arXiv subjects
Publications and source records attributed to Tomasz Grzywny.
The purpose of this paper is to find optimal estimates for the Green function and the Poisson kernel for a half-line and intervals of the geometric stable process with parameter $α\in(0,2]$. This process has an infinitesimal generator of the form $-\log(1+(-Δ)^{α/2})$. As an application we prove the scale invariant Harnack inequality as well as the boundary Harnack principle.
We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains.
We give sharp estimates for the transition density of the isotropic stable Lévy process killed when leaving a right circular cone.
The purpose of this paper is to find optimal estimates for the Green function of a half-space of {\it the relativistic $α$-stable process} with parameter $m$ on $\Rd$ space. This process has an infinitesimal generator of the form $mI-(m^{2/α}I-Δ)^{α/2},$ 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 specially for balls. The advantage of these estimates is a clarification of the relationship between the diameter of the ball and the parameter $m$ of the process. 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 $d\ge3$, it is comparable with the Green function for the isotropic $α$-stable process, provided that the points are close enough.
We prove the intrinsic ultracontractivity for the semigroup generated by a large class of symmetric Levy processes such that the Levy measure satisfies some conditions in the neighborhood of 0, killed on exiting a bounded and connected Lipschitz domain.
Suppose that Y(t) is a d-dimensional Levy symmetric process for which its Levy measure differs from the Levy measure of the isotropic alpha-stable process (0 0, we prove that the Green functions are comparable, provided D is connected. These results apply for example to alpha-stable relativistic process. This process was studied in recent years. In the paper we also considered one dimensional case for alpha<= 1 and proved that the Green functions for an open and bounded interval are comparable.