ and
$\mathcal{B}^{+}(X)$ are Borel positive functions on $X$ (that is with values
in $[0,\infty]$). Let $\mathcal{C}(X)$ be the set of all continuous functions
in $\mathcal{B}_{r}(X)$ and let $\mathcal{C}_{0}(X)$ be the set of all
functions in $\mathcal{C}(X)$ which vanish at infinity.

In the following let $U$ always be a non-empty open set in $X$ (where $U=X$ is
an interesting particular case) and let

$\mathcal{V}(U):=\\{V\colon\mbox{$V$ open and relatively compact in $U$}\\}.$

As usual, $u|_{U}$ is the restriction of the function $u$ to the set $U$.
Occasionally, we shall identify, without explicit mention, functions on $U$
with functions on $X$ which vanish on ${U^{c}}:=X\setminus U$. Further
notation is explained as we proceed.

##  2 Details for the standard examples

In this section we consider our standard examples (i), (ii), (iii) and an
arbitrary nonempty open subset $U$ of $X$. In each of the examples, let
${\mathcal{H}}^{+}(U)$ denote the set of all positive functions $h$ which are
harmonic on $U$, that is, $h\in\mathcal{B}^{+}(X)$, $h|_{U}\in\mathcal{C}(U)$,
and $Lh=0$ in the distributional sense or – equivalently – $H_{V}h=h$ for
every $V\in\mathcal{V}(U)$.222It suffices to know that every $x\in U$ has a
fundamental system of neighborhoods $V\in\mathcal{V}(U)$ with
$H_{V}h(x)=h(x)$, see [4, III.4.4]. Here $H_{V}$, for any open $V$ in $X$,
denotes the corresponding harmonic kernel for $V$. So, for every $x\in V$,
$H_{V}(x,\cdot)$ is the ($L$–)harmonic measure for $x$ with respect to $V$.
The measure is supported by $V^{c}$ (by the boundary $\partial V$ of $V$ for
(i) and (iii)). Furthermore, $H_{V}(x,\cdot)=\delta_{x}$, if $x\in X\setminus
U$.

Let us digress that, in probabilistic terms, if $(X_{t},t\geq 0)$ is the
corresponding stochastic process, that is, the Brownian motion for (i), the
isotropic $\alpha$-stable Lévy process for (ii) ([7, Section 1.1.2-3]) and the
space-time Brownian motion for (iii), then

$H_{V}f(x)=\mathbbm{E}^{x}(f\circ X_{\tau_{V}}),\qquad
f\in\mathcal{B}^{+}({\mathbbm{R}^{d}}),\,x\in{\mathbbm{R}^{d}},$

recalling that $\tau_{V}=\inf\\{t\geq 0:X_{t}\not\in V\\}$ is the time of the
first exit of $(X_{t})$ from $V$ (see, e.g., [5], [4] for more details).

Let ${}^{\ast}{\mathcal{H}}^{+}(U)$ be the set of all functions
$w\in\mathcal{B}^{+}(X)$ which are _( $L$–)hyperharmonic on $U$_, that is, are
lower semicontinuous on $U$ and satisfy $H_{V}w\leq w$ for all
$V\in\mathcal{V}(U)$. A function $s\in{}^{\ast}{\mathcal{H}}^{+}(U)$ is
_superharmonic on $U$_ if, for every $V\in\mathcal{V}(U)$, we have
$(H_{V}s)|_{V}\in\mathcal{C}(V)$ and hence $H_{V}s\in{\mathcal{H}}^{+}(V)$. It
is a _potential on $U$_, if, in addition,

(2.1) $\lim_{V\in\mathcal{V}(U),\,V\uparrow U}H_{V}s=0.$

Here, the left-hand side is the largest harmonic minorant of $s$ on $U$ and
(2.1) in particular requires that $s$ be zero on $X\setminus U$. Let
$\mathcal{P}^{\prime}(U)$ denote the set of all potentials on $U$.

We recall the _global_ Green functions $G$ (with constants $c>0$) such that

(2.2) $LG(\cdot,y)=-\delta_{y},\qquad y\in X,$

namely the Newtonian kernel for (i), if $d\geq 3$, and the Riesz kernel for
(ii):

$G(x,y)=c|x-y|^{\alpha-d},\quad x,y\in{\mathbbm{R}^{d}},$

and, for (iii) and points $x=(x^{\prime},r)$ and $y=(y^{\prime},s)$ in
${\mathbbm{R}^{d}}\times\mathbbm{R}$,

$G(x,y)=\begin{cases}\dfrac{c}{(r-s)^{d/2}}\,\exp\left(-\dfrac{|x-y|^{2}}{4(r-s)}\right)\quad\mbox{
if }r>s,\\\ 0\quad\mbox{ if }r\leq s.\end{cases}$

One can verify (2.2) by Fourier analysis, see also [8], [18] and [28]. The
functions $G(\cdot,y)$, $y\in X$, are potentials on $X$ and are harmonic on
$X\setminus\\{y\\}$. The global Green function $G$ for $X$ yields the Green
function $G_{U}$ for $U$ by

(2.3) $G_{U}(\cdot,y):=G(\cdot,y)-H_{U}G(\cdot,y),\qquad y\in U.$

We might note that, in the classical case, we can also get a Green function
for bounded $U$ by (2.3) with $G(x,y):=-c|x-y|$ if $d=1$ and
$G(x,y):=-c\ln|x-y|$ for $d=2$; see also [11] for the case of Riesz potentials
with $\alpha\geq d=1$.

Let $\mathcal{M}^{+}(U)$ denote the set of all (positive) Radon measures on
$(U,\mathcal{B}(U))$. For $\nu\in\mathcal{M}^{+}(U)$ we define the Green
potential $G_{U}\nu:=\int G_{U}(\cdot,y)\,d\nu(y)$ on $U$. Defining

$\mathcal{M}^{\prime}(U):=\\{\nu\in\mathcal{M}^{+}(U)\colon
G_{U}\nu<\infty\\},$

we have (see [24, Theorem 4.1]),

(2.4)
$\mathcal{P}_{r}^{\prime}(U)=\\{G_{U}\nu\colon\nu\in\mathcal{M}^{\prime}(U)\\}.$

By (2.2) and (2.3), $LG_{U}(\cdot,y)=-\delta_{y}$ on $U$ for every $y\in U$.
Accordingly,

(2.5) $LG_{U}\nu=-\nu\mbox{ on }U,\qquad\nu\in\mathcal{M}^{\prime}(U),$

since for $f\in C^{\infty}_{c}(U)$ we have by Fubini’s theorem,

$\int G_{U}\nu(x)Lf(x)\,dx=\int\int G_{U}(x,y)Lf(x)\,dx\,d\nu(y)=-\int
f(y)\,d\nu(y),$

because $Lf$ is bounded. See also [6, Lemma 5.3] for (ii) or [8, Proof of
Lemma 6].

Let us fix $\mu\in\mathcal{M}^{+}(U)$ which is _locally Kato_ , that is,
satisfies $G_{U}(1_{A}\mu)\in\mathcal{C}(U)$ for every compact $A$ in $U$. Let
$\varphi\colon U\times\mathbbm{R}^{+}\to\mathbbm{R}^{+}$ be Borel measurable
such that the functions $t\mapsto\varphi(x,t)$, $x\in U$, are continuous,
vanish at $0$ and increase.

In the following let $h$ always be a function in $\mathcal{B}^{+}(X)$ which is
harmonic on $U$ and not identically $0$ on $U$. Let us observe that the
“boundary conditions” in ($1$), ($1^{\prime}$), ($1^{\prime\prime}$) of the
next proposition imply, in particular, that $u=h$ on $X\setminus U$.

######  PROPOSITION 2.1.

For each $u\in\mathcal{B}^{+}(X)$ the following properties are equivalent:

  * (1)

$Lu=\varphi(\cdot,u)\mu$ on $U$ and $h-u\in\mathcal{P}^{\prime}(U)$.333See
Proposition 2.6 for a characterization under additional assumptions.

  * ($1^{\prime}$)

$Lu=\varphi(\cdot,u)\mu$ on $U$ and
$h-u\in\mathcal{P}^{\prime}_{r}(U)-\mathcal{P}^{\prime}_{r}(U)$.

  * ($1^{\prime\prime}$)

$Lu=\varphi(\cdot,u)\mu$ on $U$,
$u|_{U}+G_{U}(\varphi(\cdot,u)\mu)\in\mathcal{C}(U)$,
$\lim_{V\in\mathcal{V}(U),\,V\uparrow U}H_{V}|h-u|=0$.

  * (2)

$u+G_{U}(\varphi(\cdot,u)\mu)=h$.

The function $u\in\mathcal{B}^{+}(X)$ having these properties is uniquely
determined by $h$.

###### Proof.

(1) $\Rightarrow$ ($1^{\prime}$): Trivial.

($1^{\prime}$) $\Rightarrow$ (2): By (2.4), there exist
$\nu,\nu^{\prime}\in\mathcal{M}^{\prime}(U)$ such that $G_{U}\nu-
G_{U}\nu^{\prime}=h-u$. By (2.5),
$\nu-\nu^{\prime}=L(u-h)=Lu=\varphi(\cdot,u)\mu$ on $U$, and hence

$h=u+G_{U}\nu-G_{U}\nu^{\prime}=u+G_{U}(\varphi(\cdot,u)\mu).$

(2) $\Rightarrow$ (1) and ($1^{\prime\prime}$): Using $Lh=0$ on $U$,
$h|_{U}\in\mathcal{C}(U)$, (2.5), (2.4) and (2.1).

($1^{\prime\prime}$) $\Rightarrow$ (2): The function
$g:=h-(u+G_{U}(\varphi(\cdot,u)\mu)$ vanishes on $X\setminus U$, is real
continuous on $U$, and satisfies $Lg=0$. Hence $g$ is harmonic on $U$. In
particular, for every $V\in\mathcal{V}(U)$,

$g=H_{V}g=H_{V}(h-u)-H_{V}G_{U}(\varphi(\cdot,u)\mu),$

where the right side tends to $0$ as $V\uparrow U$. Thus $g=0$.

The uniqueness of $u$ in (2) follows by Proposition 4.3(i). ∎

As we shall see, $G_{U}(\varphi(\cdot,h)\mu)<\infty$ is sufficient for the
existence of a solution $u$. On the other hand, by Proposition 2.1(2),
$G_{U}(\varphi(\cdot,u)\mu)<\infty$ is necessary, although implicit. Inspired
by the literature of the subject, e.g., [3], we will show that
$G_{U}(\varphi(\cdot,\eta h)\mu)<\infty$, with all $0<\eta<1$, is sufficient.
In fact, in Theorem 2.3 we even prove that the existence of solutions is
_characterized_ by this being true for some measure $\mu^{\prime}\leq\mu$ such
that the difference $\mu-\mu^{\prime}$ is supported by a sufficiently small
set $A$.

To prepare Theorem 2.3, for subsets $A$ of $U$ we define the _reduced
functions_ ,

$\displaystyle{}^{U}\\!R_{h}^{A}$ $\displaystyle:=$
$\displaystyle\inf\\{w\in{}^{\ast}{\mathcal{H}}^{+}(U)\colon w\geq h\mbox{ on
}A\\},$ $\displaystyle{}^{U}\\!\hat{R}_{h}^{A}(x)$ $\displaystyle:=$
$\displaystyle\liminf\nolimits_{y\to x}{}^{U}\\!R_{h}^{A}(y)\,,\quad x\in X.$

Extending definitions in [19] we would say that $A$ is _$h$ -thin in $U$_ if
${}^{U}\\!\hat{R}_{h}^{A}\in\mathcal{P}^{\prime}(U)$, _$h$ -thick in $U$_ if
${}^{U}\\!R_{h}^{A}=h|_{U}$.

Taking into account that $h|_{U}\in{}^{\ast}\mathcal{H}^{+}(U)\cap C(U)$,
Proposition 2.1 and the next result yield the implications (i) $\Rightarrow$
in Theorems 2.3 and 2.5 below.

######  PROPOSITION 2.2.

Let $u\in\mathcal{B}^{+}(U)$ with
$u+G_{U}(\varphi(\cdot,u)\mu)=s\in{}^{\ast}\mathcal{H}^{+}(U)\cap\mathcal{C}(U)$
and let $\eta\in(0,1)$, $A:=\\{u<\eta s\\}$. Then $G_{U}(1_{U\setminus
A}\varphi(\cdot,\eta s)\mu)\leq s$ and
${}^{U}\\!\hat{R}_{s}^{A}\in\mathcal{P}^{\prime}(U)$.

###### Proof.

Clearly, $q:=G_{U}(\varphi(\cdot,u)\mu)$ is contained in
$\mathcal{P}^{\prime}(U)$ and
$(1-\eta)^{-1}q\geq{}^{U}\\!R_{s}^{A}\geq{}^{U}\\!\hat{R}_{s}^{A}$. Thus
${}^{U}\\!\hat{R}_{s}^{A}\in\mathcal{P}^{\prime}(U)$. 444In fact,
${}^{U}\\!\hat{R}_{s}^{A}={}^{U}\\!R_{s}^{A}$, since $A$ is finely open; see
(3.1). Of course, $G_{U}(1_{U\setminus A}\varphi(\cdot,\eta s)\mu)\leq
G_{U}(1_{U\setminus A}\varphi(\cdot,u)\mu)\leq s$. ∎

We say that $\varphi$ is _$h$ -proper_ (_locally $h$-Kato, respectively_), if
$G_{U}(1_{A}\varphi(\cdot,h)\mu)<\infty$
($G_{U}(1_{A}\varphi(\cdot,h)\mu)\in\mathcal{C}(U)$, respectively) for every
compact $A$ in $U$.

Provided $\varphi$ is $h$-proper, the general Theorems 6.14, 6.16 and 7.1,
proved later in the paper, yield complete answers to Problems 1 and 2 for our
standard examples, as we now present in Theorems 2.3 and 2.5.

######  THEOREM 2.3.

Let $\varphi$ be $h$-proper. The following statements are equivalent.

  * (1)

There exists a (unique) function $u\in\mathcal{B}^{+}(X)$ (continuous on $U$
if $\varphi$ is locally $h$-Kato) such that

$Lu=\varphi(\cdot,u)\mu\mbox{ on }U\quad\mbox{ and }\quad
h-u\in\mathcal{P}^{\prime}(U).$

  * (2)

There exists a (unique) $u\in\mathcal{B}^{+}(X)$ (continuous on $U$ if
$\varphi$ is locally $h$-Kato) such that $u+G_{U}(\varphi(\cdot,u)\mu)=h$.

  * (3)

For every $\eta\in(0,1)$, there exists a (finely open) Borel set $A$ in $U$
such that

(2.6) $G_{U}(1_{A^{c}}\varphi(\cdot,\eta h)\mu)<\infty\quad\mbox{ and
}\quad{}^{U}\\!\hat{R}_{h}^{A}\in\mathcal{P}^{\prime}(U).$

Further, $(1)$ holds if $G_{U}(\varphi(\cdot,\eta h)\mu)<\infty$ for all
$\eta\in(0,1)$ or $h|_{U}$ is a potential on $U$.

Of course the solutions to $(1)$ and $(2)$ are the same function, by
Proposition 2.1. The last statement in Theorem 2.3 follows on taking $A=X$ or
$A=\emptyset$ in (2.6), respectively. Moreover,
$G_{U}(1_{A^{c}}\varphi(\cdot,\eta h)\mu)<\infty$ for every $\eta\in(0,1)$ if
$G_{U}(1_{A^{c}}\varphi(\cdot,h)\mu)<\infty$, and in many cases the converse
holds as well (see Remark 6.15). See, however, the next example where the
converse fails.

######  EXAMPLES 2.4.

Let us consider the standard examples (i) and (ii).
(1) Let $U:={\mathbbm{R}^{d}}$, so that
$G_{U}(x,y)=G(x,y)\approx|x-y|^{\alpha-d}$, $0<\alpha\leq 2$ and $\alpha<d$.
Let $\mu$ be Lebesgue measure on ${\mathbbm{R}^{d}}$ and
$B_{r}:=\\{x\in{\mathbbm{R}^{d}}\colon|x|<r\\}$, $r>0$. We fix
$\gamma\in\mathbbm{R}$ and define

(2.7) $\varphi(x,t):=1_{B_{1}^{c}}(x)|x|^{-\gamma}(|x|^{c}-1),\qquad
x\in{\mathbbm{R}^{d}},\,c\geq 0.$

Since $\int_{1}^{\infty}r^{d-1}r^{\alpha-d}r^{c-\gamma}\,dr<\infty$ if and
only if $d-1+\alpha-d+c-\gamma<-1$, we see that

(2.8) $G(\varphi(\cdot,c)\mu)<\infty\quad\mbox{ if and only if }\quad
c<\gamma-\alpha.$

Let $h\in{\mathcal{H}}^{+}({\mathbbm{R}^{d}})$, $h\not\equiv 0$, to wit, $h$
is an arbitrary strictly positive constant. Obviously, $\varphi$ is locally
$h$-Kato. So, by Theorem 2.3 and (2.8), Problem 1 has a positive answer if
$\gamma>\alpha$ and $h\leq\gamma-\alpha$. We stress that in the limiting case
$h=\gamma-\alpha$ the solution exists because we have $G(\varphi(\cdot,\eta
h)\mu)<\infty$ for every $\eta\in(0,1)$, even though
$G(\varphi(\cdot,h)\mu)=\infty$. By (2.8), replacing $h$ by
$h\wedge(\gamma-\alpha)$, we also see that Problem 2 has a positive answer if
$\gamma>\alpha$.

For a converse, suppose that $\alpha>1$ and there exists
$u\in\mathcal{B}^{+}({\mathbbm{R}^{d}})$ satisfying (1) in Theorem 2.3. By
uniqueness of $u$ and radial symmetry of $\mu$ and the functions
$x\mapsto\varphi(x,t)$, for all $t\geq 0$, the function $u$ is as well
invariant under rotations of ${\mathbbm{R}^{d}}$. By Proposition 2.1,

$u+G(\varphi(\cdot,u)\mu)=h,$

where the rotationally invariant potential $q:=G(\varphi(\cdot,u)\mu)(x)$
tends to zero at infinity. The latter is trivial if $\alpha=2$. If
$1<\alpha<2$ then it follows from [30, Theorem 1], because the Riesz capacity
of the unit sphere is positive for $\alpha>1$ [32, Corollary 2.2].

So, for each $\eta\in(0,1)$, there exists $R>0$ such that $u(x)<\eta h$ for
every $x\in B_{R}^{c}$, and we obtain that

$G(1_{B_{R}^{c}}\varphi(\cdot,\eta h)\mu)\leq
G(1_{B_{R}^{c}}\varphi(\cdot,u)\mu)\leq h<\infty.$

Hence $G(\varphi(\cdot,\eta h)\mu)<\infty$, so $\eta h<\gamma-\alpha$, by
(2.8)555The same is true if $\varphi$ is radial, locally $h$-Kato.. Thus,
assuming that $\alpha>1$, Problem 1 has a positive answer if and only if
$\gamma>\alpha$ and $h\leq\gamma-\alpha$. By replacing $h$ by
$h\wedge(\gamma-\alpha)$ again, it follows that Problem 2 has a positive
answer if and only if $\gamma>\alpha$.

(2) If we still let $h$ be a constant $c>0$ on ${\mathbbm{R}^{d}}$, have
$\alpha<2$, but take $U:=B_{1}$, then, for any $\varphi$ which is $c$-proper,
Problem 1 has a positive answer (with $\sup u({\mathbbm{R}^{d}})=c$), since
$c1_{U}$ is a potential on $U$ and we may apply Theorem 2.3 with $A=U$!

(3) Finally, let us again consider the case $\alpha<2$, $U:=B_{1}$, but take
$\varphi(x,t):=t^{\gamma}$ and

$h(x):=1_{U}(x)(1-|x|^{2})^{\alpha/2-1},$

see [7, p. 56]. By [6, Corollary 3.2], for every $x\in U$,
$G_{U}(x,y)\approx(1-|y|)^{\alpha/2}$ for small $1-|y|$. Since
$\int_{0}^{\varepsilon}s^{\alpha/2}s^{(\alpha/2-1)\gamma}\,ds<\infty$ if
$\alpha/2+\gamma(\alpha/2-1)>-1$, Problem 1 has a positive answer if

$\gamma<\frac{1+\alpha/2}{1-\alpha/2},$

which reproves part of [1, Theorem 1.7].

