arXiv subjects

Wolfhard Hansen

Publications and source records attributed to Wolfhard Hansen.

At least 19 recordsLinked to original sources

arXiv2023

Positive harmonically bounded solutions for semi-linear equations

For open sets $U$ in some space $X$, we are interested in positive solutions to semi-linear equations $ Lu=φ(\cdot,u)μ$ 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), $μ$ is a positive measure on $U$ and $φ$ is an arbitrary measurable real function on $U\times \mathbb{R}^+$ such that the functions $t\mapsto φ(x,t)$, $x\in U$, are continuous, increasing and vanish at $t=0$. More precisely, given a measurable function $h\ge 0$ 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, $u\le h$ 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 $u+Kφ(\cdot,u)=g$ 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.

math.PR↗

arXiv2022

Compactness of integral operators and uniform integrability on measure spaces

Let $(E,\mathcal E,μ)$ be a measure space and $G\colon E\times E\to [0,\infty]$ be measurable. Moreover, let $\mathcal F\!_{ui}$ denote the set of all $q\in\mathcal E^+$ (measurable numerical functions $q\ge 0$ on $E$) such that $\{G(x,\cdot)q\colon x\in E\}$ is uniformly integrable, and let $\mathcal F\!_{co}$ denote the set of all $q\in\mathcal E^+$ such that the mapping $f\mapsto G(fq) :=\int G(\cdot,y) f(y) q(y)\,dμ(y)$ is a compact operator on the space $\mathcal E_b$ of bounded measurable functions on $E$ (equipped with the sup-norm). It is shown that $\mathcal F\!_{ui}=\mathcal F\!_{co} $ provided both $\mathcal F\!_{ui}$ and $\mathcal F\!_{co} $contain strictly positive functions.

math.FA↗

arXiv2020

On Evans' and Choquet's theorems on polar sets

By classical results of G.C. Evans and G. Choquet on "good kernels $G$ in potential theory", for every polar $K_σ$-set $P$, there exists a finite measure $μ$ on $P$ such that $Gμ=\infty$ on $P$, and a set $P$ admits a finite measure $μ$ on $P$ such that $\{Gμ=\infty\}=P$ if and only if $P$ is a polar $G_δ$-set. A known application of Evans' theorem yields the solutions of the generalized Dirichlet problem for open sets by the Perron-Wiener-Brelot method using only harmonic upper and lower functions. In this note it is shown that, by elementary "metric" considerations and without using any potential theory, such results can be obtained for general kernels $G$ satisfying a local triangle property. The particular case, $G(x,y)=|x-y|^{α-d}$ on $R^d$, $2<α<d$, solves a long-standing open problem.

math.AP↗

arXiv2019

Nearly hyperharmonic functions are infima of excessive functions

Let $\mathfrak X$ be a Hunt process on a locally compact space $X$ such that the set $\mathcal E_{\mathfrak X}$ of its Borel measurable excessive functions separates points, every function in $\mathcal E_{\mathfrak X}$ is the supremum of its continuous minorants in $\mathcal E_{\mathfrak X}$ and there are strictly positive continuous functions $v,w\in\mathcal E_{\mathfrak X}$ such that $v/w$ vanishes at infinity. A numerical function $u\ge 0$ on $X$ is said to be nearly hyperharmonic, if $\int^\ast u\circ X_{τ_V}\,dP^x\le u(x)$ for all $x\in X$ and relatively compact open neighborhoods $V$ of $x$, where $τ_V$ denotes the exit time of $V$. For every such function $u$, its lower semicontinous regularization $\hat u$ is excessive. The main purpose of the paper is to give a short, complete and understandable proof for the statement that every Borel measurable nearly hyperharmonic function on $X$ is the infimum of its majorants in $E_{\mathfrak X}$. The major novelties of our approach are the following: 1. A quick reduction to the special case, where starting at $x\in X$ with $u(x)<\infty$ the expected number of times the process $\mathfrak X$ visits the set of points $y\in X$, where $\hat u(y):=\liminf_{z\to y} u(z)<u(y)$, is finite. 2. The statement that the integral $\int u\,dμ$ is the infimum of all integrals $\int w\,dμ$, $w\in E_{\mathfrak X}$ and $w\ge u$, not only for measures $μ$ satisfying $\int w\,dμ<\infty$ for some excessive majorant $w$ of $u$, but also for all finite measures. At the end, the measurability assumption on $u$ is weakened considerably.

math.PR↗

arXiv2019

Equicontinuity of harmonic functions and compactness of potential kernels

Within the framework of balayage spaces (the analytical equivalent of nice Hunt processes), we prove equicontinuity of bounded families of harmonic functions and apply it to obtain criteria for compactness of potential kernels.

