Abstract not found

In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps.

Suppose that Y (t) is a d-dimensional Lévy symmetric process for which its Lévy measure differs from the Lévy measure of the isotropic α-stable process (0 < α < 2) by a finite signed measure. For a bounded Lipschitz set D we compare the Green functions of the process Y and its stable counterpart. We prove a few comparability results either one sided or two sided. Assuming an additional condition about the difference of the densities of the Lévy measures, namely that it is of order of |x | −d+̺ as |x | → 0, where ̺> 0, we prove that the Green functions are comparable, provided D is connected. These results apply for example to α-stable relativistic process. This process was studied in [R, CS3], where the bounds for its Green functions were proved for d> α and smooth sets. In the paper we also considered one dimensional case for α ≥ 1 and proved that the Green functions for an open and bounded interval are comparable. 1

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.

For any α ∈ (0, 2), a truncated symmetric α-stable process is a symmetric Lévy process in R d with a Lévy density given by c|x | −d−α 1{|x|<1} for some constant c. In this paper we study the potential theory of truncated symmetric stable processes in detail. We prove a Harnack inequality for nonnegative harmonic functions of these processes. We also establish a boundary Harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded convex domains. We give an example of a non-convex domain for which the boundary Harnack principle fails.

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 present several constructions of a \censored stable process" in an open set D R n , i.e., a symmetric stable process which is not allowed to jump outside D.

Let X t be a Cauchy process in R , d 1. We investigate some of the fine spectral theoretic properties of the semigroup of this process killed upon leaving a domain D. We establish a connection between the semigroup of this process and a mixed boundary value problem for the Laplacian in one dimension higher, known as the "Mixed Steklov Problem." Using this we derive a variational characterization for the eigenvalues of the Cauchy process in D. This characterization leads to many detailed properties of the eigenvalues and eigenfunctions for the Cauchy process inspired by those for Brownian motion. Our results are new even in the simplest geometric setting of the interval (-1, 1) where we obtain more precise information on the size of the second and third eigenvalues and on the geometry of their corresponding eigenfunctions. Such results, although trivial for the Laplacian, take considerable work to prove for the Cauchy processes and remain open for general symmetric #--stable processes. Along the way we present other general properties of the eigenfunctions, such as real analyticity, which even though well known in the case of the Laplacian, are not rarely available for more general symmetric #--stable processes. #

Let Xt be the relativistic α-stable process in Rd, α ∈ (0, 2), d>α, with infinitesimal generator H (α) 0 = −((− ∆ +m2/α) α/2 − m). We study intrinsic ultracontractivity (IU) for the Feynman-Kac semigroup Tt for this process with generator H (α) 0 − V, V ≥ 0, V locally bounded. We prove that if lim |x|→ ∞ V (x) =∞, then for every t>0 the operator Tt is compact. We consider the class V of potentials V such that V ≥ 0, lim |x|→ ∞ V (x) = ∞ and V is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For V in the class V we show that the semigroup Tt is IU if and only if lim |x|→ ∞ V (x)/|x | = ∞. If this condition is satisfied we also obtain sharp estimates of the first eigenfunction φ1 for Tt. Inparticular, when V (x) =|x | β, β>0, then the semigroup Tt is IU if and only if β>1. For β>1 the first eigenfunction φ1(x) is comparable to exp(−m 1/α |x|)(|x | +1) (−d−α−2β−1)/2.

Abstract: We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains.