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Hansen Wolfhard - One of the best experts on this subject based on the ideXlab platform.
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Nearly Hyperharmonic Functions are Infima of Excessive Functions
'Springer Science and Business Media LLC', 2020Co-Authors: Hansen Wolfhard, Netuka IvanAbstract:Hansen W, Netuka I. Nearly Hyperharmonic Functions are Infima of Excessive Functions. JOURNAL OF THEORETICAL PROBABILITY. 2020;33(3):1613-1629.Let X be a Hunt process on a Locally Compact Space X such that the set epsilon(X) of its Borel measurable excessive functions separates points, every function in epsilon(X) is the supremum of its continuous minorants in epsilon(X), and there are strictly positive continuous functions v, w is an element of epsilon(X) such that v/w vanishes at infinity. A numerical function u = 0 on X is said to be nearly hyperharmonic, if integral* u omicron X-tau V dP(x) = u} for every Borel measurable nearly hyperharmonic function on X. Principal novelties of our approach are the following: 1. A quick reduction to the special case, where starting at x is an element of X with u(x) (y) := lim inf(z) -> y u(z) = u} not only for measures mu satisfying integral w d mu < infinity for some excessive majorant w of u, but also for all finite measures. At the end, the measurability assumption on u is weakened considerably
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Semipolar Sets and Intrinsic Hausdorff Measure
'Springer Science and Business Media LLC', 2019Co-Authors: Hansen Wolfhard, Netuka IvanAbstract:Hansen W, Netuka I. Semipolar Sets and Intrinsic Hausdorff Measure. Potential Analysis. 2019;51(1):49-69.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 nu not equal 0 supported by A such that G nu:=integral G(.,y)d nu(y) is a continuous real function on X. Introducing an intrinsic Hausdorff measurem(G) using G-balls B(x, rho) := {y is an element of X : G(x, y) > 1/rho}, it is shown that every set A in X with mG(A)
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Liouville Property, Wiener's Test and Unavoidable Sets for Hunt Processes
'Springer Science and Business Media LLC', 2016Co-Authors: Hansen WolfhardAbstract:Hansen W. Liouville Property, Wiener's Test and Unavoidable Sets for Hunt Processes. Potential Analysis. 2016;44(4):639-653.Let be a balayage Space, , or - equivalently - let be the set of excessive functions of a Hunt process on a Locally Compact Space X with countable base such that separates points, every function in is the supremum of its continuous minorants and there exist strictly positive continuous such that u/v -> 0 at infinity. We suppose that there is a Green function G > 0 for X, a metric rho for X and a decreasing function having the doubling property such that G approximate to g omicron rho. Assuming that the constant function 1 is harmonic and balls of (X, rho) 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 a Z, which have a certain separation property with respect to a suitable measure lambda on X are unavoidable if and only if, for some/any point x (0) a X, the series 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. Vondraek, and the author
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H\"older continuity of harmonic functions for Hunt processes with Green function
2015Co-Authors: Hansen WolfhardAbstract: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 $\rho$ on $X$ and a decreasing function $g\colon[0,\infty)\to (0,\infty]$ having the doubling property and a mild upper decay such that $G\approx g\circ\rho$ and the capacity of balls of radius $r$ is approximately $1/g(r)$. It is shown that bounded harmonic functions are H\"older continuous, if the constant function $1$ is harmonic and jumps out of balls admit a polynomial estimate. The latter is proven if scaling invariant Harnack inequalities hold.Comment: arXiv admin note: text overlap with arXiv:1410.306
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Harnack inequalities for Hunt processes with Green function
2015Co-Authors: Hansen Wolfhard, Netuka IvanAbstract: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 $\rho$ 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 $G\approx g\circ\rho$ (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 L\'evy processes are covered by this approach.Comment: arXiv admin note: text overlap with arXiv:1409.753
J P Jurzak - One of the best experts on this subject based on the ideXlab platform.
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dominated convergence and stone weierstrass theorem
Journal of Applied Analysis, 2005Co-Authors: J P JurzakAbstract:Let C(X;R) the algebra of continuous real valued functions defined on a Locally Compact Space X. We consider linear subSpaces A ⊂ C(X;R) having the following property: there is a sequence (Φj)j∈N of positive functions in A with limx→∞ Φj(x) = +∞ for every j ∈ N, such that A consists of functions f ∈ C(X;R) bounded above for the absolute value by an homothetic of some Φn (n depends on each f). Dominated convergence of a sequence (gn)n≥1 in A is an estimation of the form |gn(x) − g(x)| ≤ en|h(x)| for all x ∈ X and all n ∈ N where gn, g, h ∈ A and en → 0 as n → ∞. We extend the Stone-Weierstrass theorem to subalgebras or lattices B ⊂ A and we show that the dominated convergence for sequences is exactly the convergence of sequences when A is endowed with a Locally convex (DF)-Space topology.