######  THEOREM 2.5.

Suppose that $\varphi$ is locally $h$-Kato and let $\mathcal{G}_{h}$ denote
the set of all functions $g\in{\mathcal{H}}^{+}(U)$ such that $g\leq h$ on
$X$, $g=h$ on $X\setminus U$ and $g\not\equiv 0$ on $U$. Then the following
statements are equivalent (with the same functions $u$ in $(1)$ and $(2)$).

  * (1)

There exists $u\in\mathcal{B}^{+}(X)$ with $u\leq h$ on $X$, $u=h$ on $U^{c}$,
$u|_{U}\in\mathcal{C}(U)\setminus\\{0\\}$ and

$Lu=\varphi(\cdot,u)\mu\quad\text{ on }U.$

  * (2)

There exists $u\in\mathcal{B}^{+}(X)$ such that $u|_{U}\in\mathcal{C}(U)$ and
$u+G_{U}(\varphi(\cdot,u)\mu)\in\mathcal{G}_{h}(U)$.

  * (3)

There exists $g\in\mathcal{G}_{h}(U)$ such that, for every $\eta\in(0,1)$,
there is a (finely open) Borel set $A$ in $U$ with

$G_{U}(1_{A^{c}}\varphi(\cdot,\eta g)\mu)<\infty\quad\mbox{ and
}\quad{}^{U}\\!\hat{R}_{g}^{A}\in\mathcal{P}^{\prime}(U).$

Moreover, the following property $(4)$ implies $(1)$ \- $(3)$ and, if $1_{U}h$
is harmonic on $U$, the converse holds as well.

  * (4)

There exists $g\in\mathcal{G}_{h}$ such that, for every $\eta\in(0,1)$, there
is a (finely open) Borel set $A$ in $U$ with

$G_{U}(1_{A^{c}}\varphi(\cdot,\eta g)\mu)<\infty\quad\mbox{ and
}\quad{}^{U}\\!\hat{R}_{g}^{A}\not\equiv g|_{U}.$

###### Proof.

$(1)\,\Rightarrow\,(2)$: Let $V_{n}$, $n\in\mathbbm{N}$, be open sets having
compact closure in $U$ such that $V_{n}\uparrow U$ as $n\to\infty$. Since
$\varphi$ is locally $h$-Kato, we have
$G_{U}(1_{V_{n}}\varphi(\cdot,u)\mu)\in\mathcal{C}(U)$, and hence
$G_{V_{n}}(\varphi(\cdot,u)\mu)=G_{U}(1_{V_{n}}\varphi(\cdot,u)\mu)-H_{V_{n}}G_{U}(1_{V_{n}}\varphi(\cdot,u)\mu)\in\mathcal{C}(V_{n})$
for every $n\in\mathbbm{N}$. The functions
$h_{n}:=u+G_{V_{n}}(\varphi(\cdot,u)\mu)$, $n\in\mathbbm{N}$, satisfy
$Lh_{n}=0$ on $V_{n}$. Therefore $h_{n}\in{\mathcal{H}}^{+}(V_{n})$. Moreover,
$h-h_{n}\geq-G_{V_{n}}(\varphi(\cdot,u)\mu)$, hence $h-h_{n}\geq 0$. The
functions $h_{n}$ do not vanish on $U$ and they increase to the (harmonic)
function $u+G_{U}(\varphi(\cdot,u)\mu)\leq h$.

The remaining implications follow from Theorem 2.3, Proposition 2.1 (observe
that $\varphi$ is locally $g$-Kato for every $g\in\mathcal{G}_{h}$) and
Theorem 6.16. ∎

As mentioned in the Introduction, we can interpret the condition on $h-u$ in
Problem 1 as a boundary condition, as follows. Recall that sequence $(x_{n})$
in $U$ converging to a point $z\in\partial U$ is called _regular_ , provided
$\lim_{n\to\infty}H_{U}f(x_{n})=f(z)$ for every $f\in\mathcal{C}(X)$ with
compact support (and hence for every $f\in\mathcal{C}(X)$ which is bounded by
a function $s\in{}^{\ast}{\mathcal{H}}^{+}(X)\cap\mathcal{C}(X)$; see Remark
3.8). The set $U$ is _regular_ , if every sequence in $U$ converging to a
point in $\partial U$ is regular.

######  PROPOSITION 2.6.

Suppose that $U$ is relatively compact, $h$ is bounded on $U$, and
$G(1_{U}\varphi(\cdot,h)\mu)\in\mathcal{C}(X)$. Let $u\in\mathcal{B}^{+}(X)$
such that $u\leq h$ on $X$, $u=h$ on $X\setminus U$, and
$Lu=\varphi(\cdot,u)\mu$ on $U$.

Then $h-u\in\mathcal{P}^{\prime}(U)$ if and only if $u$ is continuous on $U$
and

(2.9) $\lim\nolimits_{n\to\infty}(h-u)(x_{n})=0$

for every regular sequence in $U$ which converges to a point in $\partial U$.

###### Proof.

Clearly, $p:=G(1_{U}\varphi(\cdot,u)\mu)\in\mathcal{C}(X)$, since
$\varphi(\cdot,u)\leq\varphi(\cdot,h)$ on $U$. Hence

$q:=G_{U}(\varphi(\cdot,u)\mu)=p-H_{U}p\in\mathcal{C}_{b}(U)$

satisfies $\lim_{n\to\infty}q(x_{n})=0$ for every regular sequence $(x_{n})$
in $U$ (see Remark 3.8).

(a) Suppose that $h-u\in\mathcal{P}^{\prime}(U)$. Then $u+q=h$, by Proposition
2.1. So $u$ is continuous on $U$ and (2.9) holds.

(b) Suppose now that $u$ is continuous on $U$ and (2.9) holds. Then
$\widetilde{h}:=u+q$ is continuous on $U$ and $L\widetilde{h}=0$. So
$\widetilde{h}$ is harmonic on $U$. Having $\widetilde{h}\leq h+q$ this
implies that $\widetilde{h}\leq h$. Of course, $h-\widetilde{h}=0$ on
$X\setminus U$ and $h-\widetilde{h}$ is bounded on $U$. By (2.9),
$\lim_{n\to\infty}(h-\widetilde{h})(x_{n})=0$ for every regular sequence
$(x_{n})$ in $U$. Hence $h-\widetilde{h}=0$, by [4, III.4.3], so
$h-u\in\mathcal{P}^{\prime}(U)$, by Proposition 2.1. ∎

######  REMARK 2.7.

Let us consider the situation studied in [3], that is, in our standard example
(ii) (Riesz potentials, $0<\alpha<2\wedge d$) we have $U={\mathbbm{R}^{d}}$
and $\varphi(x,t)=\rho(x)\phi(t)$ with locally bounded
$\rho\in\mathcal{B}^{+}({\mathbbm{R}^{d}})$ and increasing
$\phi\in\mathcal{C}([0,\infty))$ with $\phi(0)=0$.

Moreover, let us assume that $\rho$ is radially symmetric. Theorem 1 in [3]
states that there exists a non-trivial (radially symmetric) bounded solution
$u\geq 0$ to $-(-\Delta^{\alpha/2})u=\rho(x)\phi(u)$ provided
$\int_{0}^{\infty}r^{\alpha-1}\rho(r)\,dr<\infty$. In view of our Theorem 2.3,
the result even holds under the weaker assumption that
$\int_{A}G(\cdot,y)\rho(y)\,dy<\infty$ for every compact $A$ in
${\mathbbm{R}^{d}}$.

In [3, Theorem 1] it also claimed that the converse holds. However, the proof
referring to a result of [30] is only valid if $\alpha>1$. In fact, the
converse itself fails if $\alpha\leq 1$ as the following example shows.

Suppose that $\alpha\leq 1$. For $n\in\mathbbm{N}$ and $\varepsilon\in(0,1)$
let

$B_{n}:=\\{x\in{\mathbbm{R}^{d}}\colon|x|<n\\},\quad S_{n}:=\partial
B_{n}\quad\mbox{ and }\quad A_{n}(\varepsilon):=\\{x\in
B_{n}\colon|x|>n-\varepsilon\\}.$

For the moment, let us fix $n$. Since $S_{n}$ is polar (see [32]), there is a
hyperharmonic function $w_{n}\geq 0$ on ${\mathbbm{R}^{d}}$ such that
$w_{n}>1$ on $S_{n}$ and $w_{n}\leq 2^{-n}$ on $B_{n-1}$ (taking
$B_{0}:=\emptyset$). Since $w_{n}$ is lower semicontinuous, there exists
$\varepsilon_{n}\in(0,1)$ with $w_{n}>1$ on $A_{n}:=A_{n}(\varepsilon_{n})$.
Clearly, $q_{n}:=R_{1}^{A_{n}}\leq 1$ on ${\mathbbm{R}^{d}}$, $q_{n}$ is a
potential (continuous and harmonic on the complement of $A_{n}$) and
$q_{n}\leq 2^{-n}$ on $A_{n-1}$. Therefore $q:=\sum_{n=1}^{\infty}q_{n}$ is a
potential on ${\mathbbm{R}^{d}}$, $q\leq 2$. Let

$A:=\bigcup\nolimits_{n\in\mathbbm{N}}A_{n},\quad\beta_{n}:=n^{1-\alpha}\lambda(A_{n})^{-1}\quad\mbox{
and }\quad\rho:=\sum\nolimits\beta_{n}1_{A_{n}}\lambda,$

$\lambda$ being Lebesgue measure on ${\mathbbm{R}^{d}}$. Clearly, $\rho$ is
locally bounded, radially symmetric,

$\int_{0}^{\infty}r^{\alpha-1}\rho(x)\,dr=\infty\quad\mbox{ and
}\quad\int_{A^{c}}G(\cdot,y)\rho(y)\,dy=0.$

Moreover, $p:=R_{1}^{A}\leq\sum_{n=1}^{\infty}q_{n}=q$. Hence $p$ is a
potential on ${\mathbbm{R}^{d}}$, $p\leq 1$ on ${\mathbbm{R}^{d}}$, $p=1$ on
$A$ (it is easily seen that $p$ is continuous, harmonic outside
$\overline{A}$, and $\liminf_{x\to\infty}p(x)=0$).

Then, by Theorem 2.3, there is a unique function $u\in\mathcal{B}^{+}(X)$
(continuous real and radially symmetric) such that $u+\int
G(\cdot,y)\rho(y)\phi(u(y))\,dy=1$ and hence
$-(-\Delta)^{\alpha/2}u=\rho\phi(u)$.

Thus the implication $(d)\Rightarrow(a)$ in [3, Theorem 1] indeed does not
hold for $\alpha\leq 1$.

##  3 Balayage spaces

In this section we shall recall the definition of a balayage space and various
facts which we shall use later on (for further details see [4] or [21]). The
reader who is familiar with this concept or is mainly interested in our
standard examples (or similar ones) may pass directly to Section 4 and perhaps
go back whenever needed.

###  3.1 Definition

Let $X$ be a locally compact space with countable base of open sets. For every
numerical function $g$ on $X$, let $\hat{g}$ denote the largest lower
semicontinuous minorant of $g$, that is, $\hat{g}(x):=\liminf_{y\to x}g(y)$,
$x\in X$.

Let $\mathcal{W}$ be a convex cone of positive numerical functions on $X$. We
consider the following properties that may hold for $\mathcal{W}$:

  * (C)

Continuity: Every $w\in\mathcal{W}$ is the supremum of its minorants in

$\mathcal{S}^{+}(X):=\mathcal{W}\cap\mathcal{C}(X).$

  * (S)

Separation: $\mathcal{W}$ is linearly separating, that is, for all $x\neq y$
and $\gamma>0$, there exists a function $v\in\mathcal{W}$ such that
$v(x)\neq\gamma v(y)$.

  * (T)

Transience: There exist strictly positive $u,v\in\mathcal{S}^{+}(X)$ such that
$u/v\in\mathcal{C}_{0}(X)$.

Of course, (C) implies that the functions in $\mathcal{W}$ are lower
semicontinuous, in particular, Borel measurable.

We call $(X,\mathcal{W})$ a _balayage space_ if the following hold:

  * (B0)

$\mathcal{W}$ has the properties (C), (S) and (T).

  * (B1)

If $v_{n}\in\mathcal{W}$, $v_{n}\uparrow v$, then $v\in\mathcal{W}$.

  * (B2)

If $\mathcal{V}\subset\mathcal{W}$, then
$\widehat{\inf\mathcal{V}}^{f}\in\mathcal{W}$.

  * (B3)

If $u,v^{\prime},v^{\prime\prime}\in\mathcal{W}$ and $u\leq
v^{\prime}+v^{\prime\prime}$, then there exist
$u^{\prime},u^{\prime\prime}\in\mathcal{W}$ such that
$u=u^{\prime}+u^{\prime\prime}$, $u^{\prime}\leq v^{\prime}$ and
$u^{\prime\prime}\leq v^{\prime\prime}$.

Here $(B_{1})$ means that $\mathcal{W}$ has to be closed under increasing
limits. $(B_{2})$ implies that $\mathcal{W}$ is closed under finite infima,
whereas arbitrary infima have to be regularized taking the greatest lower
semicontinuous minorant with respect to the ($\mathcal{W}$-)_fine topology_ ,
hence the superscript $f$. By definition, the ($\mathcal{W}$-)_fine topology_
is the smallest topology on $X$ that contains the initial topology and such
that every function in $\mathcal{W}$ is continuous. Since the functions in
$\mathcal{W}$ are lower semicontinuous, $(B_{2})$ eventually implies that

(3.1) $\widehat{\inf\mathcal{V}}^{f}=\widehat{\inf\mathcal{V}}\qquad\mbox{for
all }\mathcal{V}\subset\mathcal{W}.$

The subtlety of requiring the finely lower semicontinuous regularization in
$(B_{2})$ amounts to a convergence property for increasing sequences of
harmonic functions. Finally, $(B_{3})$ is some lattice property (Riesz
property).

For this and a detailed exposition of the theory of balayage spaces the reader
may consult [4, 21]. The theory covers the potential theory for a large class
of elliptic or parabolic differential equations of second order. Namely, if
$(X,{\mathcal{H}})$ is a $\mathcal{P}$-harmonic space and
${}^{\ast}\\!{\mathcal{H}}^{+}(X)$ denotes the convex cone of positive
hyperharmonic functions on $X$, then $(X,{}^{\ast}\\!{\mathcal{H}}^{+}(X))$ is
a balayage space. For details the reader may consult [19, Section 7.1], see
also below.

Moreover, the isotropic $\alpha$-stable Lévy processes with $0<\alpha<2\wedge
d$ (and the corresponding Riesz potentials) and many other transient Lévy
processes lead to balayage spaces (see the next subsection and [4, Sections
V.3 and V.4] or [21, Section 3.1]).

######  REMARK 3.1.

Occasionally, it is useful to use an _$s$ -transform (Doob’s conditioning) _
of the balayage space $(X,\mathcal{W})$ by some $s\in\mathcal{S}^{+}(X)$,
$s>0$: Defining $\widetilde{\mathcal{W}}:=\\{v/s:v\in\mathcal{W}\\}$ we
clearly get a balayage space $(X,\widetilde{\mathcal{W}})$, and have
$1\in\widetilde{\mathcal{W}}$.

###  3.2 The generic example

As we shall explain in a moment, in view of Remark 3.1 and Theorems 3.2, 3.9
below, the set ${\mathcal{E}}_{\mathbbm{P}}$ of excessive functions for a sub-
Markov semigroup $\mathbbm{P}$ with ${\mathcal{E}}_{\mathbbm{P}}$ satisfying
(B0) may be considered as the generic example of a convex cone $\mathcal{W}$
such that $(X,\mathcal{W})$ is a balayage space.

As usual, we call $K:(x,A)\mapsto[0,\infty]$ a kernel on $X$ if $K(x,\cdot)$
is a Borel measure on $X$ for every $x\in X$ and the function $K(\cdot,A)$ is
Borel measurable for every Borel measurable $A\subset X$. We then write
$Kf(x)=\int_{X}f(y)K(x,dy)$ for $x\in X$ and $f\in\mathcal{B}^{+}(X)$. For
instance, if $\psi\in\mathcal{B}^{+}(X)$ then $M_{\psi}f(x):=f(x)\psi(x)$,
$f\in\mathcal{B}^{+}(X)$, $x\in X$, defines a kernel, the multiplication
kernel.

Let $\mathbbm{P}=(P_{t})_{t>0}$ be a sub-Markov semigroup on $X$, that is, we
have kernels $P_{t}$ on $X$ such that $P_{t}1\leq 1$ and $P_{s}P_{t}=P_{s+t}$
for all $s,t>0$. We assume that $\mathbbm{P}$ is _right continuous_ , that is,
$\lim_{t\to 0}P_{t}f=f$ (pointwise) for every $f\in\mathcal{C}(X)$ with
compact support. Let ${\mathcal{E}}_{\mathbbm{P}}$ denote the corresponding
convex cone of excessive functions,

${\mathcal{E}}_{\mathbbm{P}}:=\\{u\in\mathcal{B}^{+}(X)\colon\sup\nolimits_{t>0}P_{t}u=u\\}.$

The following holds (see, for example, [21, Corollary 2.3.8]).

######  THEOREM 3.2.

$(X,{\mathcal{E}}_{\mathbbm{P}})$ is a balayage space if and only it
${\mathcal{E}}_{\mathbbm{P}}$ satisfies $(B_{0})$, that is, (C), (S) and (T)
hold.

######  REMARKS 3.3.

1\. Let $\mathbbm{V}=(V_{\lambda})_{\lambda>0}$ be the corresponding
resolvent, that is, $V_{\lambda}:=\int_{0}^{\infty}e^{-\lambda t}P_{t}\,dt$,
$\lambda>0$. For every $u\in{\mathcal{E}}_{\mathbbm{P}}$, $\lambda
V_{\lambda}u\uparrow u$ as $\lambda\uparrow\infty$. So (C) holds if the
kernels $V_{\lambda}$ (or even the kernels $P_{t}$) are _strong Feller_ , that
is, map $\mathcal{B}_{b}(X)$ into $\mathcal{C}_{b}(X)$. Further, (S) holds if
the _potential kernel
$V_{0}:=\lim_{\lambda\to\infty}V_{\lambda}=\int_{0}^{\infty}P_{t}\,dt$ of
$\mathbbm{P}$_ is _proper_ , that is, there exists $g\in\mathcal{B}(X)$ with
$g>0$ and $V_{0}g<\infty$ (see [21, Lemma 2.3.5]).

2\. Let $\mathbbm{T}:=(T_{t})_{t>0}$ be the semigroup of _uniform translation
to the left on $\mathbbm{R}$_, that is, $T_{t}(x,\cdot):=\delta_{x-t}$. It is
right continuous and has a strong Feller resolvent. Moreover,
${\mathcal{E}}_{\mathbbm{T}}$ is the set of all left-continuous increasing
functions in $\mathcal{B}^{+}(\mathbbm{R})$, the functions $1,\exp$ linearly
separate points in $\mathbbm{R}$ and $(1\wedge\exp)/(1+\exp)$ vanishes at
infinity. So $(\mathbbm{R},E_{\mathbbm{T}})$ is a balayage space.

3\. As trivial as $\mathbbm{T}$ is, its product
$\mathbbm{P}\otimes\mathbbm{T}$ with a sub-Markov semigroup $\mathbbm{P}$ is
of interest. This is because
$(X\times\mathbbm{R},{\mathcal{E}}_{\mathbbm{P}\otimes\mathbbm{T}})$ is a
balayage space provided $(X,{\mathcal{E}}_{\mathbbm{P}})$ is a balayage space
and $\mathbbm{P}$ is _strong Feller_ , that is, the kernels $P_{t}$ are strong
Feller (if $\mathbbm{P}$ is the Brownian semigroup on $X={\mathbbm{R}^{d}}$,
$d\geq 3$, then ${\mathcal{E}}_{\mathbbm{P}\otimes\mathbbm{T}}$ is the set of
positive hyperharmonic functions in our standard example (iii) given by the
heat equation). For details see Section 9.1. The use of such products may even
help getting a Hunt process associated to $\mathbbm{P}$ on $X$ if
$(X,{\mathcal{E}}_{\mathbbm{P}})$ is not a balayage space and only the
resolvent of $\mathbb{P}$ is strong Feller, see Section 9.2.

4\. A sufficient condition for (T) for $\mathbbm{P}\otimes\mathbbm{T}$ is
given in Remark 9.3 below.

