Balayage Space Research Articles

Nearly hyperharmonic functions and Jensen measures

Let $(X,\mathcal H)$ be a $\mathcal P$-harmonic space and assume for simplicity that constants are harmonic. Given a numerical function $\varphi$ on $X$ which is locally lower bounded, let \begin{equation*} J_\varphi(x):=\sup\{\int^\ast \varphi\,d\mu(x)\colon \mu\in \mathcal J_x(X)\}, \qquad x\in X, \end{equation*} where $\mathcal J_x(X)$ denotes the set of all Jensen measures $\mu$ for $x$, that is, $\mu$ is a compactly supported measure on $X$ satisfying $\int u\,d\mu\le u(x)$ for every hyperharmonic function on $X$. The main purpose of the paper is to show that, assuming quasi-universal measurability of $\varphi$, the function $J_\varphi$ is the smallest nearly hyperharmonic function majorizing $\varphi$ and that $J_\varphi=\varphi \vee \hat J_\varphi$, where $\hat J_\varphi$ is the lower semicontinuous regularization of $J_\varphi$. So, in particular, $J_\varphi $ turns out to be at least measurable as $\varphi$. This improves recent results, where the axiom of polarity was assumed. The preparations about nearly hyperharmonic functions on balayage spaces are closely related to the study of strongly supermedian functions triggered by J.-F. Mertens more than forty years ago.

Reduced functions and Jensen measures

Let $\varphi$ be a locally upper bounded Borel measurable function on a Greenian open set $\Omega$ in $R^d$ and, for every $x\in \Omega$, let $v_\varphi(x)$ denote the infimum of the integrals of $\varphi$ with respect to Jensen measures for $x$ on $\Omega$. Twenty years ago, B.J. Cole and T.J. Ransford proved that $v_\varphi$ is the supremum of all subharmonic minorants of $\varphi$ on $X$ and that the sets $\{v_\varphi<t\}$, $t\in R$, are analytic. In this paper, a different method leading to the inf-sup-result establishes at the same time that, in fact, $v_\varphi$ is the minimum of $\varphi$ and a subharmonic function, and hence Borel measurable. This is presented in the generality of harmonic spaces, where semipolar sets are polar, and the key are measurability results for reduced functions on balayage spaces which are of independent interest.

Liouville Property, Wiener’s Test and Unavoidable Sets for Hunt Processes

Let \((X,\mathcal {W})\) be a balayage space, \(1\in \mathcal {W}\), or – equivalently – let \(\mathcal W\) be the set of excessive functions of a Hunt process on a locally compact space X with countable base such that \(\mathcal {W}\) separates points, every function in \(\mathcal {W}\) is the supremum of its continuous minorants and there exist strictly positive continuous \(u,v\in \mathcal {W}\) such that u/v → 0 at infinity. We suppose that there is a Green function G > 0 for X, a metric ρ for X and a decreasing function \(g\colon [0,\infty )\to (0,\infty ]\) having the doubling property such that G ≈ g ∘ ρ. Assuming that the constant function 1 is harmonic and balls of (X, ρ) are relatively compact, it is shown that every positive harmonic function on X is constant (Liouville property) and that Wiener’s test at infinity shows, if a given set A in X is unavoidable, that is, if the process hits A with probability one, wherever it starts. An application yields that locally finite unions of pairwise disjoint balls B(z, r z ), z ∈ Z, which have a certain separation property with respect to a suitable measure λ on X are unavoidable if and only if, for some/any point x 0 ∈ X, the series \({\sum }_{z\in Z} g(\rho (x_{0},z))/g(r_{z}) \) diverges. The results generalize and, exploiting a zero-one law for hitting probabilities, simplify recent work by S. Gardiner and M. Ghergu, A. Mimica and Z. Vondracek, and the author.

