Results 11  20
of
27
Eigenfunction expansions for generators of Dirichlet forms
, 2002
"... Dedicated to the memory of Klaus Floret Abstract We present an eigenfunction expansion theorem for generators of strongly local, regular Dirichlet forms. Conditions are phrased in terms of the intrinsic metric. The result covers many cases of Hamiltonians which appear in Mathematical Physics and Geo ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
Dedicated to the memory of Klaus Floret Abstract We present an eigenfunction expansion theorem for generators of strongly local, regular Dirichlet forms. Conditions are phrased in terms of the intrinsic metric. The result covers many cases of Hamiltonians which appear in Mathematical Physics and Geometry.
Sobolev inequalities in familiar and unfamiliar settings
 In S. Sobolev Centenial Volumes, (V. Maz’ja, Ed
"... Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applica ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract The classical Sobolev inequalities play a key role in analysis in Euclidean spaces and in the study of solutions of partial differential equations. In fact, they are extremely flexible tools and are useful in many different settings. This paper gives a glimpse of assortments of such applications in a variety of contexts. 1
DECOUPLING INEQUALITIES AND INTERLACEMENT PERCOLATION ON G × Z
, 2010
"... We study the percolative properties of random interlacements on G×Z, where G is a weighted graph satisfying certain subGaussian estimates attached to the parameters α> 1 and 2 ≤ β ≤ α + 1. We develop decoupling inequalities, which are a key tool in showing that the critical level u ∗ for the percol ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
We study the percolative properties of random interlacements on G×Z, where G is a weighted graph satisfying certain subGaussian estimates attached to the parameters α> 1 and 2 ≤ β ≤ α + 1. We develop decoupling inequalities, which are a key tool in showing that the critical level u ∗ for the percolation of the vacant set of random interlacements is always finite in our setup, and that it is positive when α ≥ 1 + β 2. We also obtain several stretched exponential controls both in the percolative and nonpercolative phases of the model. Even in the case where G = Zd, d ≥ 2, several of these results are new.
ON THE CRITICAL PARAMETER OF INTERLACEMENT PERCOLATION IN HIGH DIMENSION
, 2010
"... The vacant set of random interlacements on Z d, d ≥ 3, has nontrivial percolative properties. It is known from [18], [16], that there is a nondegenerate critical value u ∗ such that the vacant set at level u percolates when u < u ∗ and does not percolate when u> u∗. We derive here an asymptotic up ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
The vacant set of random interlacements on Z d, d ≥ 3, has nontrivial percolative properties. It is known from [18], [16], that there is a nondegenerate critical value u ∗ such that the vacant set at level u percolates when u < u ∗ and does not percolate when u> u∗. We derive here an asymptotic upper bound on u∗, as d goes to infinity, which complements the lower bound from [21]. Our main result shows that u ∗ is equivalent to log d for large d, and thus has the same principal asymptotic behavior as the critical parameter attached to random interlacements on 2dregular trees, which has been explicitly computed in [23].
The Einstein relation for random walks on graphs
, 2008
"... This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the volume doubling and time comparison principles. This and the other set of conditions provide the basic vwork for the study of (sub) diffusive behavior of the random walks on weighted graphs. 1
Heat Kernel Estimates and Law of the Iterated Logarithm for Symmetric Random Walks on Fractal Graphs
"... We study twosided heat kernel estimates on a class of fractal graphs which arise from a subclass of finitely ramified fractals. These fractal graphs do not have spatial symmetry in general, and we find that there is a dependence on direction in the estimates. We will give a new form of expressio ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We study twosided heat kernel estimates on a class of fractal graphs which arise from a subclass of finitely ramified fractals. These fractal graphs do not have spatial symmetry in general, and we find that there is a dependence on direction in the estimates. We will give a new form of expression for the heat kernel estimates using a family of functions which can be thought of as a "distance for each direction". As an application, we give a law of the iterated logarithm which shows that the directional dependence leads to nonuniform behaviour in the typical paths of the random walk.
Recurrence and transience of branching diffusion processes on Riemannian manifolds
, 2001
"... The purpose of this paper is to relate the recurrence and transience properties of a branching diffusion process on a Riemannian manifold M to some properties of a linear elliptic operator on M (including spectral properties). There is a tradeoff between the tendency of the Brownian motion to e ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The purpose of this paper is to relate the recurrence and transience properties of a branching diffusion process on a Riemannian manifold M to some properties of a linear elliptic operator on M (including spectral properties). There is a tradeoff between the tendency of the Brownian motion to escape (if it is transient) and the birth process of the new particles. If the latter has a high enough intensitythenitmayoverride the transience of the Brownian motion, leading to the recurrence of the branching process, and vice versa. In the case of a spherically symmetric manifold, the critical intensity of the population growth can be found explicitly.
WEAK UNCERTAINTY PRINCIPLE FOR FRACTALS, GRAPHS AND METRIC MEASURE SPACES
, 2007
"... Abstract. We develop a new approach to formulate and prove the weak uncertainty inequality which was recently introduced by Okoudjou and Strichartz. We assume either an appropriate measure growth condition with respect to the effective resistance metric, or, in the absence of such a metric, we assum ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. We develop a new approach to formulate and prove the weak uncertainty inequality which was recently introduced by Okoudjou and Strichartz. We assume either an appropriate measure growth condition with respect to the effective resistance metric, or, in the absence of such a metric, we assume the Poincaré inequality and reverse volume doubling property. We also consider the weak uncertainty inequality in the context of Nashtype inequalities. Our results can be applied to a wide variety of metric measure spaces, including graphs, fractals and manifolds. Contents
Upper Bounds for Transition Probabilities on Graphs
, 2004
"... In this paper necessary and sufficient conditions are presented for heat kernel upper bounds for random walks on weighted graphs. Several equivalent conditions are given in the form of isoperimetric inequalities. ..."
Abstract
 Add to MetaCart
In this paper necessary and sufficient conditions are presented for heat kernel upper bounds for random walks on weighted graphs. Several equivalent conditions are given in the form of isoperimetric inequalities.