Results 1  10
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37
Random walks on infinite graphs and groups  a survey on selected topics
 Bull. London Math. Soc
, 1994
"... 2. Basic definitions and preliminaries 3 A. Adaptedness to the graph structure 4 B. Reversible Markov chains 4 ..."
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Cited by 34 (2 self)
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2. Basic definitions and preliminaries 3 A. Adaptedness to the graph structure 4 B. Reversible Markov chains 4
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 13 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Transience of percolation clusters on wedges
, 2001
"... We study random walks on supercritical percolation clusters on wedges in Z3, and show that the infinite percolation cluster is (a.s.) transient whenever the wedge is transient. This solves a question raised by O. Häggström and E. Mossel. We also show that for convex gauge functions satisfying a mild ..."
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Cited by 7 (5 self)
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We study random walks on supercritical percolation clusters on wedges in Z3, and show that the infinite percolation cluster is (a.s.) transient whenever the wedge is transient. This solves a question raised by O. Häggström and E. Mossel. We also show that for convex gauge functions satisfying a mild regularity condition, the existence of a finite energy flow on Z2 is equivalent to the (a.s.) existence of a finite energy flow on the supercritical percolation cluster. This solves a question of C. Hoffman.
The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups, arXiv:0806.0734v1 [math.AP
"... We present an invariant definition of the hypoelliptic Laplacian on subRiemannian structures with constant growth vector using the Popp’s volume form introduced by Montgomery. This definition generalizes the one of the LaplaceBeltrami operator in Riemannian geometry. In the case of leftinvariant ..."
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Cited by 7 (0 self)
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We present an invariant definition of the hypoelliptic Laplacian on subRiemannian structures with constant growth vector using the Popp’s volume form introduced by Montgomery. This definition generalizes the one of the LaplaceBeltrami operator in Riemannian geometry. In the case of leftinvariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares. We then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane.
Fundamental solutions for nondivergence form operators on stratified groups
 Trans. Amer. Math. Soc
"... Abstract. We construct the fundamental solutions Γ and γ for the nondivergence form operators ∑ i, j ai, j(x, t) XiXj − ∂t and ∑ i, j ai, j(x) XiXj, where the Xi’s are Hörmander vector fields generating a stratified group G and (ai,j)i,j is a positivedefinite matrix with Hölder continuous entries. ..."
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Cited by 6 (2 self)
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Abstract. We construct the fundamental solutions Γ and γ for the nondivergence form operators ∑ i, j ai, j(x, t) XiXj − ∂t and ∑ i, j ai, j(x) XiXj, where the Xi’s are Hörmander vector fields generating a stratified group G and (ai,j)i,j is a positivedefinite matrix with Hölder continuous entries. We also provide Gaussian estimates of Γ and its derivatives and some results for the relevant Cauchy problem. Suitable longtime estimates of Γ allow us to construct γ using both tsaturation and approximation arguments. 1. Introduction and
Hypoelliptic heat kernel inequalities on Lie groups
, 2005
"... This paper discusses the existence of gradient estimates for second order hypoelliptic heat kernels on manifolds. It is now standard that such inequalities, in the elliptic case, are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associate ..."
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Cited by 6 (1 self)
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This paper discusses the existence of gradient estimates for second order hypoelliptic heat kernels on manifolds. It is now standard that such inequalities, in the elliptic case, are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associated “Ricci curvature ” takes on the value − ∞ at points of degeneracy of the semiRiemannian metric associated to the operator. For this reason, the standard proofs for the elliptic theory fail in the hypoelliptic setting. This paper presents recent results for hypoelliptic operators. Malliavin calculus methods transfer the problem to one of determining certain infinite dimensional estimates. Here, the underlying manifold is a Lie group, and the hypoelliptic operators are invariant under left translation. In particular, “L ptype ” gradient estimates hold for p ∈ (1, ∞), and the p = 2 gradient estimate
M.: Counting Schrödinger boundstates: semiclassics and beyond
 In: Sobolev Spaces in Mathematics. II. Applications in Analysis and Partial Differential Equations, International Mathematical Series, 8, Springer and Tamara Rozhkovskaya Publisher
"... Abstract. This is a survey of the basic results on the behavior of the number of the eigenvalues of a Schrödinger operator, lying below its essential spectrum. We discuss both fast decaying potentials, for which this behavior is semiclassical, and slowly decaying potentials, for which the semiclassi ..."
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Cited by 5 (1 self)
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Abstract. This is a survey of the basic results on the behavior of the number of the eigenvalues of a Schrödinger operator, lying below its essential spectrum. We discuss both fast decaying potentials, for which this behavior is semiclassical, and slowly decaying potentials, for which the semiclassical rules are violated. The outstanding personality of Sergey Lvovich Sobolev determined the development of Analysis in XX century in many aspects. One of his most influential contributions to Mathematics is the invention of the functional spaces now named after him and the creation of the machinery of embedding theorems for these spaces. The ideology and the techniques based upon these theorems enabled S.L. Sobolev and his followers to find comprehensive and exact solutions to many key problems in Mathematical Physics. The paper to follow is devoted to a survey of results in one of such problems. This problem concerns the behavior of the discrete part of the spectrum of a Schrödinger operator with negative potential. 1.
Probability on groups: random walks and invariant diffusions
 Notices Amer. Math. Soc
, 2001
"... What do card shuffling, volume ..."
On amenability of group algebras
, 2006
"... Abstract. We study amenability of algebras and modules (based on the notion of almostinvariant finitedimensional subspace), and apply it to algebras associated with finitely generated groups. We show that a group G is amenable if and only if its group ring KG is amenable for some (and therefore fo ..."
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Cited by 4 (1 self)
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Abstract. We study amenability of algebras and modules (based on the notion of almostinvariant finitedimensional subspace), and apply it to algebras associated with finitely generated groups. We show that a group G is amenable if and only if its group ring KG is amenable for some (and therefore for any) field K. Similarly, a Gset X is amenable if and only if its span KX is amenable as a KGmodule for some (and therefore for any) field K. 1.
Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators
 Trans. Amer. Math. Soc
"... Abstract. We prove a global Harnack inequality for a class of degenerate evolution operators by repeatedly using an invariant local Harnack inequality. As a consequence we obtain an accurate Gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators. 1. ..."
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Cited by 3 (1 self)
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Abstract. We prove a global Harnack inequality for a class of degenerate evolution operators by repeatedly using an invariant local Harnack inequality. As a consequence we obtain an accurate Gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators. 1.