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14
The Theory Of Generalized Dirichlet Forms And Its Applications In Analysis And Stochastics
, 1996
"... We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers b ..."
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Cited by 18 (1 self)
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We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, socalled generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C0 semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p conditions, singular and fractional diffusion operators. Subsequently, we analyz...
INTEGRATED DENSITY OF STATES AND WEGNER ESTIMATES FOR RANDOM SCHRÖDINGER OPERATORS
, 2003
"... We survey recent results on spectral properties of random Schrödinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a selfaveraging IDS which is general enough to be applicable to random Schrödinger and LaplaceBeltrami operators ..."
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Cited by 12 (2 self)
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We survey recent results on spectral properties of random Schrödinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a selfaveraging IDS which is general enough to be applicable to random Schrödinger and LaplaceBeltrami operators on manifolds. Subsequently we study more specific models in Euclidean space, namely of alloy type, and concentrate on the regularity properties of the IDS. We discuss the role of the integrated density of states and its regularity properties for the spectral analysis of random Schrödinger operators, particularly in relation to localisation. Proofs of the central results are given in detail. Whenever there are alternative proofs, the different approaches are compared.
Shortest Paths in Distanceregular Graphs
 Europ. J. Combinatorics
"... We aim here at introducing a new point of view of the Laplacian of a graph, Γ. With this purpose in mind, we consider L as a kernel on the finite space V (Γ), in the context of the Potential Theory. Then we prove that L is a nice kernel, since it verifies some fundamental properties as maximum and e ..."
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Cited by 5 (2 self)
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We aim here at introducing a new point of view of the Laplacian of a graph, Γ. With this purpose in mind, we consider L as a kernel on the finite space V (Γ), in the context of the Potential Theory. Then we prove that L is a nice kernel, since it verifies some fundamental properties as maximum and energy principles and the equilibrium principle on any proper subset of V (Γ). If Γ is a proper set of a suitable host graph, then the equilibrium problem for Γ can be solved and the number of the different components of its equilibrium measure leads to a bound on the diameter of Γ. In particular, we obtain the structure of the shortest paths of a distanceregular graph. As a consequence, we find the intersection array in terms of the equilibrium measure. Finally, we give a new characterization of strongly regular graphs. Key words. Distanceregular graph, shortest path, equilibrium potential, capacity.
A dual characterization of length spaces with application to Dirichlet metric spaces
"... We show that under minimal assumptions, the intrinsic metric induced by a strongly local Dirichlet form induces a length space. A main input is a dual characterization of length spaces in terms of the property that the 1Lipschitz functions form a sheaf. ..."
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Cited by 4 (3 self)
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We show that under minimal assumptions, the intrinsic metric induced by a strongly local Dirichlet form induces a length space. A main input is a dual characterization of length spaces in terms of the property that the 1Lipschitz functions form a sheaf.
A Different Construction Of Gaussian Fields From Markov Chains: Dirichlet Covariances
 Ann. I. H. Poincaré
, 2002
"... . We study a class of Gaussian random fields with negative correlations. These fields are easy to simulate. They are defined in a natural way from a Markov chain that has the index space of the Gaussian field as its state space. In parallel with Dynkin's investigation of Gaussian fields having covar ..."
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Cited by 3 (1 self)
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. We study a class of Gaussian random fields with negative correlations. These fields are easy to simulate. They are defined in a natural way from a Markov chain that has the index space of the Gaussian field as its state space. In parallel with Dynkin's investigation of Gaussian fields having covariance given by the Green's function of a Markov process, we develop connections between the occupation times of the Markov chain and the prediction properties of the Gaussian field. Our interest in such fields was initiated by their appearance in random matrix theory. 1.
Dirichlet forms and stochastic completeness of graphs and subgraphs, to appear
 J. Reine Angew. Math. (Crelle’s Journal
"... Abstract. We characterize stochastic completeness for regular Dirichlet forms on discrete sets. We then study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph. We show that any graph is a subgraph of a stochastically complete graph and that stochasti ..."
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Cited by 3 (1 self)
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Abstract. We characterize stochastic completeness for regular Dirichlet forms on discrete sets. We then study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph. We show that any graph is a subgraph of a stochastically complete graph and that stochastic incompleteness of a suitably modified subgraph implies stochastic incompleteness of the whole graph. Along our way we give a sufficient condition for essential selfadjointness of generators of Dirichlet forms on discrete sets and explicitely determine the generators on all ℓ p, 1 ≤ p < ∞, in this case.
DIRICHLET FORMS ON SEPARABLE METRIC SPACES 0. Introduction
"... The general theory of Dirichlet forms on locally compact state spaces has its origin in the classical work by Beurling and Deny [6, 7] and has been developed deeply by Fukushima [11] and Silverstein [20]. Recently various investigations on Dirichlet forms on infinite dimensional, ..."
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The general theory of Dirichlet forms on locally compact state spaces has its origin in the classical work by Beurling and Deny [6, 7] and has been developed deeply by Fukushima [11] and Silverstein [20]. Recently various investigations on Dirichlet forms on infinite dimensional,
NOTE ON BASIC FEATURES OF LARGE TIME BEHAVIOUR OF HEAT KERNELS
"... Abstract. Large time behaviour of heat semigroups (and more generally, of positive selfadjoint semigroups) is studied. Convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoi ..."
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Abstract. Large time behaviour of heat semigroups (and more generally, of positive selfadjoint semigroups) is studied. Convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups. This framework includes Laplacians on manifolds, metric graphs and discrete graphs.