math.AP↗

arXiv2017

Semipolar sets and intrinsic Hausdorff measure

Given a "Green function" $G$ on a locally compact space $X$ with countable base, a Borel set $A$ in $X$ is called $G$-semipolar, if there is no measure $ν\ne 0$ supported by $A$ such that $Gν:=\int G(\cdot,y)\,dν(y)$ is a continuous real function on $X$. Introducing an intrinsic Hausdorff measure $m_G$ using $G$-balls $B(x,ρ):=\{y\in X\colon G(x,y)>1/ρ\}$, it is shown that every set $A$ in $X$ with $m_G(A)<\infty$ is contained in a $G$-semipolar Borel set. This is of interest, since $G$-semipolar sets are semipolar in the potential-theoretic sense (countable unions of totally thin sets, hit by a corresponding process at most countably many times) provided $G$ is really a Green function for a harmonic space or, more generally, a balayage space. For classical potential theory and Riesz potentials on $R^n$ or, more generally, for Green functions on a metric measure space $(X,d,μ)$ (where balls are relatively compact) given by a continuous heat kernel $(x,y,t)\mapsto p_t(x,y)$ with upper and lower bounds of the form $t^{-α/β}Φ_j(d(x,y)t^{-1/β})$, $j=1,2$, the intrinsic Hausdorff measure is equivalent to an ordinary Hausdorff measure $m_{α-β}$. It is shown that for the corresponding space-time situation on $X\times R$ (heat equation on $R^n \times R$ in the classical case of the Gauss-Weierstrass kernel) the intrinsic Hausdorff measure is equivalent to an anisotropic Hausdorff measure $m_{α,β}$ (with $α=n$ and $β=2$ for the heat equation). In particular, our result solves an open problem for the heat equation (which was the initial motivation for the paper).

math.AP↗

arXiv2017

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 $φ$ on $X$ which is locally lower bounded, let \begin{equation*} J_φ(x):=\sup\{\int^\ast φ\,dμ(x)\colon μ\in \mathcal J_x(X)\}, \qquad x\in X, \end{equation*} where $\mathcal J_x(X)$ denotes the set of all Jensen measures $μ$ for $x$, that is, $μ$ is a compactly supported measure on $X$ satisfying $\int u\,dμ\le u(x)$ for every hyperharmonic function on $X$. The main purpose of the paper is to show that, assuming quasi-universal measurability of $φ$, the function $J_φ$ is the smallest nearly hyperharmonic function majorizing $φ$ and that $J_φ=φ\vee \hat J_φ$, where $\hat J_φ$ is the lower semicontinuous regularization of $J_φ$. So, in particular, $J_φ$ turns out to be at least "as measurable as" $φ$. 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.

math.AP↗

arXiv2017

Reduced functions and Jensen measures

Let $φ$ be a locally upper bounded Borel measurable function on a Greenian open set $Ω$ in $R^d$ and, for every $x\in Ω$, let $v_φ(x)$ denote the infimum of the integrals of $φ$ with respect to Jensen measures for $x$ on $Ω$. Twenty years ago, B.J. Cole and T.J. Ransford proved that $v_φ$ is the supremum of all subharmonic minorants of $φ$ on $X$ and that the sets $\{v_φ<t\}$, $t\in R$, are analytic. In this paper, a different method leading to the inf-sup-result establishes at the same time that, in fact, $v_φ$ is the minimum of $φ$ 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.

math.AP↗

arXiv2016

Intrinsic Hölder continuity of harmonic functions

In a setting, where only exit measures are given, as they are associated with a right continuous strong Markov process on a separable metric space, we provide simple criteria for scaling invariant Hölder continuity of bounded harmonic functions with respect to a distance function which, in applications, may be adapted to the special situation. In particular, already a very weak scaling property ensures that Harnack inequalities imply Hölder continuity. Our approach covers recent results by M. Kassmann and A. Mimica as well as cases, where a Green function leads to an intrinsic metric.

math.AP↗

arXiv2016

Scaling invariant Harnack inequalities in a general setting

In a setting, where only "exit measures" are given, as they are associated with an arbitrary right continuous strong Markov process on a separable metric space, we provide simple criteria for the validity of Harnack inequalities for positive harmonic functions. These inequalities are scaling invariant with respect to a metric on the state space which, having an associated Green function, may be adapted to the special situation. In many cases, this also implies continuity of harmonic functions and Hölder continuity of bounded harmonic functions. The results apply to large classes of Lévy (and similar) processes.

