The Experts below are selected from a list of 249 Experts worldwide ranked by ideXlab platform
Michael Röckner - One of the best experts on this subject based on the ideXlab platform.
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Potential Theory of infinite dimensional Lévy processes
Journal of Functional Analysis, 2011Co-Authors: Lucian Beznea, Aurel Cornea, Michael RöcknerAbstract:Abstract We study the Potential Theory of a large class of infinite dimensional Levy processes, including Brownian motion on abstract Wiener spaces. The key result is the construction of compact Lyapunov functions, i.e., excessive functions with compact level sets. Then many techniques from classical Potential Theory carry over to this infinite dimensional setting. Thus a number of Potential theoretic properties and principles can be proved, answering long standing open problems even for the Brownian motion on abstract Wiener space, as, e.g., formulated by R. Carmona in 1980. In particular, we prove the analog of the known result, that the Cameron–Martin space is polar, in the Levy case and apply the technique of controlled convergence to solve the Dirichlet problem with general (not necessarily continuous) boundary data.
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Potential Theory of infinite dimensional L\'evy processes
arXiv: Probability, 2010Co-Authors: Lucian Beznea, Aurel Cornea, Michael RöcknerAbstract:We study the Potential Theory of a large class of infinite dimensional L\'evy processes, including Brownian motion on abstract Wiener spaces. The key result is the construction of compact Lyapunov functions, i.e. excessive functions with compact level sets. Then many techniques from classical Potential Theory carry over to this infinite dimensional setting. Thus a number of Potential theoretic properties and principles can be proved, answering long standing open problems even for the Brownian motion on abstract Wiener space, as e.g. formulated by R. Carmona in 1980. In particular, we prove the analog of the known result, that the Cameron-Martin space is polar, in the L\'evy case and apply the technique of controlled convergence to solve the Dirichlet problem with general (not necessarily continuous) boundary data.
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Analytic Potential Theory of Dirichlet Forms
Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, 1992Co-Authors: Michael RöcknerAbstract:In this chapter we develop some analytic Potential Theory of Dirichlet forms. We try to keep the amount of material as small as possible but sufficient for understanding the probabilistic part of the Theory contained in Chapters IV, V below. The corresponding probabilistic Potential Theory and the relation between the two is studied in Chapter V, Sect.5, as far as necessary for the purpose of this book. In Section 1 we consider excessive and reduced (or balayaged) functions. In Section 2 we look at the capacities corresponding to a Dirichlet form (e,D(e)) and introduce an “intrinsic” notion of e-exceptional sets. Section 3 contains all on e-quasi-continuity that we use later. In this chapter we consider the following situation: let E be a Hausdorff topological space. Let B(E) be the σ-algebra consisting of its Borel subsets and m be a σ-finite positive measure on (E,B(E)). Let (e, D(e)) be a fixed Dirichlet form on L2(E;m) with associated generator L , semigroups (T t )t>0, (T t )t>0 and resolvents (G α )α>0 , (Ĝ α )α>0 on L2(E;m) . Let K ≥ 1 be a continuity constant of (e, D(e)) and, as before, let e denote its symmetric part. Recall that e α := e + α( , ), α > 0, where ( , ) is the usual inner product in L2(E;m) and that ‖ ‖ := ( , )1/2.
Subrata Mukherjee - One of the best experts on this subject based on the ideXlab platform.
