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Abstract. We show that on a Cayley graph of a nonamenable group, a.s. the infinite clusters of Bernoulli percolation are transient for simple random walk, that simple random walk on these clusters has positive speed, and that these clusters admit bounded harmonic functions. A principal new finding on which these results are based is that such clusters admit invariant random subgraphs with positive isoperimetric constant. We also show that percolation clusters in any amenable Cayley graph a.s. admit no nonconstant harmonic Dirichlet functions. Conversely, on a Cayley graph admitting nonconstant harmonic Dirichlet functions, a.s. the infinite clusters of p-Bernoulli percolation also have nonconstant harmonic Dirichlet functions when p is sufficiently close to 1. Many conjectures and questions are presented. §1. Introduction. The question of whether various potential-theoretic properties of graphs and manifolds are preserved under perturbations or approximations has been studied for more than a

Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and

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Abstract. Let µ = (µ 1,..., µ d) be such that each µ i is a signed measure on R d belonging to the Kato class Kd,1. The existence and uniqueness of a continuous Markov process X on R d, called a Brownian motion with drift µ, was recently established by Bass and Chen. In this paper we study the potential theory of X. We show that X has a continuous density q µ and that there exist positive constants ci, i = 1, · · · , 9, such that and c1e −c2t − t d 2 e − c3 |x−y|2 2t ≤ q µ (t, x, y) ≤ c4e c5t − t d 2 e − c6 |x−y|2 2t |∇xq µ (t, x, y) | ≤ c7e c8t − t d+1 2 e − c9 |x−y|2 2t for all (t, x, y) ∈ (0, ∞) × R d × R d. We further show that, for any bounded C 1,1 domain D, the density q µ,D of X D, the process obtained by killing X upon exiting from D, has the following estimates: for any T> 0, there exist positive constants Ci, i = 1, · · · , 5, such that and C1(1 ∧ ρ(x) √ t)(1 ∧ ρ(y) √ t)t − d 2 e − C 2 |x−y|2 t ≤ q µ,D (t, x, y) ≤ C3(1 ∧ ρ(x) √)(1 ∧ t ρ(y)

We adapt the iterative scheme by Moser...

In this paper, by first deriving a global version of gradient estimates, we obtain both upper and lower bound estimates for the heat kernel satisfying Neumann boundary conditions on a compact Riemannian manifold with nonconvex boundary. 1. Introduction. Let M be a compact Riemannian manifold with boundary ∂M. In their fundamental work [L-Y], P. Li and S.T. Yau had derived a version of gradient estimates for the positive solutions to the heat equations on M. Using those estimates, they then deduced a Harnack type inequality and demonstrated

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We consider second-order linear elliptic operators of nondivergence type which are intrinsically defined on Riemannian manifolds. Cabré proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature is nonnegative. We improve Cabré’s result and, as a consequence, we give another proof to the Harnack inequality of Yau for positive harmonic functions on Riemannian manifolds with nonnegative Ricci curvature using the nondivergence structure of the Laplace operator. 1. Introduction and

We prove that a two-sided sub-Gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay. Contents