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.

Subordination of a killed Brownian motion in a domain D ⊂ R d via an α/2-stable subordinator gives rise to a process Zt whose infinitesimal generator is −(−�|D) α/2, the fractional power of the negative Dirichlet Laplacian. In this paper we establish upper and lower estimates for the density, Green function and jumping function of Zt when D is either a bounded C 1,1 domain or an exterior C 1,1 domain. Our estimates are sharp in the sense that the upper and lower estimates differ only by a multiplicative constant.

We establish a boundary Harnack principle for a large class of subordinate Brownian motions, including mixtures of symmetric stable processes, in κ-fat open sets (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the minimal Martin boundary of bounded κ-fat open sets with respect to these processes with their Euclidean boundaries.

We are interested in stationary "fluid" random evolutions with independent increments. Under some mild assumptions, we show they are solutions of a stochastic differential equation (SDE). There are situations where these evolutions are not described by flows of diffeomorphisms, but by coalescing flows or by flows of probability kernels. In an intermediate phase, for which there exists a coalescing flow and a flow of kernels solution of the SDE, a classification is given: All solutions of the SDE can be obtained by filtering a coalescing motion with respect to a sub-noise containing the Gaussian part of its noise. Thus, the coalescing motion cannot be described by a white noise.

Weak convergence of the Euler scheme for stochastic differential equations is established when coefficients are discontinuous on a set of Lebesgue measure zero. The rate of convergence is presented when coeffi-cients are Hölder continuous. Monte Carlo simulations are also discussed. 1. Introduction. We

The purpose of this paper is to find optimal estimates for the Green function of a half-space of the relativistic α-stable process with parameter m on R d space. This process has an infinitesimal generator of the form mI −(m 2/α I −∆) α/2, where 0 < α < 2, m> 0, and reduces to the isotropic α-stable process for m = 0. Its potential theory for open bounded sets has been well developed throughout the recent years however almost nothing was known about the behaviour of the process on unbounded sets. The present paper is intended to fill this gap and we provide two-sided sharp estimates for the Green function for a half-space. As a byproduct we obtain some improvements of the estimates known for bounded sets. Our approach combines the recent results obtained in [5], where an explicit integral formula for the m-resolvent of a half-space was found, with estimates of the transition densities for the killed process on exiting a half-space. The main result states that the Green function is comparable with the Green function for the Brownian motion if the points are away from the boundary of a half-space and their distance is greater than one. On the other hand for the remaining points the Green function is somehow related the Green function for the isotropic α-stable process. For example, for d ≥ 3, it is comparable with the Green function for the isotropic α-stable process, provided that the points are close enough.

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This paper is devoted to the memory of Professor Andrzej Hulanicki. We give sharp estimates for the transition density of the isotropic stable Lévy process killed when leaving a right circular cone. 1

Limit theorems for the normalized laws with respect to two kinds of weight functionals are studied for any symmetric stable Lévy process of index 1 < α ≤ 2. The first kind is a function of the local time at the origin, and the second kind is the exponential of an occupation time integral. Special emphasis is put on the role played by a stable Lévy counterpart of the universal σ-finite measure, found in [9] and [10], which unifies the corresponding limit theorems in the Brownian setup for which α = 2.