###  3.3 Potentials

Ine the following, let $(X,\mathcal{W})$ always be a balayage space and

(3.2) $\mathcal{P}(X):=\\{p\in\mathcal{S}^{+}(X)\colon\exists\
w\in\mathcal{S}^{+}(X),\,w>0,\,\hbox{with }p/w\in\mathcal{C}_{0}(X)\\}.$

We recall some basic notions and facts, often referring to the paper [22],
because we also need its results on compactness of potential kernels. In
particular, see (3.12) and (3.13) below for characterizations of
$\mathcal{P}(X)$.

Clearly, $\mathcal{P}(X)$ is a convex cone and $p\wedge w\in\mathcal{P}(X)$ if
$p\in\mathcal{P}(X)$, $w\in\mathcal{S}^{+}(X)$. Moreover, the following holds
(see [22, (1.1)] and [21, Proposition 1.2.1]): For every $w\in\mathcal{W}$,
there exists a sequence $(p_{n})$ in $\mathcal{P}(X)$ such that $p_{n}\uparrow
w$. Furthermore, if $(p_{n})$ is a sequence in $\mathcal{P}(X)$ such that
$s:=\sum_{n\in\mathbbm{N}}p_{n}\in\mathcal{C}(X)$, then $s\in\mathcal{P}(X)$.
Also, for every $p\in\mathcal{P}(X)$, there exists $q\in\mathcal{P}(X)$,
$q>0$, such that $p/q\in\mathcal{C}_{0}(X)$.

For every numerical function $f$ on $X$, we have a _reduced function_

(3.3) $R_{f}:=\inf\\{w\in\mathcal{W}\colon w\geq f\\},$

which, by $(B_{2})$, is contained in $\mathcal{W}$ if $f$ is finely lower
semicontinuous. For $A\subset X$ and $u\in\mathcal{W}$, the _reduction of $u$
on $A$_ is defined by

(3.4) $R_{u}^{A}:=R_{1_{A}u}=\inf\\{v\in\mathcal{W}\colon v\geq u\mbox{ on
}A\\},$

and we have $R_{u}^{A}\in\mathcal{W}$ if $A$ is finely open. By [4, VI.1.8],

(3.5) $R_{u}^{A}=\sup\nolimits_{n\in\mathbbm{N}}R_{f_{n}}$

for every sequence $(f_{n})$ of numerical functions on $X$ with $f_{n}\uparrow
1_{A}u$ as $n\to\infty$.

######  REMARK 3.4.

If $u\in\mathcal{W}$ is real continuous on some neighborhood of $A$, then

(3.6) $R_{u}^{A}=\inf\\{R_{u}^{W}\colon\mbox{ $W$ open, $A\subset W$}\\}.$

Indeed, let $s_{0}\in\mathcal{S}^{+}(X)$, $s_{0}>0$. If $\varepsilon>0$ and
$w\in\mathcal{W}$ with $w\geq u$ on $A$, then $w+\varepsilon s_{0}>u$ on an
open neighborhood $W$ of $A$, hence $R_{u}^{W}\leq w+\varepsilon s_{0}$ (cf.
[4, VI.1.2]).

Let $p\in\mathcal{P}(X)$. There exists a smallest closed set $C(p)$ in $X$,
called _carrier_ or _superharmonic support_ of $p$, such that

(3.7) $p=R_{p}^{C(p)},$

that is, if $w\in\mathcal{W}$ and $w\geq p$ on $C(p)$, then $w\geq p$ on $X$
(see [4, II.6.3] or [21, Proposition 4.1.6]).

There is a unique kernel $K_{p}$ on $X$, called _associated potential kernel_
or _potential kernel for $p$_, such that the following hold (see [4,
II.6.17]):

  * •

$K_{p}1=p$.

  * •

For every $f\in\mathcal{B}_{b}^{+}(X)$, $K_{p}f\in\mathcal{P}(X)$ and
$C(K_{p}f)\subset\operatorname*{supp}(f)$.

The domination principle in the next lemma will be absolutely essential later
on. For the convenience of the reader we include a proof.

######  LEMMA 3.5.

  * (i)

For every $f\in\mathcal{B}^{+}(X)$, there exists a sequence $(p_{n})$ in
$\mathcal{P}(X)$ such that $C(p_{n})$ is a compact in $\\{f>0\\}$,
$p_{n+1}-p_{n}\in\mathcal{P}(X)$ for every $n\in\mathbbm{N}$, and
$K_{p}f=\lim_{n\to\infty}p_{n}\in\mathcal{W}$. If $f,g\in\mathcal{B}^{+}(X)$,
$f\leq g$ and $K_{p}g\in\mathcal{C}(X)$, then $K_{p}f\in\mathcal{P}(X)$.

  * (ii)

_Domination principle:_ If $f,g\in\mathcal{B}^{+}(X)$ and $w\in\mathcal{W}$,
then

(3.8) $K_{p}f\leq K_{p}g+w\mbox{ on }\\{f>0\\}\quad\mbox{ implies that }\quad
K_{p}f\leq K_{p}g+w\mbox{ on }X.$

###### Proof.

(i) Let $f\in\mathcal{B}^{+}(X)$ and $\mathcal{F}=\\{(f\wedge a)1_{L}\colon
a>0,L\mbox{ compact in }\\{f>0\\}\\}$. For every $\varphi\in\mathcal{F}$,
$K_{p}\varphi\in\mathcal{P}(X)$ and
$C(K_{p}\varphi)\subset\operatorname*{supp}(\varphi)\subset\\{f>0\\}$. Since
$K_{p}$ is a kernel, we have
$\sup\\{K_{p}\varphi\colon\varphi\in\mathcal{F}\\}=K_{p}f$. Hence, by a
topological lemma of Choquet, there exist a sequence $(a_{n})$ in $(0,\infty)$
and a sequence $(L_{n})$ of compacts in $\\{f>0\\}$ such that
$\sup_{n\in\mathbbm{N}}K_{p}((f\wedge a_{n})1_{L_{n}})=K_{p}f$. Of course, we
may assume without loss of generality that both sequence are increasing. Then
the potentials $p_{n}:=K_{p}((f\wedge a_{n})1_{L_{n}})$, $n\in\mathbbm{N}$,
have the desired properties. Suppose now that $g\in\mathcal{B}^{+}(X)$ such
that $f\leq g$ and $K_{p}g\in\mathcal{C}(X)$. There exists a function
$f^{\prime}\in\mathcal{B}^{+}(X)$ with $f+f^{\prime}=g$. Then the continuous
function $K_{p}g$ is the sum of the two lower semicontinuous functions
$K_{p}f$ and $K_{p}f^{\prime}$. So these functions are continuous as well.
Thus $K_{p}f$, being a sum of functions in $\mathcal{P}(X)$, is contained in
$\mathcal{P}(X)$, see the remarks after (3.2).

(ii) Immediate consequence of (i), (3.7) and (3.4). ∎

######  REMARKS 3.6.

(1) Of course, if $1\in\mathcal{W}$, then the domination principle implies the
_complete maximum principle_ , see [4, II.7].

(2) If there is a Green function $G$ for $(X,\mathcal{W})$ and a measure
$\mu\geq 0$ on $X$ such that $p=G\mu:=\int G(\cdot,y)\,d\mu(y)$ (see [24] for
such a representation), then $K_{p}f=G(f\mu)$ for every
$f\in\mathcal{B}^{+}(X)$.

(3) Suppose that $p\in\mathcal{P}(X)$ and that $(X,\widetilde{\mathcal{W}})$
(see Remark 3.1) is obtained by an $s$-transformation of $(X,\mathcal{W})$,
that is, $\widetilde{\mathcal{W}}=(1/s)\mathcal{W}$ for some strictly positive
$s\in\mathcal{S}^{+}(X)$ (where we may assume that $p/s\in\mathcal{C}_{0}(X)$,
see (3.2)).

Then clearly $(1/s)\mathcal{P}(X)$ is the cone of continuous real potentials
for $\widetilde{\mathcal{W}}$ and the corresponding potential kernel for
$\widetilde{p}:=p/s$ is given by

${\widetilde{K}}_{\widetilde{p}}f=(1/s)K_{p}f,\qquad f\in\mathcal{B}^{+}(X).$

We also recall that for every open set $U$ in $X$ there is the  _harmonic
kernel_ $H_{U}$ determined by

(3.9) $H_{U}p=R_{p}^{X\setminus U}=\inf\\{w\in\mathcal{W}\colon w\geq p\mbox{
on }X\setminus U\\}$

for every $p\in\mathcal{P}(X)$ (see [4, p. 98 and II.5.4] or [21, Section
4.2]) and, by (3.5), satisfies $H_{U}w=R_{w}^{X\setminus U}$ for every
$w\in\mathcal{W}$ (choose $p_{n}\in\mathcal{P}(X)$, $p_{n}\uparrow w$).

######  PROPOSITION 3.7.

If $V$ is an open subset of $U$, then

$H_{V}H_{U}=H_{U},\quad\mbox{ and }\quad H_{U}(x,\cdot)\geq
H_{V}(x,\cdot)|_{U^{c}},\mbox{ for every $x\in V$}.$

###### Proof.

The equality is easily established using (3.6) and the trivial relation

$R_{p}^{U^{c}}=R_{R_{p}^{U^{c}}}^{V^{c}}\leq\inf\\{R_{R_{p}^{W}}^{V^{c}}\colon{U^{c}}\subset
W\mbox{ open}\\}\leq\inf\\{R_{p}^{W}\colon{U^{c}}\subset W\mbox{ open}\\}$

(cf. [4, VI.9.1]). Further, for every $f\in\mathcal{B}^{+}(X)$,
$H_{V}H_{U}f(x)=\int H_{U}f(y)\,H_{V}(x,dy)\geq\int_{U^{c}}f(y)\,H_{V}(x,dy$,
since $H_{U}f=f$ on ${U^{c}}$. ∎

As in our standard examples, a sequence $(x_{n})$ in $U$ which converges to a
point $z\in\partial U$ is called _regular_ if
$\lim_{n\to\infty}H_{U}f(x_{n})=f(z)$ for every $f\in\mathcal{C}(X)$ with
compact support. The open set $U$ is _regular_ if every such sequence is
regular.

######  REMARK 3.8.

Let $U$ be an open set in $X$ and let $(x_{n})$ be a regular sequence in $U$
converging to $z\in\partial U$. Then
$\lim\nolimits_{n\to\infty}H_{U}f(x_{n})=f(z)$ for every
$f\in\mathcal{B}^{+}(X)$ with $\lim_{x\to z}f(x)=f(z)$ and $f\leq s$ for some
$s\in\mathcal{S}^{+}(X)$.

Indeed, choosing $f^{\prime}\in\mathcal{C}^{+}(X)$ with compact support with
$f^{\prime}\leq f$ and $f^{\prime}(z)=f(z)$ we get
$f(z)=f^{\prime}(z)=\lim\nolimits_{n\to\infty}H_{U}f^{\prime}(x_{n})\leq\liminf_{n\to\infty}H_{U}f(x_{n})$
and

$(s-f)(z)\leq\liminf\nolimits_{n\to\infty}H_{U}(s-f)(x_{n})\\\
\leq\liminf\nolimits_{n\to\infty}(s(x_{n})-H_{U}f(x_{n}))=s(z)-\limsup\nolimits_{n\to\infty}H_{U}f(x_{n}).$

The next result collects elements of [4, II.8.6, proof of IV.8.1, VI.3.14,
VI.8.2]. We recall that $p\in\mathcal{P}(X)$ is _strict_ if and only if
$K_{p}1_{W}\neq 0$ for every finely open Borel set $W\neq\emptyset$ (see [4,
I.1.5] for the existence and [4, VI.8.2] for this characterization).

######  THEOREM 3.9.

Let $1\in\mathcal{W}$ and $p\in\mathcal{P}_{b}(X)$ be strict. Then there
exists a Hunt process $\mathfrak{X}$ on $X$ such that its transition semigroup
$\mathbbm{P}=(P_{t})_{t>0}$ satisfies

${\mathcal{E}}_{\mathbbm{P}}=\mathcal{W}\quad\mbox{ and
}\quad\int_{0}^{\infty}P_{t}\,dt=K_{p}.$

The operator $K_{p}$ and the resolvent kernels are strong Feller. If
$\tau_{V}$ is the _exit time_ of an open set $V\subset X$, that is
$\tau_{V}:=\inf\\{t\geq 0\colon X_{t}\notin V\\}$, then

(3.10) $H_{V}f(x)=\mathbbm{E}^{x}(f\circ X_{\tau_{V}}),\qquad
f\in\mathcal{B}^{+}(X),\,x\in X.$

Let ${}^{\ast}{\mathcal{H}}(U)$ denote the set of functions
$u\in\mathcal{B}(X)$ which are _hyperharmonic on $U$_, that is, are lower
semicontinuous on $U$ and satisfy

$-\infty<H_{V}u(x)\leq u(x)\quad\mbox{ for all }x\in V\in\mathcal{V}(U).$

A function $u\in{}^{\ast}{\mathcal{H}}(U)$ is called _superharmonic on $U$_ if
$(H_{V}s)|_{V}\in\mathcal{C}(V)$ for every $V\in\mathcal{V}(U)$. We note that
(see [4, II.5.5 and VI.2.6] or [21, Propositions 4.1.7 and 4.2.9])

(3.11) ${}^{\ast}{\mathcal{H}}^{+}(X)=\mathcal{W}$

and $\mathcal{S}^{+}(X)$ is the set of all continuous real superharmonic
functions on $X$. A function $u$ is called is _subharmonic on $U$_ if $-u$ is
superharmonic on $U$.

Further,
${\mathcal{H}}(U):={}^{\ast}{\mathcal{H}}(U)\cap(-{}^{\ast}{\mathcal{H}}(U))$
is the set of functions which are _harmonic on $U$_, that is,

${\mathcal{H}}(U)=\\{h\in\mathcal{B}(X)\colon h|_{U}\in\mathcal{C}(U),\
H_{V}h=h\mbox{ for every }V\in\mathcal{V}(U)\\}.$

For every $p\in\mathcal{P}(X)$, the set $C(p)$ is the smallest closed set $A$
in $X$ such that $p$ is harmonic on $X\setminus A$ (see [4, III.6.12]). By [4,
III.2.8 and III.1.2], we have the following characterizations of
$\mathcal{P}(X)$:

(3.12) $\displaystyle\mathcal{P}(X)$ $\displaystyle=$
$\displaystyle\\{p\in\mathcal{S}^{+}(X)\colon\mbox{If
$h\in{\mathcal{H}}^{+}(X)$ and $h\leq p$, then $h=0$.}\\}$ (3.13)
$\displaystyle=$
$\displaystyle\\{p\in\mathcal{S}^{+}(X)\colon\inf\\{H_{U}p\colon
U\in\mathcal{V}(X)\\}=0\\}.$

In particular,

(3.14) $(\mathcal{P}(X)-\mathcal{P}(X))\cap{\mathcal{H}}(X)=\\{0\\}.$

Indeed, if $p,q\in\mathcal{P}(X)$ and $p-q=h\in{\mathcal{H}}(X)$, then
$-H_{U}q\leq H_{U}h=h\leq H_{U}p$ for every $U\in\mathcal{V}(X)$, hence $h=0$,
by (3.13). Further, we have a _Riesz decomposition_ :

(3.15) $\mathcal{S}^{+}(X)={\mathcal{H}}^{+}(X)\oplus\mathcal{P}(X).$

By definition, a superharmonic function $s\geq 0$ on $X$ is a  _potential_ if

$\inf\\{H_{U}s\colon U\in\mathcal{V}(X)\\}=0.$

So $\mathcal{P}(X)$ is the set of all continuous real potentials on $X$. Let
$\mathcal{P}^{\prime}(X)$ denote the set of all potentials on $X$.

The following lemma will be used in Sections 9 and 8

######  LEMMA 3.10.

Let $w\in\mathcal{B}^{+}(X)$ and let $U$ be an open set in $X$ such that
$H_{U}w\leq w$ and the sets $V\in{\mathcal{U}}(U)$ satisfing $H_{V}w\leq w$
and $(H_{V}w)|_{V}\in\mathcal{C}(V)$ cover $U$. Then
$H_{U}w\in{\mathcal{H}}^{+}(U)$.

###### Proof.

Let $V\in{\mathcal{U}}(U)$. Since $H_{U}w\leq w$, we may choose
$f\in\mathcal{B}^{+}(X)$ such that $H_{U}w+f=w$. Then
$H_{V}w=H_{V}H_{U}w+H_{V}f=H_{U}w+H_{V}f$, where $H_{U}w$ and $H_{V}f$ are
lower semicontinuous on $U$. So both are real continuous on $U$ by our
assumption on $w$. Thus $(H_{U}w)|_{U}\in\mathcal{C}(U)$, and hence
$H_{U}w\in{\mathcal{H}}^{+}(U)$, by (3.7). ∎

###  3.4 Families of harmonic kernels

In this section we characterize balayage spaces by properties of their
harmonic kernels for open sets from a base for the topology of $X$. Let us
first note that, by (3.9), (3.11), [4, III. 4.2.8] or [21, Theorem 5.1.2 and
Corollary 5.2.8]), we know the following.

######  THEOREM 3.11.

If $(X,\mathcal{W})$ is a balayage space, then the family
$(H_{V})_{V\in{\mathcal{U}}(X)}$ of harmonic kernels has following properties.

  * ($H_{0}$)

$H_{V}1_{V}=0$ and $H_{V}(x,\cdot)=\delta_{x}$ for every $x\in{V^{c}}$.

  * ($H_{1}$)

For each $x\in X$, $\lim_{V\downarrow\\{x\\}}H_{V}f(x)=f(x)$ for
$f\in\mathcal{C}_{c}(X)$ or $\liminf\limits_{y\to x}R_{1}^{\\{x\\}}(y)\geq 1$.

  * ($H_{2}$)

If $\overline{V}\subset U\in{\mathcal{U}}(X)$, then $H_{V}H_{U}=H_{U}$.

  * ($H_{3}$)

For every $f\in\mathcal{B}_{b}(X)$ with compact support, $H_{V}f$ is
continuous on $V$.

  * ($H_{4}$)

For each $x\in V$, there exists $w\in{}^{\ast}{\mathcal{H}}^{+}(V)$ such that
$w(x)<\infty$ and, for every purely irregular sequence $(x_{n})$ in $V$,
$\lim_{n\to\infty}w(x_{n})=\infty$.666 A sequence $(x_{n})$ in $V$ is _purely
irregular_ if it converges to a point $z\in\partial V$ and does not contain
any regular subsequence.

  * ($H_{5}$)

The set ${}^{\ast}{\mathcal{H}}^{+}(X)$ is linearly separating and there
exists $s_{0}\in{}^{\ast}{\mathcal{H}}^{+}(X)\cap\mathcal{C}(X)$, $s_{0}>0$,
such that the functions $H_{V}s_{0}$ are continuous on $V$.

In fact, property $(H_{4})$ may be considerably improved.

######  PROPOSITION 3.12.

Let $(X,\mathcal{W})$ be a balayage space and $U$ an open set in $X$. There
exists a continuous real _Evans function_ $w$ on $U$, that is,
$w\in\mathcal{S}^{+}(U)$ such that $\lim_{n\to\infty}w(x_{n})=\infty$ for
every purely irregular sequence $(x_{n})$ in $U$ converging to a boundary
point $z\in\partial U$.

###### Proof.

Let $p\in\mathcal{P}(X)$ be strict and let $f_{n}\in\mathcal{C}(X)$,
$n\in\mathbbm{N}$, such that $0\leq f_{n}\leq p$ on $X$, $f_{n}=p$ in a
neighborhood of ${U^{c}}$ and $f_{n}\downarrow 1_{U^{c}}p$. Then
$p_{n}:=R_{f_{n}}\in\mathcal{P}(X)$ and $p_{n}\downarrow H_{U}p$. Let
$(V_{n})$ be an exhaustion of $U$. Passing to a subsequence we may assume that
$w_{n}:=p_{n}-H_{U}p\leq 2^{-n}$ on $V_{n}$. Then

$w:=\sum\nolimits_{n=1}^{\infty}(p_{n}-H_{U}p)\in\mathcal{S}^{+}(U).$

It follows as in the proof of [21, Lemma 4.2.12] that $w$ is an Evans function
for $U$. ∎

Moreover, we have the following converse to Theorem 3.11 (see [4, III.6.11 and
III.4.4] or [21, Theorem 5.3.11, Propositions 5.2.8 and 5.3.12]).

