We obtain Gaussian upper and lower bounds on the transition density qt(x, y) of the continuous time simple random walk on a supercritical percolation cluster C ∞ in the Euclidean lattice. The bounds, analogous to Aronsen’s bounds for uniformly elliptic divergence form diffusions, hold with constants ci depending only on p (the percolation probability) and d. The irregular nature of the medium means that the bound for qt(x, ·) only holds for t ≥ Sx(ω), where the constant Sx(ω) depends on the percolation configuration ω.

We show that, if a certain Sobolev inequality holds, then a scale-invariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1

We consider the symmetric non-local Dirichlet form (E, F) given by E(f, f) = (f(y) − f(x)) 2 J(x, y)dxdy Rd Rd with F the closure of the set of C 1 functions on R d with compact support with respect to E1, where E1(f, f): = E(f, f) + ∫ R d f(x) 2 dx, and where the jump kernel J satisfies κ1|y − x | −d−α ≤ J(x, y) ≤ κ2|y − x | −d−β for 0 < α < β < 2, |x − y | < 1. This assumption allows the corresponding jump process to have jump intensities whose size depends on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to (E, F). We prove a parabolic Harnack inequality for nonnegative functions that solve the heat equation with respect to E. Finally we construct an example where the corresponding harmonic functions need not be continuous.

We extend the concept of intrinsic ultracontractivity to non-symmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of non-symmetric second order elliptic operators in bounded Lipschitz domains.

In this paper, we develop a universal way of obtaining Gaussian upper bounds of the heat kernel on Riemannian manifolds. By the word "Gaussian" we mean those estimates which contain a Gaussian exponential factor similar to one which enters the explicit formula for the heat kernel of the conventional Laplace operator in R...

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

We consider Schrodinger semigroups e.- IH, H =-A+V on Iw ” with V---cIxl- ’ as 1x1--rco, O<c<[(l/2)(n-2)] * with H>O. We determine the exact power law divergence of I~e-‘Hi~p,p and of some IIe-‘Hlly,p as maps from Lp to Lq. The results are expressed most naturally in terms of the power a for which there exists a positive resonance 9 such that Hq = 0, q(x)- 1.x-‘.:Ta 1991 Academic Press, Inc. 1.

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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)

Let X be a mixture of independent Brownian motion and symmetric stable process. In this paper we establish sharp bounds for transition density of X, and prove a parabolic Harnack inequality for nonnegative parabolic functions of X.