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The extended boundary node method for three-dimensional Potential Theory
Computers & Structures, 2005Co-Authors: Srinivas Telukunta, Subrata MukherjeeAbstract:The boundary node method (BNM) [Mukherjee YX, Mukherjee S. The boundary node method for Potential problems. Int J Numer Methods Eng 1997;40:797-815] is a boundary-only mesh-free method that combines the moving least-squares (MLS) interpolation scheme with the standard boundary integral equations (BIEs). Curvilinear boundary co-ordinates were originally proposed and used in this method-for both two [Mukherjee YX, Mukherjee S. The boundary node method for Potential problems. Int J Numer Methods Eng 1997;40:797-815] and three-dimensional [Mukherjee S, Mukherjee YX. Boundary methods-elements, contours and nodes. Boca Raton, FL: CRC Press, in press] problems in Potential Theory and in linear elasticity. Li and Aluru [Li G, Aluru NR. Boundary cloud method: a combined scattered point/boundary integral approach for boundary-only analysis. Comput Methods Appl Mech Eng 2002;191:2337-70; Li G. Aluru NR. A boundary cloud method with a cloud-by-cloud polynomial basis. Eng Anal Boundary Elem 2003;27:57-71] have recently proposed an elegant improvement to the BNM (called the boundary cloud method (BCM)) that allows the use of Cartesian co-ordinates. Their novel variable basis BCM [Li G. Aluru NR. A boundary cloud method with a cloud-by-cloud polynomial basis. Eng Anal Boundary Elem 2003;27:57-71] has several advantages relative to the original BCM. It does, however, have a drawback in that continuous approximants are used for all boundary variables, even across corners. It is well known, for example, that the normal derivative of the Potential function in Potential Theory, or the traction in linear elasticity, often suffers jump discontinuities across corners in two-dimensional (2-D) and across edges and corners in three-dimensional (3-D) problems. The present authors [Telukunta S, Mukherjee S. An extended boundary node method for modeling normal derivative discontinuities in Potential Theory across edges and corners. Eng Anal Boundary Elem 2004;28:1099-110] have recently proposed a further improvement to the BNM and the variable basis BCM. This new approach is called the extended BNM (EBNM). This method employs Cartesian co-ordinates with variable bases, together with appropriate approximants for the normal derivative across edges and corners that can model discontinuities in this variable. Two-dimensional problems in Potential Theory are presented in [Telukunta S, Mukherjee S. An extended boundary node method for modeling normal derivative discontinuities in Potential Theory across edges and corners. Eng Anal Boundary Elem 2004;28:1099-110]. The present paper is concerned with far more challenging problems-3-D problems in Potential Theory.
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A ‘pure’ boundary node method for Potential Theory
Communications in Numerical Methods in Engineering, 2002Co-Authors: Ramesh Gowrishankar, Subrata MukherjeeAbstract:The standard boundary node method (BNM) uses a (meshless) diffuse interpolation (in terms of neighbouring scattered boundary points) for the primary variables. The method is not ‘truly meshless’, however, since cells on the boundary of a body are used for integration. This paper presents a ‘pure’ version of the BNM in which integration cells are dispensed with. Instead (overlapping), regions of influence (ROIs) of boundary nodes are used for integration. Initial results for 2-D Potential Theory, presented here, are encouraging. Copyright © 2002 John Wiley & Sons, Ltd.
Lucian Beznea - One of the best experts on this subject based on the ideXlab platform.
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Potential Theory of infinite dimensional Lévy processes
Journal of Functional Analysis, 2011Co-Authors: Lucian Beznea, Aurel Cornea, Michael RöcknerAbstract:Abstract We study the Potential Theory of a large class of infinite dimensional Levy processes, including Brownian motion on abstract Wiener spaces. The key result is the construction of compact Lyapunov functions, i.e., excessive functions with compact level sets. Then many techniques from classical Potential Theory carry over to this infinite dimensional setting. Thus a number of Potential theoretic properties and principles can be proved, answering long standing open problems even for the Brownian motion on abstract Wiener space, as, e.g., formulated by R. Carmona in 1980. In particular, we prove the analog of the known result, that the Cameron–Martin space is polar, in the Levy case and apply the technique of controlled convergence to solve the Dirichlet problem with general (not necessarily continuous) boundary data.