######  THEOREM 3.13.

Let $(H_{U})_{U\in{\mathcal{U}}_{0}}$ be a family of kernels on $X$, where
${\mathcal{U}}_{0}\subset{\mathcal{U}}(X)$ is a base for the topology of $X$.
For open $W$ in $X$, let
${\mathcal{U}}_{0}(W):=\\{U\in{\mathcal{U}}_{0}\colon\overline{U}\subset W\\}$
and

${}^{\ast}{\mathcal{H}}_{0}^{+}(W):=\\{w\in\mathcal{B}^{+}(X)\colon\mbox{$w$
is l.s.c.\ on $W$, $H_{U}w\leq w$ for all $U\in{\mathcal{U}}_{0}(W)$}\\}.$

Suppose that $(H_{U})_{U\in{\mathcal{U}}_{0}}$ satisfies $(H_{0})$ – $(H_{5})$
with ${\mathcal{U}}_{0}$ in place of ${\mathcal{U}}$ and
${}^{\ast}{\mathcal{H}}_{0}^{+}$ in place of ${}^{\ast}{\mathcal{H}}^{+}$.

Then $(X,{}^{\ast}\\!{\mathcal{H}}_{0}^{+}(X))$ is a balayage space such that
$H_{U}$ is the corresponding harmonic kernel for every
$U\in{\mathcal{U}}_{0}$. The convex cones
${}^{\ast}\\!{\mathcal{H}}_{0}^{+}(W)$ do not change if we replace
${\mathcal{U}}_{0}$ by a smaller base for the topology of $X$.

By the results above, it is justified to call
$(H_{U})_{U\in{\mathcal{U}}_{0}}$ in Theorem 3.13 a _family of harmonic
kernels_.

A family $(H_{U})_{U\in{\mathcal{U}}_{0}}$ is a family of _regular_ harmonic
kernels if instead of $(H_{4})$ we have the following stronger property
(absence of purely irregular sequences):

  * $(H_{4}^{\prime})$

$H_{V}f$ is continuous at $\partial V$ for every $f\in\mathcal{C}(X)$ with
compact support.

Let us note that, for every balayage space $(X,\mathcal{W})$, the intersection
of any two regular sets is regular (see [4, VII.3.2.4]). Hence the following
lemma can be useful to show that a modification of harmonic kernels for a
balayage space leads to another balayage space (see [20]).

######  LEMMA 3.14.

Suppose that ${\mathcal{U}}_{0}\subset\mathcal{V}(X)$ is a base for the
topology which is stable under finite intersections. Then the statements of
Theorem 3.11 hold if $(H_{4})$ and $(H_{5})$ are replaced by
$(H_{4}^{\prime})$ and the following property:

  * ($H_{5}^{\prime}$)

There exists $s_{0}\in\mathcal{C}^{+}(X)$ such that, for all
$V\in{\mathcal{U}}_{0}$, $H_{V}s_{0}<s_{0}$ on $V$.

###### Proof.

Let us choose $s_{0}$ according to ($H_{5}^{\prime}$). Let $x,y\in X$, $x\neq
y$. Taking $V\in{\mathcal{U}}_{0}$ with $x\in V$, $y\notin V$, the functions
$H_{V}s_{0}$ and $s_{0}$ linearly separate $x$ and $y$. So it suffices to show
that $H_{V}s_{0}\in{}^{\ast}{\mathcal{H}}^{+}(X)$.

By ($H_{4}^{\prime}$), $H_{V}s_{0}$ is lower semicontinuous. Let
$U\in{\mathcal{U}}_{0}$. By ($H_{0}$), $H_{U}H_{V}s_{0}=H_{V}s_{0}$ on $U^{c}$
and $H_{U}H_{V}s_{0}\leq H_{U}s_{0}\leq s_{0}=H_{V}s_{0}$ on $V^{c}$. Since
$U^{c}\cup V^{c}=(U\cap V)^{c}$ and $U\cap V\in{\mathcal{U}}_{0}$ by
assumption, we conclude, by ($H_{0}$) and ($H_{2}$), that

$H_{U}H_{V}s_{0}=H_{U\cap V}H_{U}H_{V}s_{0}\leq H_{U\cap
V}H_{V}s_{0}=H_{V}s_{0}.$

Thus $H_{V}s_{0}\in{}^{\ast}{\mathcal{H}}^{+}(X)$. ∎

###  3.5 Simple example: The discrete case

Let $P$ be a sub-Markov kernel on a countable, discrete space $X$, with the
base of topology ${\mathcal{U}}_{0}:=\bigl{\\{}\\{x\\}\colon x\in X\\}$. We
define kernels $H_{\\{x\\}}$ by

$H_{\\{x\\}}(x,A):=\begin{cases}\dfrac{P(x,A\setminus\\{x\\})}{1-P(x,\\{x\\})},&P(x,\\{x\\})<1,\\\
0,&P(x,\\{x\\})=1,\end{cases}$

and $H_{\\{x\\}}(y,\cdot)=\delta_{y}$, if $x,y\in X$ and $x\neq y$ (see [4,
III.1.1.3]). Then

${}^{\ast}{\mathcal{H}}_{0}^{+}(X)=S_{P}:=\\{u\in\mathcal{B}^{+}(X)\colon
Pu\leq u\\}.$

Obviously $(H_{0})$ – $(H_{4}^{\prime})$ are satisfied. By Theorems 3.11, 3.13
and Lemma 3.14, we immediately get the following.

######  PROPOSITION 3.15.

The following statements are equivalent.

  * (1)

$(X,S_{P})$ is a balayage space.

  * (2)

$S_{P}$ separates the points of $X$.

  * (2)

There exists a real function $s_{0}\geq 0$ on $X$ such that $Ps_{0}<s_{0}$.

Of course, we could also use Theorem 3.2 and the (strong Feller) semigroup
$\mathbbm{P}:=(P_{t})_{t>0}$ given by

(3.16) $P_{t}:=e^{-t}\sum\nolimits_{k\geq 0}\frac{t^{k}}{k!}\,P^{k},\qquad
t>0.$

for which ${\mathcal{E}}_{\mathbbm{P}}=S_{P}$ and the potential kernel is
$\sum_{k\geq 0}P^{k}$ (see Remark 3.3).

###  3.6 Restriction on open subsets and lifting of potentials

We shall need the following (see [4, V.1.1]).

######  PROPOSITION 3.16.

Let $(X,\mathcal{W})$ be a balayage space, $U$ an open set in $X$ and
$\mathcal{W}_{U}:={}^{\ast}{\mathcal{H}}(U)^{+}|_{U}$. Then
$(U,\mathcal{W}_{U})$ is a balayage space called the _restriction of
$(X,\mathcal{W})$ on $U$_, with the corresponding continuous real potentials
$\mathcal{P}(U)$. The following holds.

  * (1)

$(H_{V}|_{U})_{V\in\mathcal{V}(U)}$ is an associated family of harmonic
kernels.

  * (2)

The $\mathcal{W}_{U}$-fine topology is the relative $W$-fine topology of $U$.

  * (3)

For every $p\in\mathcal{P}(X)$, $p-H_{U}p\in\mathcal{P}(U)$.

A partial converse to (3) in Proposition 3.16 is the following _lifting of
potentials_ , which will be important in our discussion of Green functions
later on, see Proposition 8.4 and Remark 8.5.

######  THEOREM 3.17.

Let $U$ be an open subset of $X$, $A$ compact in $U$ and $q\in\mathcal{P}(U)$,
harmonic on $U\setminus A$. Then there is a unique $p\in\mathcal{P}(X)$ (the
_lifting of $q$ on $X$_), harmonic on $X\setminus A$, such that $p-q$ is
harmonic on $U$, equivalently, $p-H_{U}p=q$.

This is essentially [20, Theorem 10.1], where unfortunately the last part is
not stated explicitly. Here is its (obvious) proof: Of course, $p-H_{U}p=q$
implies that $p-q=H_{U}p$ is harmonic on $U$. Suppose conversely that $p-q$ is
harmonic on $U$. Then $H_{V}(p-q)=p-q$ for every $V\in{\mathcal{U}}(U)$.
Exhausting $U$ by a sequence $(V_{n})\subset{\mathcal{U}}(U)$ we have (for
example, by [4, VI.1.2]; if $U$ is not relatively compact, we use in addition
that $\inf\\{R_{p}^{K^{c}}\colon K\mbox{ compact}\\}=0$)

$\lim\nolimits_{n\to\infty}H_{V_{n}}q=0\quad\mbox{ and
}\quad\lim\nolimits_{n\to\infty}H_{V_{n}}p=\lim\nolimits_{n\to\infty}R_{p}^{V_{n}^{c}}=R_{p}^{U^{c}}=H_{U}p.$

######  REMARK 3.18.

Let us note the following (for details see Remark 8.5): If $(X,\mathcal{W})$
has a Green function $G$, then $p=G\nu$ leads to $p-H_{U}p=G_{U}(1_{U}\nu)$,
$G_{U}$ being the associated Green function on $U$, and the lifting
$q=G_{U}\nu\in\mathcal{P}(U)$, with $\nu$ having compact support in $U$, is
$G\nu$.

##  4 Consequences of the domination principle

For the sake of generality (which is needed in [10]), let us assume in the
next two sections that $\varphi$ is a Borel measurable real function on
$X\times\mathbbm{R}$ such that the functions $t\mapsto\varphi(x,t)$, $x\in X$,
are continuous. We fix a potential $p\in\mathcal{P}(X)$ and are interested in
functions $u$ which, for a given function $v$, satisfy

(4.1) $u+K_{p}\varphi(\cdot,u)=v.$

To simplify our considerations we assume once and for all that

(4.2) $\varphi(\cdot,0)=0$

(if $\varphi$ does not satisfy (4.2) we may replace $\varphi$ by
$\varphi-\varphi(\cdot,0)$ and $v$ by $v-K_{p}\varphi(\cdot,0)$).

Then let us say that $\varphi$ is _sign-preserving_ if $t\varphi(\cdot,t)\geq
0$ for all $t\in\mathbbm{R}$. Of course, $\varphi$ is sign-preserving if
$\varphi$ is _increasing_ , that is, the functions $t\mapsto\varphi(x,t)$,
$x\in X$, are increasing.

The following consequences of the domination principle (see Lemma 3.5) will be
crucial for estimating functions $u$ satisfying (4.1), and some of them will
be needed to show that solutions of (4.1) obtained after a truncation of
$\varphi$ are, in fact, solutions of the original equation.

######  PROPOSITION 4.1.

[2, Lemma 3.3] Let $u,v,f\in\mathcal{B}_{r}(X)$ such that $K_{p}|f|<\infty$,
$u+K_{p}f=v$ and $\\{u<0\\}\subset\\{f\leq 0\\}$. Then $v\leq u+R_{v}$.777For
the definition of $R_{v}$ see (3.3).

###### Proof.

Let $w\in\mathcal{W}$, $w\geq v$. Obviously, $K_{p}f^{+}-K_{p}f^{-}=K_{p}f\leq
v\leq w$ on $\\{u\geq 0\\}$, hence on $\\{f>0\\}$. Thus $K_{p}f\leq w$ on $X$,
by (3.8), and $v=u+K_{p}f\leq u+w$. ∎

######  PROPOSITION 4.2.

Suppose that $\varphi$ is sign-preserving and let $u,v\in\mathcal{B}_{r}(X)$
such that $K_{p}|\varphi(\cdot,u)|<\infty$ and $u+K_{p}\varphi(\cdot,u)=v$.

  * (i)

Then $|u|\leq R_{v^{+}}+R_{v^{-}}$.

  * (ii)

If $w^{\prime},w^{\prime\prime}\in\mathcal{W}_{r}$ such that
$v=w^{\prime}-w^{\prime\prime}$, then $-w^{\prime\prime}\leq u\leq
w^{\prime}$.
In particular, $0\leq u\leq v$ if $v\in\mathcal{W}$, and $v\leq u\leq 0$ if
$v\in-\mathcal{W}$.

###### Proof.

We have $\\{u\leq 0\\}\subset\\{\varphi(\cdot,u)\leq 0\\}$ and $\\{-u\leq
0\\}\subset\\{-\varphi(\cdot,u)\leq 0\\}$. Hence, by Proposition 4.1,

(4.3) $v\leq u+R_{v}\quad\mbox{ and }\quad-v\leq-u+R_{-v}.$

(i) Trivially, $v\leq R_{v}=R_{v^{+}}$ and $-v\leq R_{-v}=R_{v^{-}}$. Thus
(4.3) implies that $-R_{v^{-}}\leq u+R_{v^{+}}$ and
$-R_{v^{+}}\leq-u+R_{v^{-}}$, that is, $|u|\leq R_{v^{+}}+R_{v^{-}}$.

(ii) Clearly, $R_{w^{\prime}-w^{\prime\prime}}\leq w^{\prime}$,
$R_{w^{\prime\prime}-w^{\prime}}\leq w^{\prime\prime}$. So, by (4.3),
$-w^{\prime\prime}\leq u$ and $-w^{\prime}\leq-u$. If $w^{\prime\prime}=0$,
then $0\leq u$ and $u=v-K_{p}\varphi(\cdot,u)\leq v$. If $w^{\prime}=0$, then
$u\leq 0$ and $u=v-K_{p}\varphi(\cdot,u)\geq v$. ∎

######  PROPOSITION 4.3.

Suppose that $\varphi$ is increasing. Then the following hold.

  1. (i)

Let $u_{j},v_{j}\in\mathcal{B}_{r}(X)$ such that
$K_{p}|\varphi(\cdot,u_{j})|<\infty$ and

$u_{j}+K_{p}\varphi(\cdot,u_{j})=v_{j},\qquad j=1,2.$

Then $v_{2}-v_{1}\leq u_{2}-u_{1}+R_{v_{2}-v_{1}}$. If
$v_{2}-v_{1}\in\mathcal{W}$, then $0\leq u_{2}-u_{1}\leq v_{2}-v_{1}$.
In particular, $u_{1}=u_{2}$ if $v_{1}=v_{2}$.

  2. (ii)

Let $\widetilde{\varphi}\in\mathcal{B}_{r}(X\times\mathbbm{R})$ with
$\widetilde{\varphi}\leq\varphi$, and suppose that
$u,\widetilde{u}\in\mathcal{B}_{r}(X)$ satisfy

$K_{p}(|\varphi(\cdot,u)|+|\widetilde{\varphi}(\cdot,\widetilde{u})|)<\infty\quad\mbox{
and }\quad
u+K_{p}\varphi(\cdot,u)=\widetilde{u}+K_{p}\widetilde{\varphi}(\cdot,\widetilde{u}).$

Then $u\leq\widetilde{u}$.

###### Proof.

(i) We define $u:=u_{2}-u_{1}$, $v:=v_{2}-v_{1}$ and
$f:=\varphi(\cdot,u_{2})-\varphi(\cdot,u_{1})$. Then $u+K_{p}f=v$ and
$\\{u\leq 0\\}\subset\\{f\leq 0\\}$. Therefore $v\leq u+R_{v}$, by Proposition
4.1. If $v\in\mathcal{W}$, then $R_{v}=v$, hence $0\leq u$, $f\geq 0$ and
$v-u=K_{p}f\geq 0$.

(ii) We have
$\widetilde{u}-u+K_{p}(\widetilde{\varphi}(\cdot,\widetilde{u})-\varphi(\cdot,u))=0$,
where

$\\{\widetilde{u}-u\leq
0\\}\subset\\{\varphi(\cdot,\widetilde{u})-\varphi(\cdot,u)\leq
0\\}\subset\\{\widetilde{\varphi}(\cdot,\widetilde{u})-\varphi(\cdot,u)\leq
0\\}.$

Thus $0\leq\widetilde{u}-u$, by Proposition 4.1. ∎

##  5 Application of Schauder’s theorem

Again let $\varphi\colon X\times\mathbbm{R}\to\mathbbm{R}$ be Borel measurable
such that, for every $x\in X$, the function $(x,t)\mapsto\varphi(x,t)$ is
continuous.

######  PROPOSITION 5.1.

Let $q\in\mathcal{P}_{b}(X)$ such that $K_{q}$ is a compact operator on
$\mathcal{B}_{b}(X)$ and suppose that $|\varphi|\leq c$ for some
$c\in(0,\infty)$. For every $u\in\mathcal{B}_{r}(X)$, let

(5.1) $K^{\varphi}u:=K_{q}\varphi(\cdot,u).$

  * (i)

For every $u\in\mathcal{B}_{r}(X)$,
$K^{\varphi}u\in\mathcal{P}_{b}(X)-\mathcal{P}_{b}(X)$ and $|K^{\varphi}u|\leq
cq$.

  * (ii)

Let $(u_{n})$ be a sequence in $\mathcal{B}_{r}(X)$. Then there exists a
subsequence $(u_{n_{k}})$ of $(u_{n})$ such that the sequence
$(K^{\varphi}u_{n_{k}})$ in $\mathcal{B}_{b}(X)$ converges uniformly. Further,
if $(u_{n})$ converges pointwise to $u\in\mathcal{B}_{r}(X)$, then
$(K^{\varphi}u_{n})$ converges uniformly to $K^{\varphi}u$.

  * (iii)

For every $v\in\mathcal{B}_{r}(X)$, there exists $u\in\mathcal{B}_{r}(X)$
(continuous if $v$ is continuous) such that $u+K^{\varphi}u=v$.

###### Proof.

(i) Obvious.

(ii) Since $|\varphi(\cdot,u_{n})|\leq c$ for all $n\in\mathbbm{N}$, the first
statement follows from the compactness of $K_{q}$. If
$\lim_{n\to\infty}u_{n}=u\in\mathcal{B}_{r}(X)$, then, by continuity of
$t\mapsto\varphi(\cdot,t)$, the functions $\varphi(\cdot,u_{n})$ converge
pointwise to $\varphi(\cdot,u)$, and hence the sequence $(K^{\varphi}u_{n})$
converges pointwise to $K^{\varphi}u$, by Lebesgue’s convergence theorem. A
standard argument on subsequences finishes the proof of (ii).

(iii) We fix $v\in\mathcal{B}_{r}(X)$ and consider the mapping
$S\colon\mathcal{B}_{b}(X)\to\mathcal{B}_{b}(X)$ defined by

$Su:=-K^{\varphi}(v+u).$

By (ii), $S$ is continuous and $S(\mathcal{B}_{b}(X))$ is relatively compact.
By Schauder’s fixed point theorem, there exists $u_{0}\in\mathcal{B}_{b}(X)$
such that $u_{0}=Su_{0}$, hence $u:=v+u_{0}$ satisfies
$u=v+Su_{0}=v-K^{\varphi}u$. If $v\in\mathcal{C}(X)$, then
$u\in\mathcal{C}(X)$, since $K^{\varphi}u\in\mathcal{C}(X)$. ∎

In this paper we shall need only a very simple consequence of the preceding
result. For a stronger version see [10, Theorem 6.2].

######  COROLLARY 5.2.

Let $p\in\mathcal{P}(X)$ and suppose that $|\varphi|$ is bounded and there
exists a compact $A$ in $X$ such that $\varphi(x,\cdot)=0$ if $x\in A^{c}$.
Then, for every $v\in\mathcal{B}_{r}(X)$, there exists
$u\in\mathcal{B}_{r}(X)$ (continuous if $v$ is continuous) such that

$K_{p}|\varphi(\cdot,u)|<\infty\quad\mbox{ and }\quad
u+K_{p}\varphi(\cdot,u)=v.$

###### Proof.

By Remarks 3.6,(3), we may assume that $p\leq 1$. Then
$q:=K_{p}1_{A}\in\mathcal{P}_{b}(X)$ and $C(q)\subset A$. Hence $K_{q}$ is a
compact operator on $B_{b}(X)$, by [22, Proposition 4.1]. Since clearly
$K_{p}\varphi^{\pm}(\cdot,u)=K_{q}\varphi^{\pm}(\cdot,u)$ for every
$u\in\mathcal{B}_{r}(X)$, an application of Proposition 5.1 completes the
proof. ∎

##  6 Semilinear equations for general balayage spaces

Let $(X,\mathcal{W})$ be a balayage space and let us fix an open set $U$ in
$X$. We shall identify functions on $U$ with functions on $X$ which vanish on
$X\setminus U$. Let $(V_{n})$ be an exhaustion of $U$ by open sets such that,
for every $n\in\mathbbm{N}$, the closure of $V_{n}$ is a compact in $V_{n+1}$.

Defining

$\mathcal{W}_{U}:={}^{\ast}\\!{\mathcal{H}}^{+}(U)|_{U}=\\{w\in{}^{\ast}\\!{\mathcal{H}}^{+}(U)\colon
w=0\mbox{ on }X\setminus U\\}$

we know that $(U,\mathcal{W}_{U})$ is a balayage space (and
$\mathcal{W}_{U}=\mathcal{W}$ if $U=X$, see Section 3.6). Let $p$ be a
continuous real potential on $U$ and let $K:=K_{p}$ be the associated
potential kernel (with respect to $(U,\mathcal{W}_{U})$).