Netuka Ivan - One of the best experts on this subject based on the ideXlab platform.
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Nearly Hyperharmonic Functions are Infima of Excessive Functions
'Springer Science and Business Media LLC', 2020Co-Authors: Hansen Wolfhard, Netuka IvanAbstract:Hansen W, Netuka I. Nearly Hyperharmonic Functions are Infima of Excessive Functions. JOURNAL OF THEORETICAL PROBABILITY. 2020;33(3):1613-1629.Let X be a Hunt process on a Locally Compact Space X such that the set epsilon(X) of its Borel measurable excessive functions separates points, every function in epsilon(X) is the supremum of its continuous minorants in epsilon(X), and there are strictly positive continuous functions v, w is an element of epsilon(X) such that v/w vanishes at infinity. A numerical function u = 0 on X is said to be nearly hyperharmonic, if integral* u omicron X-tau V dP(x) = u} for every Borel measurable nearly hyperharmonic function on X. Principal novelties of our approach are the following: 1. A quick reduction to the special case, where starting at x is an element of X with u(x) (y) := lim inf(z) -> y u(z) = u} not only for measures mu satisfying integral w d mu < infinity for some excessive majorant w of u, but also for all finite measures. At the end, the measurability assumption on u is weakened considerably
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Semipolar Sets and Intrinsic Hausdorff Measure
'Springer Science and Business Media LLC', 2019Co-Authors: Hansen Wolfhard, Netuka IvanAbstract:Hansen W, Netuka I. Semipolar Sets and Intrinsic Hausdorff Measure. Potential Analysis. 2019;51(1):49-69.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 nu not equal 0 supported by A such that G nu:=integral G(.,y)d nu(y) is a continuous real function on X. Introducing an intrinsic Hausdorff measurem(G) using G-balls B(x, rho) := {y is an element of X : G(x, y) > 1/rho}, it is shown that every set A in X with mG(A)
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Harnack inequalities for Hunt processes with Green function
2015Co-Authors: Hansen Wolfhard, Netuka IvanAbstract: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 $\rho$ 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 $G\approx g\circ\rho$ (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 L\'evy processes are covered by this approach.Comment: arXiv admin note: text overlap with arXiv:1409.753
Natalia Zorii - One of the best experts on this subject based on the ideXlab platform.
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balayage of measures on a Locally Compact Space
arXiv: Classical Analysis and ODEs, 2020Co-Authors: Natalia ZoriiAbstract:We develop a theory of inner balayage of a positive Radon measure $\mu$ of finite energy on a Locally Compact Space $X$ to arbitrary $A\subset X$, generalizing Cartan's theory of Newtonian inner balayage on $\mathbb R^n$, $n\geqslant3$, to a suitable function kernel on $X$. As an application of the theory thereby established, we show that if the Space $X$ is perfectly normal and of class $K_\sigma$, then a recent result by Bent Fuglede (Anal. Math., 2016) on outer balayage of $\mu$ to quasiclosed $A$ remains valid for arbitrary Borel $A$. We give in particular various alternative definitions of inner (outer) balayage, provide a formula for evaluation of its total mass, and prove convergence theorems for inner (outer) swept measures and their potentials. The results obtained do hold (and are new in part) for most classical kernels on $\mathbb R^n$, $n\geqslant2$, which is important in applications.
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interior capacities of condensers with infinitely many plates in a Locally Compact Space
arXiv: Classical Analysis and ODEs, 2009Co-Authors: Natalia ZoriiAbstract:The study deals with the theory of interior capacities of condensers in a Locally Compact Space, a condenser being treated here as a countable, Locally finite collection of arbitrary sets with the sign +1 or -1 prescribed such that the closures of opposite-signed sets are mutually disjoint. We are motivated by the known fact that, in the nonCompact case, the main minimum-problem of the theory is in general unsolvable, and this occurs even under very natural assumptions (e.g., for the Newtonian, Green, or Riesz kernels in an Euclidean Space and closed condensers of finitely many plates). Therefore it was particularly interesting to find statements of variational problems dual to the main minimum-problem (and hence providing some new equivalent definitions of the capacity), but now always solvable (e.g., even for nonclosed, unbounded condensers of infinitely many plates). For all positive definite kernels satisfying B. Fuglede's condition of consistency between the strong and weak-star topologies, problems with the desired properties are posed and solved. Their solutions provide a natural generalization of the well-known notion of interior capacitary distributions associated with a set. We give a description of those solutions, establish statements on their uniqueness and continuity, and point out their characteristic properties.
