Results 1  10
of
28
Drift transforms and Green function estimates for discontinuous processes
 JOURNAL OF FUNCTIONAL ANALYSIS
, 2003
"... ..."
Harnack inequality for some classes of Markov processes
"... In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps. ..."
Abstract

Cited by 25 (13 self)
 Add to MetaCart
In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps.
Estimates of Green function for some perturbations of fractional Laplacian
"... Suppose that Y (t) is a ddimensional Lévy symmetric process for which its Lévy measure differs from the Lévy measure of the isotropic αstable 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 ..."
Abstract

Cited by 23 (3 self)
 Add to MetaCart
Suppose that Y (t) is a ddimensional Lévy symmetric process for which its Lévy measure differs from the Lévy measure of the isotropic αstable 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 αstable 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
Potential theory of truncated stable processes
 MATHEMATISCHE ZEITSCHRIFT
, 2007
"... For any α ∈ (0, 2), a truncated symmetric αstable process is a symmetric Lévy process in R d with a Lévy density given by cx  −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 nonneg ..."
Abstract

Cited by 22 (18 self)
 Add to MetaCart
For any α ∈ (0, 2), a truncated symmetric αstable process is a symmetric Lévy process in R d with a Lévy density given by cx  −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 nonconvex domain for which the boundary Harnack principle fails.
INTRINSIC ULTRACONTRACTIVITY OF NONSYMMETRIC DIFFUSION SEMIGROUPS IN BOUNDED DOMAINS
 TOHOKU MATH. J.
, 2008
"... We extend the concept of intrinsic ultracontractivity to nonsymmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of nonsymmetric second order elliptic operators in bounded Lipschitz domains. ..."
Abstract

Cited by 21 (18 self)
 Add to MetaCart
We extend the concept of intrinsic ultracontractivity to nonsymmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of nonsymmetric second order elliptic operators in bounded Lipschitz domains.
Intrinsic ultracontractivity of the Feynman–Kac semigroup for relativisitic stable processes
 ZBL 1112.47034 MR 2231884
, 2006
"... Let Xt be the relativistic αstable 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 FeynmanKac semigroup Tt for this process with generator H (α) 0 − V, V ≥ 0, V locally bounded. We prove that if lim ..."
Abstract

Cited by 21 (3 self)
 Add to MetaCart
Let Xt be the relativistic αstable 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 FeynmanKac 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.
Censored Stable Processes
 Probab. Theory Relat. Fields
"... 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. ..."
Abstract

Cited by 19 (11 self)
 Add to MetaCart
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.
Heat kernel estimates for the fractional Laplacian
, 2009
"... We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains. ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains.
Sharp bounds on the density, Green function and jumping function of subordinate killed BM
 PROBAB. THEORY RELAT. FIELDS
, 2004
"... Subordination of a killed Brownian motion in a domain D ⊂ R d via an α/2stable 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 ..."
Abstract

Cited by 15 (12 self)
 Add to MetaCart
Subordination of a killed Brownian motion in a domain D ⊂ R d via an α/2stable 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.
The Cauchy Process and the Steklov Problem
"... 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 dimen ..."
Abstract

Cited by 15 (9 self)
 Add to MetaCart
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. #