Let us observe that, for every $f\in\mathcal{B}^{+}(U)$,

(6.1)
$\lim\nolimits_{n\to\infty}H_{V_{n}}Kf=\inf\nolimits_{n\in\mathbbm{N}}H_{V_{n}}Kf=0\qquad\mbox{
if }Kf<\infty.$

Indeed, suppose that $Kf<\infty$ and let $x\in U$. Then there exists $a>0$
such that $q^{\prime}:=K(f-f\wedge a)$ satisfies $q^{\prime}(x)<\varepsilon$.
Since $q^{\prime\prime}:=K(f\wedge a)$ is a continuous real potential on $U$,
there exists $n\in\mathbbm{N}$ such that
$H_{V_{n}}q^{\prime\prime}(x)<\varepsilon$ and hence

$H_{V_{n}}Kf(x)=H_{V_{n}}q^{\prime}(x)+H_{V_{n}}q^{\prime\prime}(x)\leq\varepsilon+q^{\prime\prime}(x)<2\varepsilon.$

Let $\Phi$ denote the set of all Borel measurable functions $\varphi\colon
U\times[0,\infty)\to[0,\infty)$ such that $\varphi(\cdot,0)=0$ and the
functions $t\mapsto\varphi(x,t)$, $x\in U$, are continuous and increasing. To
apply the results of Sections 4 and 5 we extend the functions $\varphi\in\Phi$
to real functions on $U\times\mathbbm{R}$ taking the value $0$ on
$(-\infty,0)$.

Given $A\in\mathcal{B}(U)$ and $\varphi\in\Phi$, let us define
$\varphi_{A}\in\Phi$ by

$\varphi_{A}(x,t):=1_{A}(x)\varphi(x,t),\qquad x\in U,\,t\in\mathbbm{R}.$

We define

(6.2)
$K^{\varphi}f:=K\varphi(\cdot,f|_{U}),\qquad\varphi\in\Phi,f\in\mathcal{B}_{r}^{+}(X).$

######  REMARK 6.1.

To apply the general results below to our standard examples it suffices to
choose a strictly positive (bounded) function $f_{0}\in\mathcal{B}^{+}(U)$
such that $G_{U}(f_{0}\mu)\in\mathcal{C}(U)$ (for this much less is needed
than $G_{U}(1_{A}\mu)\in\mathcal{C}(U)$ for every compact $A$ in $U$), take
$p:=G_{U}(f_{0}\mu)$ and replace $\varphi(x,t)$ by
$\widetilde{\varphi}(x,t):=f_{0}(x)^{-1}\varphi(x,t)$. Indeed, we then have
$K^{\widetilde{\varphi}}f=G_{U}(f\mu)$ for every $f\in\mathcal{B}^{+}(U)$.

In the following let $\varphi,\psi\in\Phi$ and $h\in{\mathcal{H}}^{+}(U)$
(vanishing outside of $U$ or not). As in our standard examples we immediately
get the following (see Proposition 2.2 and its proof).

######  PROPOSITION 6.2.

Let $u\in\mathcal{B}^{+}(U)$,
$u+K^{\varphi}u=s\in{}^{\ast}\mathcal{H}^{+}(U)\cap C(U)$, $\eta\in(0,1)$.
Then $A:=\\{u<\eta s\\}$ is finely open, $K^{\varphi_{U\setminus A}}(\eta
s)\leq s$ and
${}^{U}\\!\hat{R}_{s}^{A}={}^{U}\\!R_{s}^{A}\in\mathcal{P}^{\prime}(U)$.

For $\varphi\in\Phi$ and $n\in\mathbbm{N}$, we define $\varphi_{n}\in\Phi$ by

$\varphi_{n}(x,t):=1_{V_{n}}(x)\varphi(x,t)\wedge n=(\varphi_{V_{n}}\wedge
n)(x,t),\qquad x\in U,\,t\in\mathbbm{R}.$

Applying Corollary 5.2 and Proposition 4.3(ii) to the balayage space
$(U,\mathcal{W}_{U})$, we get a (unique) decreasing sequence
$(u_{n}^{\prime})$ in $\mathcal{C}^{+}(U)$ satisfying
$u_{n}^{\prime}+K^{\varphi_{n}}u_{n}^{\prime}=h|_{U}$ for every
$n\in\mathbbm{N}$. Taking $u_{n}:=u_{n}^{\prime}+1_{X\setminus U}h$ we obtain
that

(6.3) $u_{n}+K^{\varphi_{n}}u_{n}=h,\qquad n\in\mathbbm{N}.$

Let

(6.4)
$T^{\varphi}h:=\inf\nolimits_{n\in\mathbbm{N}}u_{n}=\lim\nolimits_{n\to\infty}u_{n}.$

By (6.3), the functions $u_{n}$ are subharmonic on $U$, hence the function
$T^{\varphi}h$ is subharmonic on $U$ as well and the sequence
$(H_{V_{n}}T^{\varphi})$ is increasing. So

(6.5)
$P^{\varphi}h:=\sup\nolimits_{n\in\mathbbm{N}}H_{V_{n}}T^{\varphi}h=\lim\nolimits_{n\to\infty}H_{V_{n}}T^{\varphi}h\leq
h$

is harmonic on $U$ and equal to $h$ on $X\setminus U$. Clearly, $P^{\varphi}h$
is the smallest majorant of $T^{\varphi}h$ which is harmonic on $U$.

######  PROPOSITION 6.3.

$T^{\varphi}h+K^{\varphi}T^{\varphi}h\leq P^{\varphi}h$, and $T^{\varphi}h$ is
the largest $f\in\mathcal{B}^{+}(X)$ such that $f\leq h$ and $f+K^{\varphi}f$
is subharmonic on $U$.

###### Proof.

(1) Let $u:=T^{\varphi}h$ and $v:=u+K^{\varphi}u$. The positive functions
$u_{n}+K^{\varphi_{n}}u=h-K^{\varphi_{n}}(u_{n}-u)$ are subharmonic on $U$,
bounded by $h$ and converge to the finely continuous function $v$. Hence
$v\leq h$ and $v$ is subharmonic on $U$. In particular,
$v\leq\lim_{n\to\infty}H_{V_{n}}v$, where
$\lim_{n\to\infty}H_{V_{n}}v=P^{\varphi}h$, by (6.5) and (6.1).

(2) Let $f\in\mathcal{B}^{+}(X)$, $f\leq h$, such that $f+K^{\varphi}f$ is
subharmonic on $U$, in particular, $q:=K^{\varphi}f<\infty$. Then $f+q\leq h$.

Indeed, $s:=h-(f+q)$ is superharmonic on $U$ and $s=(h-f)-q\geq-q$. Therefore
$s\geq\lim_{n\to\infty}H_{V_{n}}s\geq 0$ on $U$, by (6.1). Of course,
$s=h-f\geq 0$ on $X\setminus U$.

Moreover,
$s_{n}:=h-(f+K^{\varphi_{n}}f)=h-(f+K^{\varphi}f)+K^{\varphi-\varphi_{n}}f\geq
0$ for every $n\geq 0$, and $s_{n}$ is superharmonic on $U$. Since
$f+K^{\varphi_{n}}f+s_{n}=h=u_{n}+K^{\varphi_{n}}u_{n}$, we obtain that $f\leq
u_{n}$, by Proposition 4.3,(i). Thus $f\leq u$. ∎

######  LEMMA 6.4.

The mappings $(\varphi,h)\mapsto T^{\varphi}h$ and $(\varphi,h)\mapsto
P^{\varphi}h$ are decreasing in $\varphi$ and increasing in $h$.

###### Proof.

Suppose that $\varphi,\psi\in\Phi$ and $g,h\in{\mathcal{H}}^{+}(U)$, such that
$\varphi\leq\psi$ and $g\leq h$. Let
$u_{n}^{\prime},v_{n}^{\prime},w_{n}^{\prime}\in\mathcal{C}^{+}(U)$,
$n\in\mathbbm{N}$, such that

$u_{n}^{\prime}+K^{\varphi_{n}}u_{n}^{\prime}=h|_{U},\quad
v_{n}^{\prime}+K^{\varphi_{n}}v_{n}^{\prime}=g|_{U},\quad
w_{n}^{\prime}+K^{\psi_{n}}w_{n}^{\prime}=g|_{U}.$

By Proposition 4.3, $w_{n}^{\prime}\leq v_{n}^{\prime}\leq u_{n}^{\prime}$ for
every $n\in\mathbbm{N}$. Therefore $T^{\psi}g\leq T^{\varphi}g\leq
T^{\varphi}h$. Using (6.5) the proof is completed. ∎

######  DEFINITION 6.5.

We shall say that $\varphi$ is _$h$ -proper_ (_locally $h$-Kato_,
respectively) if $K^{\varphi_{A}}h<\infty$
($K^{\varphi_{A}}h\in\mathcal{C}(U)$, respectively) for every compact $A$ in
$U$.

######  PROPOSITION 6.6.

  * (1)

If $K^{\varphi}h<\infty$, then $P^{\varphi}h=h$.

  * (2)

If $\varphi$ is $h$-proper, then
$T^{\varphi}h+K^{\varphi}T^{\varphi}h=P^{\varphi}h$ and
$P^{\varphi}(P^{\varphi}h)=P^{\varphi}h$.

If $\varphi$ is even locally $h$-Kato, then $T^{\varphi}h$ is continuous on
$U$.

###### Proof.

Let $u_{n}$, $n\in\mathbbm{N}$, be as in (6.3) and $u:=T^{\varphi}h$,
$v:=u+K^{\varphi}u$.

(1) Obviously, (6.3) and dominated convergence imply that $u+K^{\varphi}u=h$.
Therefore $P^{\varphi}h=\lim_{n\to\infty}H_{V_{n}}u=h$, by (6.1).

(2) By Proposition 6.3, $v\leq P^{\varphi}h$. To prove the reverse inequality
it suffices to show that $v\in{\mathcal{H}}^{+}(U)$, since $P^{\varphi}h$ is
the smallest majorant of $u$ which is harmonic on $U$. Let
$m,n\in\mathbbm{N}$, $n\geq m$. Then

(6.6) $u_{n}+K(1_{V_{m}}\varphi(\cdot,u_{n})\wedge n)=h-K(1_{V_{n}\setminus
V_{m}}\varphi(\cdot,u_{n})\wedge n)\in{\mathcal{H}}^{+}(V_{m}),$

where $1_{V_{m}}\varphi(\cdot,u_{n})\wedge n\leq 1_{V_{m}}\varphi(\cdot,h)$.
If $\varphi$ is $h$-proper, the left side in (6.6) (which is majorized by $h$)
converges to

$v_{m}:=u+K(1_{V_{m}}\varphi(\cdot,u))$

as $n\to\infty$, and $v_{m}$ is harmonic on $V_{m}$. The sequence
$(v_{m})_{m\in\mathbbm{N}}$ is increasing to $v$. So $v$ is harmonic on $U$.

Further, applying Proposition 6.3 to $P^{\varphi}h$ instead of $h$ and $f:=u$,
we obtain that $u\leq T^{\varphi}(P^{\varphi}h)$, hence $P^{\varphi}h\leq
P^{\varphi}(P^{\varphi}h)$, where $P^{\varphi}(P^{\varphi}h)\leq
P^{\varphi}h$, by Lemma 6.4.

Finally, suppose that $\varphi$ is $h$-Kato and let $m\in\mathbbm{N}$. Then
$q_{m}:=K(1_{V_{m}}\varphi(\cdot,u))$ and hence $u=v_{m}-q_{m}$ is continuous
on $V_{m}$. ∎

######  COROLLARY 6.7.

If $\varphi$ is $h$-proper, the following statements are equivalent:

  * (1)

$P^{\varphi}h=h$.

  * (2)

$T^{\varphi}h+K^{\varphi}T^{\varphi}h=h$.

  * (3)

There exists $u\in\mathcal{B}^{+}(X)$ such that $u+K^{\varphi}u=h$.

###### Proof.

By Proposition 6.6,(3), it remains to show that (3) implies (1). If (3) holds,
then $u\leq T^{\varphi}h$, by Proposition 6.6,(2), and therefore $h\leq
T^{\varphi}h+K^{\varphi}T^{\varphi}h=P^{\varphi}h$. ∎

We consider _reduced functions_ for $(U,\mathcal{W}(U))$, see (3.4).

######  DEFINITION 6.8.

Given $A\subset U$, let

${}^{U}\\!R_{h}^{A}:=\inf\\{w\in\mathcal{W}_{U}\colon w\geq h\mbox{ on
}A\\},\qquad{}^{U}\\!\hat{R}_{h}^{A}(x):=\liminf\nolimits_{y\to
x}{}^{U}\\!R_{h}^{A}(y).$

######  LEMMA 6.9.

Let $A\in\mathcal{B}(U)$. Then $T^{\varphi_{A}}h\geq
h-{}^{U}\\!\hat{R}_{h}^{A}$. In particular, $P^{\varphi_{A}}h\not\equiv 0$ on
$U$ if ${}^{U}\\!\hat{R}_{h}^{A}\not\equiv h$ on $U$, and $P^{\varphi_{A}}h=h$
if ${}^{U}\\!\hat{R}_{h}^{A}\in\mathcal{P}^{\prime}(U)$.

###### Proof.

Let $\psi:=\varphi_{A}$ and $u_{n}\in\mathcal{B}^{+}(X)$ such that
$u_{n}+K^{\psi_{n}}u_{n}=h$, $n\in\mathbbm{N}$. Let $w\in\mathcal{W}_{U}$ such
that $w\geq h$ on $A$. Then $p_{n}:=K^{\psi_{n}}u_{n}\leq h\leq w$ on $A$, and
hence $p_{n}\leq w$ on $U$, by the domination principle. So $u_{n}=h-p_{n}\geq
h-w$ for every $n\in\mathbbm{N}$. Therefore $T^{\psi}h\geq
h-{}^{U}\\!R_{h}^{A}$. Since $T^{\psi}h$ is finely continous on $U$, we obtain
that even $T^{\psi}h\geq h-{}^{U}\\!\hat{R}_{h}^{A}$.

If ${}^{U}\\!\hat{R}_{h}^{A}\not\equiv h$ on $U$, then, of course,
$P^{\varphi_{A}}h\not\equiv h$ on $U$, since $P^{\varphi_{A}}h\geq
T^{\varphi_{A}}h$. If ${}^{U}\\!\hat{R}_{h}^{A}\in\mathcal{P}^{\prime}(U)$,
then $\lim_{n\to\infty}H_{V_{n}}{}^{U}\\!\hat{R}_{h}^{A}=0$ whence
$P^{\psi}h\geq h$, by (6.5). ∎

Here is a remarkable relation between the perturbations by $\varphi$, $\psi$
and $\varphi+\psi$.

######  PROPOSITION 6.10.

$T^{\varphi}+T^{\psi}\leq I+T^{\varphi+\psi}$ und $P^{\varphi}+P^{\psi}\leq
I+P^{\varphi+\psi}$.
If $P^{\varphi}h=h$, then $P^{\varphi+\psi}h=P^{\psi}h$. In particular,
$P^{\varphi+\psi}h=h$, if $P^{\varphi}h=P^{\psi}h=h.$

###### Proof.

Let $h\in{\mathcal{H}}^{+}(U)$ and $u_{n},v_{n},w_{n}\in\mathcal{C}^{+}(U)$,
$n\in\mathbbm{N}$, such that

$u_{n}+K^{\varphi_{n}}u_{n}=h,\quad v_{n}+K^{\psi_{n}}u_{n}=h,\quad
w_{n}+K^{(\varphi+\psi)_{n}}w_{n}=h.$

Obviously, $(\varphi+\psi)_{n}\leq\varphi_{n}+\psi_{n}$. Hence, by Proposition
4.3,(2), $w_{n}\leq u_{n}\wedge v_{n}$ and

$\displaystyle(h-u_{n})+(h-v_{n})$ $\displaystyle=$ $\displaystyle
K^{\varphi_{n}}u_{n}+K^{\psi_{n}}v_{n}$ $\displaystyle\geq$ $\displaystyle
K^{\varphi_{n}+\psi_{n}}w_{n}\geq K^{(\varphi+\psi)_{n}}w_{n}=h-w_{n}.$

Consequently, $(h-T^{\varphi}h)+(h-T^{\psi}h)\geq h-T^{\varphi+\psi}h$, that
is, $T^{\varphi}h+T^{\psi}h\leq h+T^{\varphi+\psi}h$. This clearly implies
$P^{\varphi}h+P^{\psi}h\leq h+P^{\varphi+\psi}h$ and the additional
statements. ∎

The next consequence will not be used, but certainly is of independent
interest.

######  COROLLARY 6.11.

If $\varphi$ is $h$-proper, then $P^{c\varphi}=P^{\varphi}$ for every $c>0$.

###### Proof.

By Proposition 6.6,(3) and Proposition 6.10,

$2P^{\varphi}h=2P^{\varphi}(P^{\varphi}h)\leq
P^{\varphi}h+P^{2\varphi}(P^{\varphi}h),$

hence $P^{\varphi}h\leq P^{2\varphi}(P^{\varphi}h)\leq P^{2\varphi}$. The
proof is completed using Lemma 6.4. ∎

The following consequence, however, is crucial for us.

######  PROPOSITION 6.12.

Let $A\in\mathcal{B}(U)$ such that $K^{\varphi_{U\setminus A}}h<\infty$. If
${}^{U}\\!\hat{R}_{h}^{A}\in\mathcal{P}^{\prime}(U)$, then $P^{\varphi}h=h$.
If ${}^{U}\\!\hat{R}_{h}^{A}\not\equiv h$ on $U$, then $P^{\varphi}h\not\equiv
0$ on $U$.

###### Proof.

Clearly, $\varphi=\varphi_{U\setminus A}+\varphi_{A}$, where
$P^{\varphi_{U\setminus A}}h=h$, by Proposition 6.6,(1). The proof is
completed by Lemma 6.9. ∎

To get to Theorem 6.14, the main result of this section, we only need the
following additional simple observation.

######  LEMMA 6.13.

Let $u_{n},v_{n}\in\mathcal{B}^{+}(U)$, $u_{n}+K^{\varphi}u_{n}=v_{n}$ and
$v_{n+1}-v_{n}\in\mathcal{W}_{U}$, $n\in\mathbbm{N}$. Then the sequence
$(u_{n})$ is increasing.

Let $v:=\lim_{n\to\infty}v_{n}$. If $u:=\lim_{n\to\infty}u_{n}<\infty$, then
$u+K^{\varphi}u=v$. Moreover, $u\in\mathcal{C}(X)$ if $v\in\mathcal{C}(X)$ and
$u_{n}\in\mathcal{C}(X)$ for every $n\in\mathbbm{N}$.

###### Proof.

By [4, Proposition 4.3,(ii)], the sequence $(u_{n})$ is increasing. Supposing
that $u<\infty$, we get $u+K^{\varphi}u=v$, where of course $K^{\varphi}u$ is
lower semicontinuous. If $u_{n}\in\mathcal{C}(X)$, $n\in\mathbbm{N}$, we get
that $u$ is lower semicontinuous as well. Then the continuity of their sum $v$
implies that both $u$ and $K^{\varphi}u$ are continuous. ∎

######  THEOREM 6.14.

If $\varphi$ is $h$-proper, the following statements are equivalent.

  * (1)

There exists $u\in\mathcal{B}^{+}(X)$ (continuous on $U$ if $\varphi$ is
locally $h$-Kato) such that

$u+K^{\varphi}u=h.$

  * (2)

$P^{\varphi}h=h$.

  * (3)

For every $\eta\in(0,1)$, there exists a finely open set $A\in\mathcal{B}(U)$
such that

$K\varphi_{U\setminus A}(\cdot,\eta h)\leq h\quad\mbox{ and
}\quad{}^{U}\\!R_{h}^{A}\in\mathcal{P}^{\prime}(U).$

  * (4)

For every $\eta\in(0,1)$, there exists a set $A\in\mathcal{B}(U)$ such that

$K\varphi_{U\setminus A}(\cdot,\eta h)<\infty\quad\mbox{ and
}\quad{}^{U}\\!\hat{R}_{h}^{A}\in\mathcal{P}^{\prime}(U).$

###### Proof.

(1) $\Leftrightarrow$ (2): Corollary 6.7.