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interior capacities of condensers in Locally Compact Spaces
arXiv: Classical Analysis and ODEs, 2009Co-Authors: Natalia ZoriiAbstract:The study is motivated by the known fact that, in the nonCompact case, the main minimum-problem of the theory of interior capacities of condensers in a Locally Compact Space is in general unsolvable, and this occurs even under very natural assumptions (e.g., for the Newton, Green, or Riesz kernels in an Euclidean Space and closed condensers). Therefore it was particularly interesting to find statements of variational problems dual to the main minimum-problem (and hence providing some new equivalent definitions of the capacity), but always solvable (e.g., even for nonclosed condensers). For all positive definite kernels satisfying B. Fuglede's condition of consistency between the strong and vague topologies, problems with the desired properties are posed and solved. Their solutions provide a natural generalization of the well-known notion of interior capacitary distributions associated with a set. We give a description of those solutions, establish statements on their uniqueness and continuity, and point out their characteristic properties. A condenser is treated as a finite collection of arbitrary sets with sing +1 or -1 prescribed, such that the closures of opposite-signed sets are mutually disjoint.
Wolfhard Hansen - One of the best experts on this subject based on the ideXlab platform.
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semipolar sets and intrinsic hausdorff measure
Potential Analysis, 2019Co-Authors: Wolfhard Hansen, Ivan NetukaAbstract: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 ≠ 0 supported by A such that \(G\nu :=\int G(\cdot ,y)\,d\nu (y)\) is a continuous real function on X. Introducing an intrinsic Hausdorff measuremG using G-balls B(x, ρ) := {y ∈ X : 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), if G is a genuine Green function. The result has immediate consequences for classical potential theory, Riesz potentials and the heat equation (where it solves an open problem). More generally, it is applied to metric measure Spaces (X, d, μ), where a continuous heat kernel with upper and lower bounds of the form t−α/βΦj(d(x,y)t− 1/β), j = 1, 2, is given. Then the intrinsic Hausdorff measure on X is equivalent to an ordinary Hausdorff measure mα−β. For the corresponding Space-time structure on X × ℝ, the intrinsic Hausdorff measure turns out to be equivalent to an anisotropic Hausdorff measure mα,β.
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semipolar sets and intrinsic hausdorff measure
arXiv: Analysis of PDEs, 2017Co-Authors: Wolfhard Hansen, Ivan NetukaAbstract: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 $\nu\ne 0$ supported by $A$ such that $G\nu:=\int G(\cdot,y)\,d\nu(y)$ is a continuous real function on $X$. Introducing an intrinsic Hausdorff measure $m_G$ using $G$-balls $B(x,\rho):=\{y\in X\colon G(x,y)>1/\rho\}$, 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 $(X,d,\mu)$ (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 $t^{-\alpha/\beta}\Phi_j(d(x,y)t^{-1/\beta})$, $j=1,2$, the intrinsic Hausdorff measure is equivalent to an ordinary Hausdorff measure $m_{\alpha-\beta}$. 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 $m_{\alpha,\beta}$ (with $\alpha=n$ and $\beta=2$ for the heat equation). In particular, our result solves an open problem for the heat equation (which was the initial motivation for the paper).
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Semipolar sets and intrinsic Hausdorff measure
arXiv: Analysis of PDEs, 2017Co-Authors: Wolfhard Hansen, Ivan NetukaAbstract: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 $\nu\ne 0$ supported by $A$ such that $G\nu:=\int G(\cdot,y)\,d\nu(y)$ is a continuous real function on $X$. Introducing an intrinsic Hausdorff measure $m_G$ using $G$-balls $B(x,\rho):=\{y\in X\colon G(x,y)>1/\rho\}$, it is shown that every set $A$ in $X$ with $m_G(A)
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Unavoidable sets and Wiener’s test for Hunt processes
2016Co-Authors: Wolfhard HansenAbstract:Let (X,W) be a balayage Space, 1 ∈ W, or – equivalently – let W be the set of excessive functions of a Hunt process on a Locally Compact Space X with countable base such that W separates points, every function in W is the supremum of its continuous minorants and there exist strictly positive continuous u, v ∈ W such that u/v → 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: [0,∞) → (0,∞] having the doubling property and a mild upper decay at infinity such that G ≈ g ◦ ρ (which is equivalent to a 3G-inequality). Then the corresponding capacity for balls of radius R is bounded by a con-stant multiple of 1/g(R). Assuming that the constant function 1 is harmonic and the capacity of large balls satisfies a reverse estimate or that bounded functions are harmonic if and only if they are constant (Liouville property), it is proven 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, rz), 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 x0 ∈ X, the series z∈Z g(ρ(x0, z))/g(rz) diverges. The results generalize and, exploiting a zero-one law for hitting proba-bilities, simplify recent work by S. Gardiner and M. Ghergu, A. Mimica and Z. Vondraček, and the author