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Potential Theory of infinite dimensional L\'evy processes
arXiv: Probability, 2010Co-Authors: Lucian Beznea, Aurel Cornea, Michael RöcknerAbstract:We study the Potential Theory of a large class of infinite dimensional L\'evy processes, including Brownian motion on abstract Wiener spaces. The key result is the construction of compact Lyapunov functions, i.e. excessive functions with compact level sets. Then many techniques from classical Potential Theory carry over to this infinite dimensional setting. Thus a number of Potential theoretic properties and principles can be proved, answering long standing open problems even for the Brownian motion on abstract Wiener space, as e.g. formulated by R. Carmona in 1980. In particular, we prove the analog of the known result, that the Cameron-Martin space is polar, in the L\'evy case and apply the technique of controlled convergence to solve the Dirichlet problem with general (not necessarily continuous) boundary data.
Tao Zhou - One of the best experts on this subject based on the ideXlab platform.
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Correction: Potential Theory for Directed Networks
PLoS ONE, 2013Co-Authors: Qian-ming Zhang, Wen-qiang Wang, Yu-xiao Zhu, Tao ZhouAbstract:The name of the fourth author was given incorrectly. The correct name is: Yu-Xiao Zhu. The correct citation is: "Zhang Q-M, Lu L, Wang W-Q, Zhu YX, Zhou T (2013) Potential Theory for Directed Networks. PLoS ONE 8(2): e55437. doi:10.1371/journal.pone.0055437." The correct abbreviation in contributions is: YXZ.
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Potential Theory for directed networks.
PloS one, 2013Co-Authors: Qian-ming Zhang, Wen-qiang Wang, Yu-xiao, Tao ZhouAbstract:Uncovering factors underlying the network formation is a long-standing challenge for data mining and network analysis. In particular, the microscopic organizing principles of directed networks are less understood than those of undirected networks. This article proposes a hypothesis named Potential Theory, which assumes that every directed link corresponds to a decrease of a unit Potential and subgraphs with definable Potential values for all nodes are preferred. Combining the Potential Theory with the clustering and homophily mechanisms, it is deduced that the Bi-fan structure consisting of 4 nodes and 4 directed links is the most favored local structure in directed networks. Our hypothesis receives strongly positive supports from extensive experiments on 15 directed networks drawn from disparate fields, as indicated by the most accurate and robust performance of Bi-fan predictor within the link prediction framework. In summary, our main contribution is twofold: (i) We propose a new mechanism for the local organization of directed networks; (ii) We design the corresponding link prediction algorithm, which can not only testify our hypothesis, but also find out direct applications in missing link prediction and friendship recommendation.
Ivan Netuka - One of the best experts on this subject based on the ideXlab platform.
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Jensen Measures in Potential Theory
Potential Analysis, 2011Co-Authors: Wolfhard Hansen, Ivan NetukaAbstract:It is shown that, for open sets in classical Potential Theory and—more generally—for elliptic harmonic spaces Y, the set Jx(Y) of Jensen measures (representing measures with respect to superharmonic functions on Y) for a point x ∈ Y is a simple union of closed faces of the compact convex set \(M_x(\mathcal P(Y))\) of representing measures with respect to Potentials on Y, a set which has been thoroughly studied a long time ago. In particular, the set of extreme Jensen measures can be immediately identified. The results hold even without ellipticity (thus capturing also many examples for the heat equation) provided a rather weak approximation property for superharmonic functions holds. Equally sufficient are a certain transience property and a weak regularity property. More important, each of these properties turns out to be necessary and sufficient for obtaining (in the classical case) that Jx(Y) coincides with the set of all compactly supported probability measures in \(M_x(\mathcal P(Y))\).
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Jensen measures in Potential Theory
arXiv: Analysis of PDEs, 2010Co-Authors: Wolfhard Hansen, Ivan NetukaAbstract:It is shown that, for open sets in classical Potential Theory and - more generally - for elliptic harmonic spaces, the set of Jensen measures for a point is a simple union of closed faces of a compact convex set which has been thoroughly studied a long time ago. In particular, the set of extreme Jensen measures can be immediately identified. The results hold even without ellipticity (thus capturing also many examples for the heat equation) provided a rather weak approximation property for superharmonic functions or a certain transience property holds.