(1) $\Rightarrow$ (3): Proposition 6.2.

(3) $\Rightarrow$ (4): Trivial.

(4) $\Rightarrow$ (2): Let us note that ${}^{U}\\!\hat{R}_{\eta
h}^{A}=\eta{}^{U}\\!\hat{R}_{h}^{A}$, $\eta\geq 0$, and fix
$\eta_{n}\in(0,1)$, $\eta_{n}\uparrow 1$. Propositions 6.6,(3) and 6.12 imply
that the functions $u_{n}:=T^{\varphi}(\eta_{n}h)$, $n\in\mathbbm{N}$, satisfy
$u_{n}+K^{\varphi}u_{n}=\eta_{n}h$. By Lemma 6.4, the sequence $(u_{n})$ is
increasing. By Lemma 6.13, $u:=\lim_{n\to\infty}u_{n}$ satisfies
$u+K^{\varphi}u=h$. ∎

######  REMARK 6.15.

Let us note that (4) is weaker than the following property.

  * (4′)

There exists a set $A\in\mathcal{B}(U)$ such that

$K\varphi_{U\setminus A}(\cdot,h)<\infty\quad\mbox{ and
}\quad{}^{U}\\!\hat{R}_{h}^{A}\in\mathcal{P}^{\prime}(U).$

Provided $\varphi(\cdot,h)\leq c\varphi(\cdot,\eta h)$ for some $\eta\in(0,1)$
it is equivalent to $(4)$ and we do not need Lemma 6.13. This is the case if,
for example, $\varphi(x,t)=r(x)t^{\gamma}$, $\gamma>0$, or
$\varphi(x,t)=r(x)f(t)$ with $f(0)=0$, $f>0$ on $(0,\infty)$ (continuous
increasing) and $a\leq h\leq b$ on $U$ for some $a,b\in(0,\infty)$.

Finally, let us suppose that $h\not\equiv 0$ on $U$ and define

$\mathcal{G}_{h}(U):=\\{g\in{\mathcal{H}}^{+}(U)\colon g\leq h\mbox{ on }U,\
g=h\mbox{ on }X\setminus U,\ g\not\equiv 0\mbox{ on }U\\}.$

If $\varphi$ is $h$-proper (locally $h$-Kato, respectively), then $\varphi$ is
$g$-proper (locally $g$-Kato, respectively) for every $g\in\mathcal{G}_{h}$.
Therefore the equivalences of Theorem 6.14 hold for each
$g\in\mathcal{G}_{h}$. Moreover, we get the following.

######  THEOREM 6.16.

Assume $\varphi$ is $h$-proper and consider the following two properties.

  * (1)

There exists $g\in\mathcal{G}_{h}(U)$ such that, for every $\eta\in(0,1)$,
there is a set $A\in\mathcal{B}(U)$ with $K\varphi_{U\setminus A}(\cdot,\eta
g)<\infty$ and ${}^{U}\\!R_{g}^{A}\not\equiv g|_{U}$.

  * (2)

There exist $g\in\mathcal{G}_{h}(U)$ and $u\in\mathcal{B}^{+}(X)$ (continuous
on $U$ if $\varphi$ is locally $h$-Kato) such that $u+K^{\varphi}u=g$.

Then $(1)$ implies $(2)$. If $1_{U}h$ is harmonic on $U$, then $(1)$ and $(2)$
are equivalent.

###### Proof.

(1) $\Rightarrow$ (2): Let $\eta_{n}\in(0,1)$ such that $\eta_{n}\uparrow 1$.
By Proposition 6.6,(2), the functions $u_{n}:=T^{\varphi}(\eta_{n}g)$,
$n\in\mathbbm{N}$, satisfy
$u_{n}+K^{\varphi}u_{n}=P^{\varphi}(\eta_{n}g)\in{\mathcal{H}}^{+}(U)$, where
$P^{\varphi}(\eta_{n}g)\not\equiv 0$ on $U$, by Proposition 6.12. By Lemma
6.4, the sequences $(u_{n})$ and $(P^{\varphi}(\eta_{n}g))$ are increasing. By
Lemma 6.13, $u:=\lim_{n\to\infty}u_{n}$ satisfies

$u+K^{\varphi}u=\lim\nolimits_{n\to\infty}P^{\varphi}(\eta_{n}g)=:\widetilde{g},$

where $\widetilde{g}$ is harmonic on $U$, $\widetilde{g}\not\equiv 0$ on $U$
and $\widetilde{g}=h$ on $X\setminus U$, hence
$\widetilde{g}\in\mathcal{G}_{h}$.

Suppose now that $1_{U}h$ is harmonic on $U$ and that $(2)$ holds. Since $g=h$
on $X\setminus U$, we see that also $1_{U}g=g-1_{X\setminus U}h=g-(h-1_{U}h)$
is harmonic on $U$. Let $\eta\in(0,1)$. By Proposition 6.2, there exists
$A\in\mathcal{B}(U)$ with $K\varphi_{U\setminus A}(\cdot,\eta g)<\infty$ and
${}^{U}\\!\hat{R}_{g}^{A}\in\mathcal{P}^{\prime}(U)$. In particular,
${}^{U}\\!\hat{R}_{g}^{A}$ cannot be equal to the function
$g|_{U}\in{\mathcal{H}}^{+}(U)\setminus\\{0\\}$. ∎

##  7 Cases, where $1_{U}h$ is not harmonic on $U$

As before, let $h\in{\mathcal{H}}^{+}(U)$. If $h=0$ on $X\setminus U$ or the
harmonic measures $H_{V}(x,\cdot)$, $x\in V$, are supported by the boundary
$\partial V$ for every $V\in\mathcal{V}(U)$ (as in our standard examples (i)
and (iii)), then $h|_{U}=1_{U}h$ is harmonic on $U$. However, in our example
(ii) (Riesz potentials) the constant $1$ is harmonic on $U$, but $1_{U}$ is
not harmonic on $U$ if $X\setminus U$ has strictly positive Lebesgue measure.

In our situation of an open set $U$ in a general balayage space
$(X,\mathcal{W})$, we can state the following. Let $h\in{\mathcal{H}}^{+}(U)$
such that $1_{U}h$ is not harmonic on $U$. Then $u_{0}:=1_{X\setminus
U}h=h-1_{U}h$ is not harmonic on $U$. So there exists $m\in\mathbbm{N}$ and
$x\in U$ such $H_{V_{m}}u_{0}(x)>0$. But, of course, $u_{0}$ is subharmonic on
$U$, and hence the sequence $(H_{V_{n}}u_{0})$ is increasing to a minorant
$h_{0}$ of $h$ which is harmonic on $U$ and satisfies $h_{0}(x)\geq
H_{V_{m}}u_{0}(x)>0$. Therefore $h_{0}\in\mathcal{G}_{h}$,

$\mathcal{G}_{h}=\\{g\in{\mathcal{H}}^{+}(U)\colon h_{0}\leq g\leq h\\},$
(7.1) $h_{0}=h\quad\mbox{ if and only if }\quad 1_{U}h\mbox{ is a potential on
}U.$

Our definition of $P^{\varphi}h$ immediately leads to

$P^{\varphi}h\in\mathcal{G}_{h}\qquad\mbox{ for every }\varphi\in\Phi.$

Moreover, $h-h_{0}$ is the largest minorant of $h$ in ${\mathcal{H}}^{+}(U)$
which vanishes on $X\setminus U$.

By Proposition 6.6, we get the following.

######  THEOREM 7.1.

Suppose that $\varphi$ is $h$-proper and $1_{U}h$ is _not_ harmonic on $U$.

Then $T^{\varphi}h+K^{\varphi}T^{\varphi}h=P^{\varphi}h\in\mathcal{G}_{h}$,
$u:=T^{\varphi}h$ answers Problem 2, and it answers Problem 1 if $1_{U}h$ is a
potential on $U$.

So, in our standard case (ii) (Riesz potentials), the answer to Problem 2 is
“Yes” if the Lebesgue measure of $\\{x\in X\setminus U\colon h(x)>0\\}$ is
strictly positive. Moreover, using the next result and (7.1) we get further
criteria for a positive answer to Problem 1 in the general case.

######  PROPOSITION 7.2.

Suppose that $H_{U}(x,\partial U)=0$ for every $x\in U$, let
$f\in\mathcal{B}^{+}(X)$ such that $h:=H_{U}f\in{\mathcal{H}}^{+}(U)$.
Moreover, let us assume that

  * (i)

there exists a strictly positive harmonic function on $X$ or

  * (ii)

$U$ is relatively compact.

Then $1_{U}h$ is a potential on $U$.

###### Proof.

Using an $s$-transformation (see [10]) we may assume without loss of
generality that $1\in\mathcal{W}$ (even $1\in{\mathcal{H}}(U)$ if (i) holds).

Let us first consider $h:=H_{U}1$ and fix $x\in U$. By [4, VI.1.9, 10.2], the
measures $\nu_{n}:=H_{V_{n}}(x,\cdot)$, $n\in\mathbbm{N}$, converge weakly to
$\nu:=H_{U}(x,\cdot)$ as $n\to\infty$. To prove that $h|_{U}$ is a potential
on $U$ we have to show that $\lim_{n\to\infty}\nu_{n}(1_{U}h)=0$.

Let us first consider that case (i) so that $h=1$ and the measures $\nu_{n}$
and $\nu$ are probability measures. Of course, $\nu(U)=0$ and, by assumption,
also $\nu(\partial U)=0$. Given $\varepsilon>0$, there hence exists a function
$v\in\mathcal{C}^{+}(X)$ with compact support in $X\setminus\overline{U}$ such
that $v\leq 1$ and $\nu(v)>1-\varepsilon$. There exists $m\in\mathbbm{N}$ such
that, for every $n\geq m$, $\nu_{n}(v)>\nu(v)-\varepsilon$, and therefore
$\nu_{n}(U)\leq 1-\nu_{n}(v)<2\varepsilon$.

Let us now suppose that $U$ is relatively compact. We know that $h\leq 1$. By
[4, VI.9.4], the sequence $(1_{U^{c}}\nu_{n})$ is increasing. Hence also the
measures $\nu_{n}^{\prime}:=1_{U}\nu_{n}$ (which are supported by the compact
$\overline{U}\setminus V_{n}$) converge weakly. Their limit $\nu^{\prime}$ is
a measure on $\partial U$ which, of course, satisfies $\nu^{\prime}\leq\nu$.
Therefore $\nu^{\prime}=0$, and hence
$\lim_{n\to\infty}\nu_{n}^{\prime}(1)=0$,
$\lim_{n\to\infty}\nu_{n}(1_{U}h)=0$.

Finally, let $f\in\mathcal{B}^{+}(X)$. If $f$ is bounded, then obviously
$1_{U}H_{U}f$, being bounded by a multiple of $1_{U}H_{U}1$, is a potential on
$U$. In fact, it is now easily seen that this holds as long as
$H_{U}f\in{\mathcal{H}}^{+}(U)$ on $U$ (approximation of $f$ by $f\wedge n$,
$n\in\mathbbm{N}$). ∎

##  8 Application to balayage spaces admitting
a Green function

It should be more or less clear from Section 2 how to apply our results to
situations which are more general than our standard examples. To be precise,
let us consider again a general balayage space $(X,\mathcal{W})$.

######  DEFINITION 8.1.

In the following we say that a function $G\in\mathcal{B}^{+}(X\times X)$ is a
_Green function_ for $(X,\mathcal{W})$, if the functions $G(\cdot,y)$, $y\in
X$, are non-vanishing potentials on $X$ which are harmonic on
$X\setminus\\{y\\}$.

######  LEMMA 8.2.

Let $y\in X$ and $g\in{\mathcal{H}}^{+}(X\setminus\\{y\\})$.

  * (1)

Then $g\in\mathcal{W}$ if and only if $g$ is l.s.c. at $y$ and $H_{V}g(y)\leq
g(y)$ for a fundamental system of neighborhoods $V$ of $y$.

  * (2)

Moreover, $g\in\mathcal{P}^{\prime}(X)$, if $g\in\mathcal{W}$ and
$H_{W}g\downarrow 0$ locally uniformly as $W\in{\mathcal{U}}(X)$ increases to
$X$.

###### Proof.

(1) Immediate consequence of [4, III.4.4].

(2) It remains to show that $(H_{U}g)|_{U}\in\mathcal{C}(U)$ for every
$U\in{\mathcal{U}}(X)$. This holds trivially if $y\notin\overline{U}$, since
then $H_{U}g=g$. Hence, by Lemma 3.10, it suffices to consider the case, where
$y\in U\in\mathcal{V}(U)$.

Fixing $\varepsilon>0$ we may choose a neighborhood $W\in{\mathcal{U}}(X)$ of
$\overline{U}$ such that $H_{W}g<\varepsilon$ on $\overline{U}$, and hence
$w:=H_{U}(1_{W^{c}}g)<\varepsilon$ on $\overline{U}$ (see, for example, [4,
VI.9.4]). Further, we choose $\varphi\in\mathcal{C}^{+}(X)$ with compact
support such that $\varphi\leq g$ on $X$ and $\varphi=g$ on $W\setminus U$.
Since $H_{U}\varphi\in{\mathcal{H}}^{+}(U)$, we obtain that
$h:=H_{U}(1_{W\setminus U}g)\in{\mathcal{H}}^{+}(U)$. Having $H_{U}g=h+w$ and
$0\leq w<\varepsilon$ on $U$, we finally conclude that
$(H_{U}g)_{U}\in\mathcal{C}(U)$. ∎

The equation in the following result is often called _Hunt’s formula_.

######  PROPOSITION 8.3.

If $G$ is a Green function for $(X,\mathcal{W})$, then, for every open $U$ in
$X$, the function $G_{U}\in\mathcal{B}^{+}(U\times U)$, defined by

(8.1) $G_{U}(x,y):=G(x,y)-(H_{U}G(\cdot,y))(x),\qquad x,y\in U,$

is a Green function for $(U,\mathcal{W}(U))$.

###### Proof.

Let $y\in U$ and $g:=G(\cdot,y)$. By Lemma 3.10 and (3.7),
$H_{U}g\in{\mathcal{H}}^{+}(U)$. Hence $G_{U}(\cdot,y)$ is superharmonic on
$U$ and harmonic on $U\setminus\\{y\\}$. It does not vanish, since otherwise
$g$ would be harmonic on $X$.

Let $z\in U$ and $\varepsilon>0$. We first choose $V\in{\mathcal{U}}(X)$ with
$R_{g}^{V^{c}}(z)=H_{V}g(z)<\varepsilon/2$. By Remark 3.4, there exists
$W\in{\mathcal{U}}(U\cap V)$ such that $R_{g}^{W^{c}}(z)<R_{g}^{(U\cap
V)^{c}}(z)+\varepsilon/2$. Trivially, $R_{g}^{(U\cap V)^{c}}\leq
R_{g}^{U^{c}}+R_{g}^{V^{c}}$. So
$R_{g}^{W^{c}}(z)<R_{g}^{U^{c}}(z)+\varepsilon$, that is,
$H_{W}g(z)<H_{U}g(x)+\varepsilon$. Since $H_{W}H_{U}=H_{U}$, we see that

$H_{W}(G_{U}(\cdot,y))(z)=H_{W}g(z)-H_{U}g(z)<\varepsilon.$

Thus $G_{U}(\cdot,y)$ is a potential on $U$. ∎

Given a Green function $G$ for $X$, let $\mathcal{M}_{C}^{+}(X)$ be the set of
all positive Radon measures $\nu$ on $X$ such that

(8.2) $G\nu:=\int G(\cdot,y)\,d\nu(y)\in\mathcal{C}(X).$

By Fubini’s theorem, we immediately see that, for every measure
$\nu\in\mathcal{M}_{C}^{+}(U)$,

$G_{U}\nu\in{\mathcal{H}}(U\setminus\operatorname*{supp}(\nu))\cap\mathcal{P}(U),$

and hence $f\mapsto G_{U}(f\nu)$ is the potential kernel $K_{p}$ for
$p:=G_{U}\nu$. Note that $\nu$ in (8.2) is uniquely determined by the
potential $G\nu$ (see [24, Proposition 5.2]).

In many cases, $\mathcal{P}(X)=\\{G\nu\colon\nu\in\mathcal{M}_{C}^{+}(X)\\}$,
for example, if the functions $G(x,\cdot)$, $x\in X$, are continuous on
$X\setminus\\{x\\}$, continuous on $X$ if $x$ is finely isolated, and there
exists $\nu\in\mathcal{M}_{C}^{+}(X)$ charging every non-empty finely open set
(see [24]). Let us note the following.

######  PROPOSITION 8.4.

If $G$ is a Green function for $(X,\mathcal{W})$ such that
$\mathcal{P}(X)=\\{G\nu\colon\nu\in\mathcal{M}_{C}^{+}(X)\\}$, then
$\mathcal{P}(U)=\\{G_{U}\nu\colon\nu\in\mathcal{M}_{C}^{+}(U)\\}$ for every
open $U$ in $X$.

###### Proof.

Let $p\in\mathcal{P}(U)$. There exists a sequence $(p_{n})$ in
$\mathcal{P}(U)$ such that $p=\sum_{n=1}^{\infty}p_{n}$ and every $p_{n}$,
$n\in\mathbbm{N}$, has a compact support in $U$. By Theorem 3.17, there exist
$q_{n}\in\mathcal{P}(X)$ such that $q_{n}-H_{U}q_{n}=p_{n}$. Assuming that
there are $\nu_{n}\in\mathcal{M}_{C}^{+}(X)$ such that $G\nu_{n}=q_{n}$,
$n\in\mathbbm{N}$, we have $\nu_{n}(X\setminus U)=0$ and get
$\nu:=\sum_{n=1}^{\infty}\nu_{n}\in\mathcal{M}_{C}(U)$ with $G_{U}\nu=p$. ∎

######  REMARK 8.5.

Let $G$ be a Green function on $X$,
$\mathcal{P}(X)=\\{G\nu\colon\nu\in\mathcal{M}_{C}^{+}(X)\\}$.

(1) If $p=G\nu\in\mathcal{P}(X)$, then clearly (8.1) and
$H_{U}G(1_{U^{c}}\nu)=G(1_{U^{c}}\nu)$ imply that $p-H_{U}p=G_{U}(1_{U}\nu)$.

(2) Consider now $q\in\mathcal{P}(U)$ which is harmonic outside a compact $A$
in $U$. By Proposition 8.4, $q=G_{U}\nu$ for some
$\nu\in\mathcal{M}_{C}^{+}(U)$ which is supported by $A$. By Proposition 3.17,
we have a lifting $p\in\mathcal{P}(X)$ of $q$. By our assumption on $G$ there
exists $\widetilde{\nu}\in\mathcal{M}^{+}(X)$ such that $p=G\widetilde{\nu}$.
Of course, $\widetilde{\nu}$ is supported by $A$, and we conclude from (1)
that $q=p-H_{U}p=G_{U}\widetilde{\nu}$. By uniqueness, $\widetilde{\nu}=\nu$.

In the remainder of this section we assume that $U$ is an open set in $X$ and
that we have a Green function $G_{U}$ for $(U,\mathcal{W}(U))$ such that

(8.3) $\mathcal{P}(U)=\\{G_{U}\nu\colon\nu\in\mathcal{M}_{C}^{+}(U)\\}.$

Further, suppose that we have an additive mapping
$L\colon\mathcal{P}(U)+{\mathcal{H}}^{+}(U)\to\mathcal{M}_{C}(U)$ satisfying

  * (1)

$LG_{U}\nu=-\nu$ on $U$ for every $\nu\in\mathcal{M}_{C}(U)$,

  * (2)

$Lg=0$ on $U$ for every $g\in{\mathcal{H}}^{+}(U)$.

We observe that we could _define_ $L$ by (1) and (2), since
$\mathcal{P}(U)+{\mathcal{H}}^{+}(U)$ is a direct sum by (3.14) and, by [24,
Proposition 5.2], the measure $\nu$ is uniquely determined by the potential
$G\nu$.

Let us fix a Borel measurable function $\varphi\colon
U\times[0,\infty)\to[0,\infty)$ and $h\in{\mathcal{H}}^{+}(U)$ such that the
functions $(x,t)\to\varphi(x,t)$, $x\in U$, are continuous, increasing and
$\varphi$ is locally $h$-Kato (with respect to $\mu$), that is,
$G_{U}(1_{A}\varphi(\cdot,h)\mu)\in\mathcal{C}(U)$. We now get the following
(cf. Proposition 2.1).