Unavoidable sets and harmonic measures living on small sets

Given a connected open set $U\ne\emptyset$ in $ R^d$, $d\ge 2$, a relatively closed set $A$ in $U$ is called \emph{unavoidable in $U$}, if Brownian motion, starting in $x\in U\setminus A$ and killed when leaving $U$, hits $A$ almost surely or, equivalently, if the harmonic measure for $x$ with respect to $U\setminus A$ has mass $1$ on $A$. First a new criterion for unavoidable sets is proven which facilitates the construction of smaller and smaller unavoidable sets in $U$. Starting with an arbitrary champagne subdomain of $U$ (which is obtained omitting a locally finite union of pairwise disjoint closed balls $\overline B(z, r_z)$, $z\in Z$, satisfying $\sup_{z\in Z} r_z/\mbox{dist}(z,U^c)<1$), a combination of the criterion and the existence of small nonpolar compact sets of Cantor type yields a set $A$ on which harmonic measures for $U\setminus A$ are living and which has Hausdorff dimension $d-2$ and, if $d=2$, logarithmic Hausdorff dimension $1$. This can be done as well for Riesz potentials (isotropic $\alpha$-stable processes) on Euclidean space and for censored stable processes on $C^{1,1}$ open subsets. Finally, in the very general setting of a balayage space $(X,\mathcal W)$ on which the function $1$ is harmonic (which covers not only large classes of second order partial differential equations, but also non-local situations as, for example, given by Riesz potentials, isotropic unimodal L\'evy processes or censored stable processes) a construction of champagne subsets $X\setminus A$ of $X$ with small unavoidable sets $A$ is given which generalizes (and partially improves) recent constructions in the classical case.

Restricted Mean Value Property for Balayage Spaces with Jumps

It is shown that, for α-stable processes (Riesz potentials) or—more generally—for balayage spaces with jumps, “one-radius” results for harmonicity can be obtained under fairly weak assumptions.

The Condenser Problem

The condenser problem is studied in the potential-theoretic framework of balayage spaces. New proofs of existence and uniqueness are presented. Furthermore, a probabilistic interpretation is given.

Modification of balayage spaces by transitions with application to coupling of PDE’s

AbstractModifications of balayage spaces are studied which, in probabilistic terms, correspond to killing and transitions (creation of mass combined with jumps). This is achieved by a modification of harmonic kernels for sufficiently small open sets. Applications to coupling of elliptic and parabolic partial differential equations of second order are discussed.

Duality of two balayage-space structures and Dirichlet integrals on balayage spaces

We discuss the duality (mutual adjointness) of two balayage-space structures ℌ and $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathfrak{H}} $$ on a locally compact spaceX via measure representations σ for ℌ and ĝs for $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathfrak{H}} $$ . Then we show that some properties of the Dirichlet integral δ f (X) of functionsf onX, which have been shown for harmonic spaces having an adjoint harmonic structure, still hold for a balayage space having an adjoint balayage-space structure. In particular, we obtain the relations 0≤δ f (X)+1/2∫f 2 dσ(1)≤∫fdσ(f)<∞ forf=p 1−p 2 with bounded continuous potentialsp j such that ∫pjdσ(p j)<∞,j=1,2.

On Continuous Capacities

Let (X, W) be a balayage space, \gamma a Choquet capacity on X , \beta(E) the essential base of E \subset X and, for a compact set K \subset X, \alpha (K) = \gamma (\beta(K)) . Then some properties of the set function \alpha are investigated. In particular, it is shown when \alpha is the Choquet capacity. Further, some relation a to the so-called continuous capacity deduced from a kernel on X is given. At last, some open problems from the book [1] by G. Anger are solved.

Kernels and Choquet capacities

M. Brelot showed that the capacity corresponding to a function-kernel is a Choquet capacity, provided that the kernel satisfies the principle of equilibrium, the weak domination principle and the adjoint kernel satisfies the weak principle of equilibrium. This result is not applicable for a series of important kernels in potential theory (e.g. the fundamental solution of the heat equation, or the Kolmogorov equation), since the above principles no longer hold in this situation. New principles for function kernels guaranteeing that the capacity is a Choquet capacity are introduced and applied in the framework of balayage spaces. In particular, polar and adjoint polar sets are shown to coincide in this context.

Limits of balayage measures

LetA be a subset of a balayage space (X,W) and ν a measure onX. It is shown that for every sequence νn of measures such that limn→∞νn and limn→∞ν n A =λ the limit measure λ is of the formfν+[(1-f)ν]A for some (unique) Borel function 0≤f≤1Cb(A). Furthermore, conditions are given such that any such functionf occurs.