math.AP↗

arXiv2016

Darning and gluing of diffusions

We introduce darning of compact sets (darning and gluing of finite unions of compact sets), which are not thin at any of their points, in a potential-theoretic framework which may be described, analytically, in terms of harmonic kernels/harmonic functions or, probabilistically, in terms of a diffusion. This is accomplished without leaving our kind of setting so that the procedure can be iterated without any problem. It applies to darning and gluing of compacts in Euclidean spaces (manifolds) of different dimensions, which is of interest pertaining to recent studies on heat kernels.

math.PR↗

arXiv2015

Hölder continuity of harmonic functions for Hunt processes with Green function

math.AP↗

arXiv2015

Harnack inequalities for Hunt processes with Green function

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 $u/v\to 0$ at infinity. We suppose that there is a Green function $G>0$ for $X$, a metric $ρ$ on $X$ and a decreasing function $g\colon[0,\infty)\to (0,\infty]$ having the doubling property and a mild upper decay near $0$ such that $G\approx g\circρ$ (which is equivalent to a $3G$-inequality). Then the corresponding capacity for balls of radius $r$ is bounded by a constant multiple of $1/g(r)$. Assuming that reverse inequalities hold as well and that jumps of the process, when starting at neighboring points, are related in a suitable way, it is proven that positive harmonic functions satisfy scaling invariant Harnack inequalities. Provided that the Ikeda-Watanabe formula holds, sufficient conditions for this relation are given. This shows that rather general Lévy processes are covered by this approach.

math.AP↗

arXiv2015

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 $u/v\to 0$ at infinity. We suppose that there is a Green function $G>0$ for $X$, a metric $ρ$ on $X$ and a decreasing function $g\colon[0,\infty)\to (0,\infty]$ having the doubling property such that $G\approx g\circρ$. Assuming that the constant function $1$ is harmonic and balls are relatively compact, is is shown that every positive harmonic function 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 $B(z,r_z)$, $z\in Z$, 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 $x_0\in X$, the series $\sum_{z\in Z} g(ρ(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. Vondra\v cek, and the author.

math.AP↗

arXiv2014

Hunt's hypothesis (H) and the triangle property of the Green function

Let $X$ be a locally compact abelian group with countable base and let $\mathcal W$ be a convex cone of positive numerical functions on $X$ which is invariant under the group action and such that $(X,\mathcal W)$ is a balayage space or (equivalently, if $1\in \mathcal W$) such that $\mathcal W$ is the set of excessive functions of a Hunt process on $X$, $\mathcal W$ separates points, every function in $\mathcal W$ is the supremum of its continuous minorants in $\mathcal W$, and there exist strictly positive continuous $u,v\in \mathcal W$ such that $u/v\to 0$ at infinity. Assuming that there is a Green function $G>0$ for $X$ which locally satisfies the triangle inequality $G(x,z)\wedge G(y,z)\le C G(x,y)$ (true for many Lévy processes), it is shown that Hunt's hypothesis (H) holds, that is, every semipolar set is polar.

math.AP↗

arXiv2014

Unavoidable collections of balls for processes with isotropic unimodal Green function

Let us suppose that we have a right continuous Markov semigroup on $R^d$, $d\ge 1$, such that its potential kernel is given by convolution with a function $G_0=g(|\cdot|)$, where $g$ is decreasing, has a mild lower decay property at zero, and a very weak decay property at infinity. This captures not only the Brownian semigroup (classical potential theory) and isotropic $α$-stable semigroups (Riesz potentials), but also more general isotropic Lévy processes, where the characteristic function has a certain lower scaling property, and various geometric stable processes. There always exists a corresponding Hunt process. A subset $A$ of $R^d$ is called unavoidable, if the process hits $A$ with probability $1$, wherever it starts. It is known that, for any locally finite union of pairwise disjoint balls $B(z,r_z)$, $z\in Z$, which is unavoidable, $\sum_{z\in Z} g(|z|)/g(r_z)=\infty$. The converse is proven assuming, in addition, that, for some $\varepsilon>0$, $|z-z'|\ge \varepsilon |z| (g(|z|)/g(r_z))^{1/d}$, whenever $z,z'\in Z$, $z\ne z'$. It also holds, if the balls are regularly located, that is, if their centers keep some minimal mutual distance, each ball of a certain size intersects $Z$, and $r_z=g(ϕ(|z|))$, where $ϕ$ is a decreasing function. The results generalize and, exploiting a zero-one law, simplify recent work by A. Mimica and Z. Vondracek.

math.AP↗

arXiv2014

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 $α$-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 Lévy 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.

math.AP↗

arXiv2012

Localization and Schrödinger perturbations of kernels

We study iterations of integral kernels satisfying a transience-type condition and we prove exponential estimates analogous to Gronwall\rq{}s inequality. As a consequence we obtain estimates of Schrödinger perturbations of integral kernels, including Markovian semigroups.

math.FA↗