######  PROPOSITION 8.6.

For every $u\in\mathcal{B}^{+}(X)$ the following properties are equivalent:

  * (1)

$Lu=\varphi(\cdot,u)\mu$ on $U$ and $h-u\in\mathcal{P}(U)$.

  * (2)

$u+G_{U}(\varphi(\cdot,u)\mu)=h$.

The function $u\in\mathcal{B}^{+}(X)$ having these properties is uniquely
determined by $h$.

Thus the results of Section 6 yield the following.

######  THEOREM 8.7.

The equivalences of Theorems 2.3 and 2.5 hold in the present setting (where we
may replace $h-u\in\mathcal{P}^{\prime}(U)$, $g-u\in\mathcal{P}^{\prime}(U)$
by $h-u\in\mathcal{P}(U)$, $g-u\in\mathcal{P}(U)$, respectively).

##  9 Appendix

###  9.1 Products of sub-Markov semigroups

Let $\mathbbm{P}:=(P_{t})_{t>0}$ be a sub-Markov semigroup on $X$, and let
$\widetilde{\mathbbm{P}}:=(\widetilde{P}_{t})_{t>0}$ be a sub-Markov semigroup
on $\widetilde{X}$ ($\widetilde{X}$ being a locally compact space with
countable base). We define a sub-Markov semigroup
$\mathbbm{P}\otimes\widetilde{\mathbbm{P}}=(P_{t}\otimes\widetilde{P}_{t})_{t>0}$
on $X\times\widetilde{X}$ by

$(P_{t}\otimes\widetilde{P}_{t})f(x,\widetilde{x}):=\int\int
f(y,\widetilde{y})\,P_{t}(x,dy)\,\widetilde{P}_{t}(\widetilde{x},d\widetilde{y})$

for $f\in\mathcal{B}^{+}(X\times\widetilde{X})$, $(x,\widetilde{x})\in
X\times\widetilde{X}$ (see [4, Section V.5] and [21, Section 3.3.2]).

If $(X,{\mathcal{E}}_{\mathbbm{P}})$ and
$(\widetilde{X},{\mathcal{E}}_{\widetilde{\mathbbm{P}}})$ are balayage spaces,
then $(X\times\widetilde{X},\mathbbm{P}\otimes\widetilde{\mathbbm{P}})$ is a
balayage space if and only if
${\mathcal{E}}_{\mathbbm{P}\otimes\widetilde{\mathbbm{P}}}$ satisfies (C)
(which, in particular, holds if one of the semigroups is strong Feller and the
other has a strong Feller resolvent; see [4, V.5.9] and [21, Proposition
3.2.1]).

An important case for $\widetilde{\mathbbm{P}}$ is the semigroup
$\mathbbm{T}:=(T_{t})_{t>0}$ of uniform translation to the left on
$\mathbbm{R}$ (see Remark 3.3,2). We note that

$(P_{t}\otimes T_{t})f(x,s):=P_{t}f(\cdot,s-t)(x),\qquad
f\in\mathcal{B}^{+}(X\times\mathbbm{R}),\,t>0,\,x\in X,\,s\in\mathbbm{R},$

and that $\mathbbm{T}$ has a strong Feller resolvent. The semigroup
$\mathbb{T}$ is often a testing case, also for product semigroups; by [4,
V.5.10], the following holds.

######  THEOREM 9.1.

If $\mathbbm{P}$ is a sub-Markov semigroup on $X$ such that
$(X,{\mathcal{E}}_{\mathbbm{P}})$ is a balayage space, the following
statements are equivalent:

  * (1)

$\mathbbm{P}$ is strong Feller.

  * (2)

$(X\times\mathbbm{R},{\mathcal{E}}_{\mathbbm{P}\otimes\mathbbm{T}})$ is a
balayage space.

  * (3)

$(X\times\widetilde{X},{\mathcal{E}}_{\mathbbm{P}\otimes\widetilde{\mathbbm{P}}})$
is a balayage space for _every_ balayage space
$(\widetilde{X},{\mathcal{E}}_{\widetilde{\mathbbm{P}}})$ such that
$\widetilde{\mathbbm{P}}$ is a sub-Markov semigroup on $\widetilde{X}$ with
strong Feller resolvent.

The next result is borrowed from [4, V.5.6]. For the proof of
$(2)\Rightarrow(1)$, which is not optimal in [4], we note that, for all $r>0$
and $f\in\mathcal{C}_{b}(X)$, $\lim_{t\downarrow r}P_{t}f=P_{r}f$ locally
uniformly, by [4, V.2.8 and V.2.11], and hence the resolvent of
$\mathbbm{P}\otimes\mathbbm{T}$ is strong Feller, by [4, V.5.9].

######  THEOREM 9.2.

For every sub-Markov semigroup $\mathbbm{P}$ on $X$, the following statements
are equivalent:

  * (1)

$(X\times\mathbbm{R},{\mathcal{E}}_{\mathbbm{P}\otimes\mathbbm{T}})$ is a
balayage space.

  * (2)

$\mathbbm{P}$ is a strong Feller semigroup, $\lim_{t\to 0}P_{t}f=f$ locally
uniformly for every $f\in\mathcal{C}(X)$ with compact support, and
${\mathcal{E}}_{\mathbbm{P}\otimes\mathbbm{T}}$ satisfies (T).

As usual, given a sub-Markov semigroup $\mathbbm{P}$ and $\lambda\geq 0$, a
function $u\in\mathcal{B}^{+}(X)$ is called _$\lambda$ -supermedian_ (with
respect to $\mathbbm{P}$) if $e^{-\lambda t}P_{t}u\leq u$ for all $t>0$. If
$\lambda=0$ we speak of supermedian functions.

######  REMARK 9.3.

If $\mathbbm{P}$ is right continuous, then
${\mathcal{E}}_{\mathbbm{P}\otimes\mathbbm{T}}$ satisfies (T) provided there
are $\lambda\geq 0$ and strictly positive $\lambda$-supermedian functions
$u,v\in\mathcal{C}(X)$ with $u/v\in\mathcal{C}_{0}(X)$.

Indeed, defining $w_{0}(s):=e^{\lambda s}$, $s\in\mathbbm{R}$, we have
$T_{t}w_{0}(s)=w_{0}(s-t)=e^{-\lambda t}w_{0}(s)$. If $w$ is any
$\lambda$-supermedian function on $X$, then $w^{\prime}\colon(x,s)\mapsto
w(x)w_{0}(s)$ clearly satisfies

(9.1) $(P_{t}\otimes T_{t})w^{\prime}(x,s)=P_{t}w(x)T_{t}w_{0}(s)\leq
e^{\lambda t}w(x)e^{-\lambda t}w_{0}(x)=w^{\prime}(x,s),$

hence it is supermedian with respect to
$\mathbbm{P}^{\prime}:=\mathbbm{P}\otimes\mathbbm{T}$.

Now let $u,v\in\mathcal{C}(X)$ be strictly positive $\lambda$-supermedian
functions on $X$ with $u/v\in\mathcal{C}_{0}(X)$. We may assume that $u\leq
1\leq v$ (replace $u$ by $u\wedge 1$ and $v$ by $v+1$). Then
$\widetilde{u}:=u^{\prime}\wedge 1$ and $\widetilde{v}:=1+v^{\prime}$ are
strictly positive continuous supermedian functions on $X^{\prime}$. By right
continuity of $\mathbbm{P}^{\prime}$, both are excessive. Moreover,
$f:=\widetilde{u}/\widetilde{v}\in\mathcal{C}_{0}(X^{\prime})$. Indeed, for
every $s\in\mathbbm{R}$,

$f(\cdot,s)\leq\frac{u^{\prime}(\cdot,s)}{v^{\prime}(\cdot,s)}=\frac{u}{v}\in\mathcal{C}_{0}(X)\quad\mbox{
and }\quad f(\cdot,s)\leq\frac{e^{\lambda s}\wedge 1}{1+e^{\lambda s}}\leq
e^{-\lambda|s|},$

which clearly implies that $f$ tends to $0$ at infinity.

###  9.2 Existence of corresponding Hunt processes

By [4, IV.7.4 and IV.7.5]), the following holds (for a converse, see Theorem
3.2).

######  PROPOSITION 9.4.

Let $\mathbbm{P}=(P_{t})_{t>0}$ be a sub-Markov semigroup on $X$ and
$\mathfrak{X}$ be a Markov process on $X$ with transition semigroup
$\mathbbm{P}$ (such a process always exists). If
$(X,{\mathcal{E}}_{\mathbbm{P}})$ is a balayage space, then there is a Hunt
process on $X$ equivalent to $\mathfrak{X}$.

######  COROLLARY 9.5.

Let $\mathbbm{P}=(P_{t})_{t>0}$ be a right continuous sub-Markov semigroup on
$X$ such that its resolvent is strong Feller and, for some $\lambda\geq 0$,
there exist strictly positive $\lambda$-supermedian functions
$u,v\in\mathcal{C}(X)$ with $u/v\in\mathcal{C}_{0}(X)$.

Then there is a Hunt process $\mathfrak{X}$ on $X$ having transition semigroup
$\mathbbm{P}$.

Due to the previous results the statement is easily obtained if even
$\mathbbm{P}$ is strong Feller. Indeed, then, by Theorem 9.2 and Remark 9.3,
$(X\times\mathbbm{R},{\mathcal{E}}_{\mathbbm{P}\otimes\mathbbm{T}})$ is a
balayage space. So, by Theorem 3.9, there exists a Hunt process
$\mathfrak{X}^{\prime}$ on $X\times\mathbbm{R}$ having transition semigroup
$\mathbbm{P}\otimes\mathbbm{T}$. Using the projection $(x,s)\mapsto x$ from
$X\times\mathbbm{R}$ to $X$ we get a Hunt process $\mathfrak{X}$ on $X$ with
transition semigroup $\mathbbm{P}$.

In the general case we use the following for the _random walk to the left on
$\mathbbm{Z}$_.

######  LEMMA 9.6.

Let $X=\mathbbm{Z}$ and $P(m,\cdot):=\delta_{m-1}$, $m\in\mathbbm{Z}$.
Further, let $\alpha\in(0,1)$, $a>(1-\alpha)^{-1}$ and $u(m):=a^{m}$,
$m\in\mathbbm{Z}$. Then $P_{t}u\leq e^{-\alpha t}u$ for every $t>0$.

###### Proof.

Let $g(t):=e^{(\alpha-1)t}+(e^{\alpha t}-e^{(\alpha-1)t})a^{-1}$, $t>0$. Since
$u(m^{\prime})\leq u(m-1)$ for every $m^{\prime}\leq m-1$, (3.16) implies
that, for all $t>0$ and $m\in\mathbbm{Z}$,

$P_{t}u(m)\leq e^{-t}u(m)+(1-e^{-t})u(m-1)=e^{-\alpha t}g(t)u(m).$

Clearly, $g(0)=1$ and $g^{\prime}(0)=\alpha-1+a^{-1}<0$. So there exists
$t_{0}>0$ such that, for all $0<t\leq t_{0}$, $g(t)\leq 1$, and hence
$P_{t}u\leq e^{-\alpha t}u$. By the semigroup property of $\mathbbm{P}$, we
finally obtain that $P_{t}u\leq e^{-\alpha t}u$ for all $t>0$. ∎

###### Proof of Corollary 9.5.

Let $Q(m,\cdot):=\delta_{m-1}$, $m\in\mathbbm{Z}$, and
$Q_{t}:=e^{-t}\sum_{k\geq 0}t^{k}/k!Q^{k}$, $t>0$. Let $a>2$ and
$w_{0}(m):=a^{m}$, $m\in\mathbbm{Z}$. Of course, $w_{0}$ separates
$\mathbbm{Z}$. By Lemma 9.6, $Q_{t}w_{0}\leq e^{-t/2}w_{0}$ for every $t>0$.
Defining $\widetilde{P}_{t}:=Q_{2\lambda t}$ we then have

$\widetilde{P}_{t}w_{0}\leq e^{-\lambda t}w_{0}\qquad\mbox{for every }t>0.$

Proceeding as in Remark 9.3, we get strictly positive functions
$\widetilde{u},\widetilde{v}\in\mathcal{C}(X^{\prime})\cap{\mathcal{E}}_{\mathbbm{P}^{\prime}})$
such that $\widetilde{u}/\widetilde{v}\in\mathcal{C}_{0}(X^{\prime})$.

Of course, $\widetilde{\mathbbm{P}}:=(\widetilde{P}_{t})_{t>0}$ is a strong
Feller Markov semigroup on $\mathbbm{Z}$ with
${\mathcal{E}}_{\widetilde{\mathbbm{P}}}=S_{Q}$, and hence
$(\mathbbm{Z},{\mathcal{E}}_{\widetilde{\mathbbm{P}}})$ is a balayage space,
by Proposition 3.15. The sub-Markov semigroup
$\mathbbm{P}^{\prime}:=\mathbbm{P}\otimes\widetilde{\mathbbm{P}}$ on
$X\times\mathbbm{Z}$ is right continuous, its resolvent is strong Feller and
its potential kernel is proper (see Section 9.1 and [4, V.5.2 and V.5.9] or
[21, Propositions 3.2.2 and 3.2.11]). Thus
$(X\times\mathbbm{Z},{\mathcal{E}}_{\mathbbm{P}\otimes\widetilde{\mathbbm{P}}})$
is a balayage space and we may finish now using the projection $(x,m)\mapsto
x$ from $X\times\mathbbm{Z}$ on $X$. ∎

###  9.3 Harmonic measures for space-time semigroups

Let $\widetilde{X}:=X\times\mathbbm{R}$ and let $\mathbbm{P}=(P_{t})_{t>0}$ be
a strong Feller sub-Markov semigroup on $X$ such that
$\widetilde{\mathbbm{P}}:=\mathbbm{P}\otimes\mathbbm{T}$ yields a balayage
space $(\widetilde{X},{\mathcal{E}}_{\widetilde{\mathbbm{P}}})$ (see Section
9.1 for necessary and sufficient conditions). We recall that
$\widetilde{P}_{t}((x,r),\cdot)=P_{t}(x,\cdot)\otimes\delta_{r-t}$, that is,

$\widetilde{P}_{t}f(x,r)=P_{t}f(\cdot,r-t)(x),\qquad
f\in\mathcal{B}^{+}(\widetilde{X}),\,x\in X,\,r\in\mathbbm{R}.$

######  PROPOSITION 9.7.

Let $a,r\in\mathbbm{R}$, $a<r$, and $x\in X$. Then

(9.2) $H_{X\times(a,\infty)}((x,r),\cdot)=\widetilde{P}_{r-a}((x,r),\cdot).$

Let $\widetilde{U}$ be an open neighborhood of $(x,r)$ in $X\times(a,\infty)$.
Then the harmonic measure $H_{\widetilde{U}}((x,r),\cdot)$ is supported by
$X\times[a,r]$.

###### Proof.

Given $w\in{\mathcal{E}}_{\widetilde{\mathbbm{P}}}$, we define a function
$w^{\prime}\leq w$ on $\widetilde{X}$ by

$w^{\prime}(y,s):=\begin{cases}\widetilde{P}_{s-a}w(y,s),&\quad\mbox{ if
}s>a,\\\ w(y,s),&\quad\mbox{if }s\leq a.\end{cases}$

Let $q\in\mathcal{P}(\widetilde{X})$. If
$w\in{\mathcal{E}}_{\widetilde{\mathbbm{P}}}$ with $w\geq q$ on
$X\times(-\infty,a]$, then clearly $w^{\prime}\geq q^{\prime}$, since the
measures $\widetilde{P}_{s-a}((y,s),\cdot)$, $s>a$, are supported by
$X\times\\{a\\}$. Further, it is easily verified that
$q^{\prime}\in{\mathcal{E}}_{\widetilde{\mathbbm{P}}}$. Therefore
$H_{X\times(a,\infty)}q=R_{q}^{X\times(-\infty,a]}=q^{\prime}$. Thus (9.2)
holds.

Moreover, $\widetilde{A}:=X\times(-\infty,r]$ is an absorbing set, since
obviously $1_{X\times(r,\infty)}\in{\mathcal{E}}_{\widetilde{\mathbbm{P}}}$,
(see [4, V.1.2]). Hence $H_{\widetilde{U}}((x,r),\cdot)$ is supported by
$\widetilde{A}$. By Proposition 3.7,

$H_{\widetilde{U}}((x,r),X\times(-\infty,a))\geq
H_{X\times(a,\infty)}((x,r),X\times(-\infty,a))=0.$

∎

The following results will be used in Sections 9.4 and 9.5.

######  LEMMA 9.8.

Suppose that $x\in X$ and $t>0$ such that $P_{t}(x,\cdot)\neq\delta_{x}$. Then
there exist $t>0$, $\eta\in(0,1)$ and a neighborhood $V\in{\mathcal{U}}(X)$
such $P_{t}1_{V}\leq\eta$ on $V$.

###### Proof.

Since $P_{t}1\leq 1$, we get that $a:=P_{t}(x,\\{x\\})<1$. Let $\eta\in(a,1)$.
There exists $V_{0}\in{\mathcal{U}}(X)$ containing $x$ such that
$P_{t}(x,V_{0})<\eta$. Since $P_{t}(\cdot,V_{0})\in\mathcal{C}(X)$, there
exists $V\in{\mathcal{U}}(x)$ such that $x\in V\subset V_{0}$ such that
$P_{t}1_{V_{0}}<\eta$ on $V$. Since $P_{t}1_{V}\leq P_{t}1_{V_{0}}$, the proof
is finished. ∎

######  PROPOSITION 9.9.

Let $V$ be open in $X$, $t>0$ and $\eta\in(0,1)$ with $P_{t}1_{V}\leq\eta$ on
$V$. Then $\lim_{a\to\infty}H_{V\times(-a,a)}1_{V\times\\{-a\\}}=0$ and

(9.3)
$\lim\nolimits_{a\to\infty}H_{V\times(-a,a)}\widetilde{w}=H_{V\times\mathbbm{R}}\widetilde{w}$

for every $\widetilde{w}\in{\mathcal{E}}_{\widetilde{\mathbbm{P}}}$ such that
$\widetilde{w}$ is bounded on $V\times\mathbbm{R}$.

###### Proof.

Let $\widetilde{V}:=V\times\mathbbm{R}$, $a>0$,
$\widetilde{V}_{a}:=V\times(-a,a)$,
$h_{a}:=H_{\widetilde{V}_{a}}1_{V\times\\{-a\\}}$,

$\widetilde{B}_{r}:=V\times\\{r\\}\quad\mbox{ and }\quad\gamma_{a}(r):=\sup
h_{a}(\widetilde{B}_{r}),\qquad-a<r<a.$

Of course, $\gamma_{a}\leq 1$. Suppose that $\widetilde{x}=(x,r)$, $x\in V$,
$r<a$, and $s:=r-t>-a$. Defining $\widetilde{W}:=V\times(s,a)$ and using
Proposition 3.7, we obtain that

$\displaystyle h_{a}(\widetilde{x})=H_{\widetilde{W}}h_{a}(\widetilde{x})$
$\displaystyle\leq$
$\displaystyle\gamma_{a}(s)H_{\widetilde{W}}(\widetilde{x},\widetilde{B}_{s})$
$\displaystyle\leq$
$\displaystyle\gamma_{a}(s)H_{X\times(s,a)}(\widetilde{x},\widetilde{B}_{s})=\gamma_{a}(s)P_{t}(x,V)\leq\eta\gamma_{a}(s).$

Thus $\gamma_{a}(s)\leq\eta\gamma_{a}(r)$, and it is straightforward to prove
that

$\lim\nolimits_{a\to\infty}\gamma_{a}=0.$

Obviously,
$H_{\widetilde{V}}\widetilde{w}=H_{\widetilde{V}_{a}}H_{\widetilde{V}}\widetilde{w}\leq
H_{\widetilde{V}_{a}}\widetilde{w}$. On the other hand, defining
$c:=\sup\widetilde{w}(\widetilde{V})$,

$H_{\widetilde{V}_{a}}\widetilde{w}(\widetilde{x})\leq
H_{\widetilde{V}_{a}}(1_{{V^{c}}\times(-a,a)})\widetilde{w}(\widetilde{x})+c\gamma_{a}(r)\leq
H_{\widetilde{V}}\widetilde{w}(\widetilde{x})+c\gamma_{a}(r),$

where the last inequality holds by Proposition 3.7. Since
$\lim_{a\to\infty}\gamma_{a}(r)=0$, we finally conclude that (9.3) holds. ∎

Let $\pi$ be the canonical projection $(x,r)\mapsto x$ from $\widetilde{X}$ to
$X$.

