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.

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...

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The aim of this article is to provide a scheme for simulating diffusion processes evolving in one-dimensional discontinuous media. This scheme does not rely on smoothing the coefficients that appear in the infinitesimal generator of the diffusion processes, but uses instead an exact description of the behavior of their trajectories when they reach the points of discontinuity. This description is supplied with the local comparison of the trajectories of the diffusion processes with those of a skew Brownian motion. 1. Introduction. The

this paper is to estimate the hitting probability function / K (t; x) := P x (9 s 2 [0; t] ; X s 2 K) where K ae M is a fixed compact set. In words, / K (t; x) is the probability that Brownian motion started at x hits K by time t. Our goal is to obtain precise estimates on / K for all t ? 0 and x outside a neighborhood of K, hence avoiding the somewhat different question of the behavior of / K near the boundary of K. In the context of Riemannian manifolds, this natural question has been considered only in a handful of papers including [2], [4]. We were led to study / K in our attempt to develop sharp heat kernel estimates on manifolds with more than one end. Indeed, the proof of the heat kernel estimates announced in [20] depends in a crucial way on the results of the present paper (see [21]). In this context, it turns out to be important to estimate also the time derivative @ t / K (t; x) which is a positive function. We develop a general approach which allows to obtain estimates of / K in terms of the heat kernel p(t; x; y) or closely related objects such as the Dirichlet heat kernel p U (t; x; y) of some open set U . In the case when X t is transient, that is, M is non-parabolic, we show that the behavior of / K (t; x), away from K, is comparable to that of Z t 0 p(s; x; y)ds; where y is a reference point on @K. If (X t ) t?0 is recurrent, that is, M is parabolic, we obtain similar estimates through Z t 0 p U (s; x; y)ds where U is a certain region slightly larger than\Omega := M n K. We also show that @ t / K (t; x) is comparable to p\Omega (t; x; y) where y stays at a certain distance from @K. For precise statements, see Theorems 3.3, 3.5, 3.7 and Corollaries 3.9, 3.10. Using the known results concerning the heat kernel p(t; x; y) and the results of [23...

. We prove the doubling property of L-caloric measure corresponding to the second order parabolic equation in the whole space and in Lipschitz domains. For parabolic equations in the divergence form, a weaker form of the doubling property follows easily from a recent result, the backward Harnack inequality, and known estimates of the Green's function. Our method works for both the divergence and non-divergence cases. Moreover, the backward Harnack inequality and estimates of the Green's function are not needed in the course of proof.

We consider linear nonautonomous second order parabolic equations on R^N. Under an instability condition, we prove the existence of two complementary Floquet bundles, one spanned by a positive entire solution- the principal Floquet bundle, the other one consisting of sign-changing solutions. We establish an exponential separation between the two bundles, showing in particular that a class of signchanging solutions are exponentially dominated by positive solutions.

Abstract. Some continuum models for dislocation interactions in a simple crystal geometry are studied. The simplest models are mixed systems ofconservation laws which are shown to exhibit singularities and instabilities. These are then regularized, leading to parabolic free-boundary problems. In both cases, solutions describing the formation of structures such as dislocation pile-ups are discussed.

Abstract. Does a minimal harmonic function h remain minimal when it is viewed as a parabolic function? The question is answered for a class of long thin semi-infinite tubes D ⊂ R d of variable width and minimal harmonic functions h corresponding to the boundary point of D “at infinity. ” Suppose f(u) is the width of the tube u units away from its endpoint and f is a Lipschitz function. The answer to the question is affirmative if and only if ∫ ∞ f 3 (u)du = ∞. If the test fails, there exist parabolic h-transforms of space-time Brownian motion in D with infinite lifetime which are not time-homogenous.