######  COROLLARY 9.10.

Suppose that $(X,{\mathcal{E}}_{\mathbbm{P}})$ is a balayage space, $U$ is an
open set in $X$ and $w\in{\mathcal{E}}_{\mathbbm{P}}$ such that $w\circ\pi$ is
harmonic on $U\times\mathbbm{R}$. Assume also that, for every $x\in U$, there
exists $t>0$ such that $P_{t}(x,\cdot)\neq\delta_{x}$. Then $w$ is harmonic on
$U$.

###### Proof.

Clearly, $w\circ\pi\in\mathcal{W}_{\widetilde{\mathbbm{P}}}$. Let $x\in U$ and
let $V_{0}\in{\mathcal{U}}(U)$, $x\in V_{0}$. By Lemma 9.8, there exist $t>0$
and a neighborhood $V$ of $x$ in $V_{0}$ such that $\sup P_{t}1_{V}(V)<1$. By
assumption, $H_{V\times(-a,a)}(w\circ\pi)=w\circ\pi$ for every $a>0$. So
$H_{V\times\mathbbm{R}}(w\circ\pi)=w\circ\pi$, by Proposition 9.9. By [4,
V.8.2], $H_{V\times\mathbbm{R}}(w\circ\pi)\widetilde{w}=(H_{V}w)\circ\pi$.
Thus $H_{V}w(x)=w(x)$. By [4, III.4.4], we finally conclude that $w$ is
harmonic on $U$. ∎

###  9.4 Space-time Green function

In this section we intend to show that, in many cases, a sub-Markov semigroup
$\mathbbm{P}$ on a space $X$ with “heat kernel” $(x,y,t)\mapsto p_{t}(x,y)$
leads to a balayage space
$(X\times\mathbbm{R},{\mathcal{E}}_{\mathbbm{P}\otimes\mathbbm{T}})$ with
Green function $((x,r),(y,s))\mapsto 1_{(s,\infty)}(r)p_{r-s}(x,y)$.

Let us suppose that we are in the setting of Section 9.3 and that

$P_{t}(x,\cdot)=p_{t}(x,\cdot)m$

where $m$ is a Radon measure on $X$ and $(x,y,t)\mapsto p_{t}(x,y)$ is a
positive real function on $X\times X\times\mathbbm{R}$ having the following
properties:

  * (i)

It is continuous outside $\\{(x,x)\colon x\in X\\}\times\\{0\\}$ and $p_{t}=0$
for $t\leq 0$.

  * (ii)

For all $s,t\in(0,\infty)$ and $x,y\in X$,

$p_{s+t}(x,y)=\int p_{s}(x,z)p_{t}(z,y)\,m(dz)\qquad\mbox{(\emph{Chapman-
Kolmogorov equations})}.$

  * (iii)

For all $y\in X$ and $a>0$, $\lim_{x\to\infty}\sup\\{p_{t}(x,y)\colon 0<t\leq
a\\}=0$.

######  REMARK 9.11.

The strong Feller property of $\mathbbm{P}$, which we assumed, can as well be
easily derived from (i) and (iii).

######  EXAMPLES 9.12.

Let $X:={\mathbbm{R}^{d}}$, $d\geq 1$. Let $\mathbbm{P}$ be one of the
following sub-Markov semigroups on $X$:

1\. Brownian semigroup, given by the transition density

$p_{t}(x,y)=(4\pi t)^{-d/2}\exp\bigl{(}-\dfrac{|x-y|^{2}}{4t}\bigr{)},\qquad
x,y\in{\mathbbm{R}^{d}},\ t>0.$

This yields the harmonic space given by the heat equation, see, e.g., [4,
Section V.6] or [18, Section 2.3].

2\. Isotropic $\alpha$-stable semigroup with $0<\alpha<2$, having a transition
density

$p_{t}(x,y)\approx t^{-d/\alpha}\land\frac{t}{|x-y|^{d+\alpha}},\qquad
x,y\in{\mathbbm{R}^{d}},\ t>0,$

see, e.g., [27, (2.11)].

For
$\widetilde{\mathbbm{P}}:=\mathbbm{P}\otimes\mathbbm{T}=(\widetilde{P}_{t})_{t>0}$
on $\widetilde{X}:=X\times\mathbbm{R}$ we now have

(9.4) $\widetilde{P}_{t}f(x,r)=\int p_{t}(x,z)f(z,r-t)\,dm(z),\qquad
f\in\mathcal{B}^{+}(\widetilde{X}).$

We define $\widetilde{G}\colon\widetilde{X}\times\widetilde{X}\to[0,\infty)$
by

(9.5) $\widetilde{G}((x,r),(y,s)):=p_{r-s}(x,y).$

######  THEOREM 9.13.

$\widetilde{G}$ is a Green function for the balayage space
$(\widetilde{X},{\mathcal{E}}_{\widetilde{\mathbbm{P}}})$.

###### Proof.

Let us fix $\widetilde{y}=(y,s)\in\widetilde{X}$ and define

$\widetilde{g}:=\widetilde{G}(\cdot,\widetilde{y}).$

By (i), $\widetilde{g}$ is continuous on $X\setminus\\{\widetilde{y}\\}$,
l.s.c. at $\widetilde{y}$. For $t>0$, $z\in X$ and $r\in\mathbbm{R}$,
$\widetilde{g}(z,r-t)=p_{r-t-s}(z,y)$, and hence, by (9.4) and (ii),

(9.6)
$\widetilde{P}_{t}\widetilde{g}(x,r)=1_{(s+t,\infty)}(r)\,\widetilde{g}(x,r),\qquad
t>0,\,(x,r)\in X.$

Since $\widetilde{g}(x,r)=0$ for $r\leq s$, we conclude that
$\widetilde{g}\in{\mathcal{E}}_{\widetilde{\mathbbm{P}}}$.

We next show that $\widetilde{g}$ is harmonic on
$X\setminus\\{\widetilde{y}\\}$. If $\widetilde{x}\in X\times(-\infty,s]$ and
$V$ is an open neighborhood of $\widetilde{x}$, then $0\leq
H_{V}\widetilde{g}(\widetilde{x})\leq\widetilde{g}(\widetilde{x})=0$ whence
$H_{V}\widetilde{g}(\widetilde{x})=\widetilde{g}(\widetilde{x})=0$. So let us
fix $\widetilde{x}=(x,r)\in X$, $r>s$. There exists $t>0$ such that $r>s+t$.
Let $\widetilde{V}$ be an open neighborhood of $\widetilde{x}$ such that its
closure is a compact in $\widetilde{U}:=X\times(r-t,\infty)$. We have
$\widetilde{g}\geq H_{\widetilde{V}}\widetilde{g}\geq
H_{\widetilde{V}}H_{\widetilde{U}}\widetilde{g}=H_{\widetilde{U}}\widetilde{g}$,
where
$H_{\widetilde{U}}\widetilde{g}(\widetilde{x})=\widetilde{P}_{t}\widetilde{g}(\widetilde{x})=\widetilde{g}(\widetilde{x})$,
by Proposition 9.7 and (9.6). So $H_{V}\widetilde{g}(x)=\widetilde{g}(x)$. By
[4, III.4.4], we conclude that
$\widetilde{g}\in{\mathcal{H}}^{+}(\widetilde{X}\setminus\\{\widetilde{y}\\})$.

Finally, let $a,b\in\mathbbm{R}$, $a<s\leq r<b$, and let
$V\in{\mathcal{U}}(X)$, $\varepsilon>0$. By (iii), there exists
$W\in{\mathcal{U}}(X)$ containing $V$ such that $\widetilde{g}\leq\varepsilon$
on $\widetilde{A}:=(X\setminus W)\times[a,b]$. Let
$\widetilde{W}=W\times(a,b)$. By Proposition 9.7, the measures
$H_{\widetilde{W}}(\widetilde{x},\cdot)$, $\widetilde{x}\in V\times(a,b)$, are
supported by the union of $\widetilde{A}$ and $W\times\\{a\\}$. Since
$\widetilde{g}=0$ on $X\times\\{a\\}$, we obtain that
$H_{\widetilde{W}}\widetilde{g}=H_{\widetilde{W}}(1_{\widetilde{A}}\widetilde{g})\leq\varepsilon$
on $V\times(a,b)$. Thus $\widetilde{g}\in\mathcal{P}(X)$, by Lemma 8.2. ∎

Let $\widetilde{m}:=m\otimes\lambda_{\mathbbm{R}}$ and let $\widetilde{V}$ be
the potential kernel of $\widetilde{\mathbbm{P}}$, that is,
$\widetilde{V}:=\int_{0}^{\infty}\widetilde{P}_{t}\,dt$. For all
$f\in\mathcal{B}^{+}(\widetilde{X})$ and
$\widetilde{x}=(x,r)\in\widetilde{X}$,

$\displaystyle\widetilde{V}f(x)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\widetilde{P}_{t}f(\cdot,r-t)(x)\,dt=\int_{0}^{\infty}\int_{X}p_{t}(x,y)f(y,r-t)\,dm(y)\,dt$
$\displaystyle=$
$\displaystyle\int\widetilde{G}((x,r),(y,s))f(y,s)\,d\widetilde{m}(y,s)=:\widetilde{G}(f\widetilde{m})(x).$

In particular, $\widetilde{V}$ is proper, since for every bounded interval
$(a,b)$ in $\mathbbm{R}$,

$\widetilde{V}1_{X\times(a,b)}(x,r)=\int_{(r-b)^{+}}^{(r-a)^{+}}\widetilde{P}_{t}1(x)\,dt\leq
b-a.$

Thus, for every $w\in{\mathcal{E}}_{\widetilde{\mathbbm{P}}}$, there exists a
sequence $(f_{n})$ in $\mathcal{B}_{b}^{+}(\widetilde{X})$ such that

$\widetilde{G}(f_{n}\widetilde{m})\uparrow w$

(see [4, II.3.11]). So, by Theorem 9.13 and [24, Theorem 1.1], we obtain the
following.

######  COROLLARY 9.14.

For every $p\in\mathcal{P}^{\prime}_{r}(\widetilde{X})$, there exists a unique
$\mu\in\mathcal{M}^{+}(\widetilde{X})$ such that $p=\widetilde{G}\mu$.

Further, using Propositions 8.3 and 8.4 we get the following.

######  THEOREM 9.15.

For every open $U$ in $\widetilde{X}$, the definition

$\widetilde{G}_{U}(\cdot,\widetilde{y}):=\widetilde{G}(\cdot,\widetilde{y})-H_{U}\widetilde{G}(\cdot,\widetilde{y}),\qquad\widetilde{y}\in
U,$

yields a Green function $\widetilde{G}_{U}$ on $U$ and, for every continuous
real potential $p$ on $U$, there exists a unique $\mu\in\mathcal{M}^{+}(U)$
such that $p=\widetilde{G}_{U}\mu$.

###  9.5 Green function for semigroups with densities

Let us suppose that we are in the setting of the previous subsection, that
$(X,{\mathcal{E}}_{\mathbbm{P}})$ is a balayage space and, for every $x\in X$,
there exists $t>0$, such that $P_{t}(x,\cdot)\neq\delta_{x}$, that is, $X$
does not contain absorbing points. We define

$G_{y}(x):=G(x,y):=\int_{0}^{\infty}p_{t}(x,y)\,dt,\qquad x,y\in X.$

######  THEOREM 9.16.

Suppose that for every neighborhood $V$ of a point $y\in X$, the function
$G_{y}$ is bounded and continuous on $X\setminus V$ and that $G_{y}(y)<\infty$
if $y$ is isolated. Then $G$ is a Green function for
$(X,{\mathcal{E}}_{\mathbbm{P}})$.

###### Proof.

Let $\widetilde{G}$ be the Green function from the previous section. Clearly,
for all $x,y\in X$ and $r,s\in\mathbbm{R}$,

$G(x,y)=\int_{\mathbbm{R}}\widetilde{G}((x,r),(y,s))\,ds.$

Let us fix $y\in X$ and let $\widetilde{L}:=\\{y\\}\times\mathbbm{R}$,

$g:=G(\cdot,y)\quad\mbox{ and }\quad\widetilde{g}:=g\circ\pi.$

By Fubini, $g\in{\mathcal{E}}_{\mathbbm{P}}$,
$\widetilde{g}\in{\mathcal{E}}_{\mathbbm{\widetilde{P}}}$ and, using that the
functions $\widetilde{G}(\cdot,(y,s))$, $s\in\mathbbm{R}$, are harmonic on
$\widetilde{X}\setminus\widetilde{L}$,

$H_{\widetilde{V}}\widetilde{g}=\widetilde{g}\qquad\mbox{ for every
}\widetilde{V}\in{\mathcal{U}}(\widetilde{X}\setminus\widetilde{L}).$

So $\widetilde{g}$ is harmonic on $\widetilde{X}\setminus L$ and, by Corollary
9.10, $g$ is harmonic on $X\setminus\\{y\\}$.

If $V\in{\mathcal{U}}(X)$ with $y\in V$, then
$(H_{V}g)|_{V}\in\mathcal{C}_{b}(V)$, since $g$ is bounded on ${V^{c}}$ and
$1\in\mathcal{W}$. Hence $g$ is superharmonic on $X$, by Lemma 3.10.

Let $U_{n}\in{\mathcal{U}}(X)$ exhaust $X$. Then the functions
$h_{n}:=H_{U_{n}}g$, $n\in\mathbbm{N}$, are decreasing to a harmonic function
$h$ on $X$. It remains to show that $h=0$. Defining
$\widetilde{V}_{n}:=U_{n}\times(-n,n)$ we get

$H_{\widetilde{V}_{n}}\widetilde{g}\downarrow 0\quad\text{ on
}\\{\widetilde{g}<\infty\\}.$

as $n\to\infty$. We have
$h_{n}\circ\pi=H_{U_{n}\times\mathbbm{R}}\widetilde{g}\leq
H_{\widetilde{V}_{n}}\widetilde{g}$. Hence $h=0$ on $X\setminus\\{y\\}$. By
continuity, $h=0$ on $X$, unless $y$ is isolated. If $y$ is isolated, then we
get directly that $h(y)=0$. ∎

By the same arguments as at the end of the previous section, we now get the
following.

######  COROLLARY 9.17.

Assume, in addition, that the functions $G(x,\cdot)$, $x\in X$, are continuous
on $X\setminus\\{x\\}$ and that the potential kernel of $\mathbbm{P}$ is
proper. Then the following holds:

  * (1)

For every $p\in\mathcal{P}^{\prime}_{r}(X)$, there exists a unique
$\mu\in\mathcal{M}^{+}(X)$ such that $p=G\mu$.

  * (2)

For every open $U$ in $X$, the definition

$G_{U}(\cdot,y):=G(\cdot,y)-H_{U}G(\cdot,y),\qquad y\in U,$

yields a Green function $G_{U}$ on $U$ and, for every continuous real
potential $p$ on $U$, there exists a unique $\mu\in\mathcal{M}^{+}(U)$ such
that $p=G_{U}\mu$.

## References

  * [1] N. Abatangelo.  Large $S$-harmonic functions and boundary blow-up solutions for the fractional Laplacian.  Discrete Contin. Dyn. Syst., 35: 5555-5607, 2015.
  * [2] A. Baalal and W. Hansen.  Nonlinear perturbation of balayage spaces.  Ann. Acad. Sci. Fenn. Math., 27: 163–172, 2002.
  * [3] M. Ben Chrouda and M. Ben Fredj.  Nonnegative entire bounded solutions to some semilinear equations involving the fractional Laplacian.  Potential Anal., 48: 495–513, 2018.
  * [4] J. Bliedtner and W. Hansen.  Potential Theory – An Analytic and Probabilistic Approach to Balayage.  Universitext. Springer, Berlin-Heidelberg-New York-Tokyo, 1986.
  * [5] K. Bogdan and T. Byczkowski.  Potential theory for the $\alpha$-stable Schrödinger operator on bounded Lipschitz domains.  Studia Math., 133(1): 53–92, 1999.
  * [6] K. Bogdan and T. Byczkowski.  Potential theory of Schrödinger operator based on fractional Laplacian.  Probab. Math. Statist., 20(2): 293–335, 2000.
  * [7] K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song and Z. Vondraček.  Potential analysis of stable processes and its extensions, LNM 1980, Springer-Verlag, Berlin, 2009.
  * [8] K. Bogdan, T. Jakubowski.  Estimates of the Green function for the fractional Laplacian perturbed by gradient.  Potential Anal., 36(3): 455–481, 2012.
  * [9] K. Bogdan, S. Jahros, and E. Kania.  Semilinear Dirichlet problem for the fractional Laplacian.  Nonlinear Anal., 193: 111512 (20pp.), 2020.
  * [10] K. Bogdan and W. Hansen.  Solving semi-linear equations by potential-theoretic methods.  In preparation.
  * [11] K. Bogdan and T. Żak.  On Kelvin transformation.  J. Theoret. Probab., 19(1):89–120, 2006.
  * [12] H. Brezis and W. Strauss.  Semilinear elliptic equations in L1.  J. Math. Soc. Japan, 25: 565–590, 1973.
  * [13] Z.-Q. Chen, R.J. Williams, and Z. Zhao.  On the existence of positive solutions of semilinear elliptic equations with Dirichlet boundary conditions.  Math. Ann., 298: 543–556, 1994.
  * [14] Kh. El Mabrouk.  Semilinear perturbations of harmonic spaces, Liouville property and a boundary value problem.  Potential Anal., 19: 35–50, 2003.
  * [15] Kh. El Mabrouk.  Entire bounded solutions for a class of sublinear elliptic equations.  Nonlinear Anal., 58: 205–218, 2004.
  * [16] Kh. El Mabrouk.  Positive solutions to singular semilinear elliptic problems.  Positivity, 10: 665–680, 2006.
  * [17] Kh. El Mabrouk and W. Hansen.  Nonradial large solutions of sublinear elliptic problems.  J. Math. Anal. Appl., 330: 1025–1041, 2007.
  * [18] L. Evans.  Partial differential equations. Second edition. Graduate Studies in Mathematics, 19., American Mathematical Society, Providence, RI, 2010.
  * [19] A. Grigor’yan and W. Hansen.  A Liouville property for Schrödinger operators.  Math. Ann., 312: 659–716, 1998.
  * [20] W. Hansen.  Modification of balayage spaces by transitions with applications to coupling of PDE’s.  Nagoya Math. J., 169: 77–118, 2003.
  * [21] W. Hansen.  Three views on potential theory.  A course at Charles University (Prague), Spring 2008.  http://www.karlin.mff.cuni.cz/ hansen/lecture/ course-07012009.pdf.
  * [22] W. Hansen.  Equicontinuity of harmonic functions and compactness of potential kernels.  Rev. Roumaine Math. Pures Appl., 66 (1), 179–189 (2021).
  * [23] W. Hansen and I. Netuka.  Density of extremal measures in parabolic potential theory.  Math. Ann., 345: 657–684 (2009).
  * [24] W. Hansen and I. Netuka.  Representation of potentials.  Rev. Roumaine Math. Pures Appl., 59: 93–104, 2014.
  * [25] P. Hartman and G. Stampacchia.  On some non-linear elliptic differential-functional equations.  Acta Math., 115: 271–310, 1966.
  * [26] J. Keller.  On solutions of $\Delta u=f(u)$.  Comm. Pure Appl. Math. 10: 503–510, 1957.
  * [27] M. Kwaśnicki.  Ten equivalent definitions of the fractional Laplace operator.  J. Fract. Calc. Appl. Anal. 20(1):7–51, 2017.
  * [28] N. S. Landkof, _Foundations of modern potential theory_. Springer-Verlag, New York-Heidelberg 1972.
  * [29] M. Marcus and L. Véron, _Nonlinear second order elliptic equations involving measures._ De Gruyter, Berlin 2014.
  * [30] Y. Mizuta.  On the radial limits of Riesz potentials at infinity.  HIroshima Math. J., 7: 165–177, 1977.
  * [31] R. Osserman.  On the inequality $\Delta u\geq f(u)$.  Pacific J. Math. 7: 1641–1647, 1957.
  * [32] S. C. Port,  The first hitting distribution of a sphere for symmetric stable processes.  Trans. Amer. Math. Soc., 135: 115–125, 1969.

Krzysztof Bogdan, Department of Pure and Applied Mathematics, Wroclaw
University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370 Wroclaw,
Poland, email<EMAIL_ADDRESS>
Wolfhard Hansen, Fakultät für Mathematik, Universität Bielefeld, 33501
Bielefeld, Germany, e-mail<EMAIL_ADDRESS>