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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 ..."
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Cited by 40 (16 self)
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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. #
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. ..."
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Cited by 36 (14 self)
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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.
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 ..."
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Cited by 32 (5 self)
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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.
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. ..."
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Cited by 30 (23 self)
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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.
General gauge and conditional gauge theorems
 Ann. Probab
, 2002
"... General gauge and conditional gauge theorems are established for a large class of (not necessarily symmetric) strong Markov processes, including Brownian motions with singular drifts and symmetric stable processes. Furthermore, new classes of functions are introduced under which the general gauge an ..."
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Cited by 28 (14 self)
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General gauge and conditional gauge theorems are established for a large class of (not necessarily symmetric) strong Markov processes, including Brownian motions with singular drifts and symmetric stable processes. Furthermore, new classes of functions are introduced under which the general gauge and conditional gauge theorems hold. These classes are larger than the classical Kato class when the process is Brownian motion in a bounded C 1,1 domain. 1. Introduction. Given a strong Markov process X and a potential q, the conditional expectation u(x, y) of the Feynman–Kac transform of X by q is called the conditional gauge function. (The precise definition will be given later.) The function u is important in studying the potential theory of the Schrödingertype operator L + q, as it is the ratio of the Green’s function of L + q and that
Conditional gauge theorem for nonlocal FeynmanKac transforms
 PROBAB. THEORY RELAT. FIELDS
, 2003
"... ..."
Symmetric stable processes in cones
 Potential Anal
"... Abstract. We study exponents of integrability of the first exit time from generalized cones for conditioned rotation invariant stable Lévy processes. Along the way, we introduce the “spherical fractional Laplacian ” and derive some of its spectral properties. 1. ..."
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Cited by 16 (6 self)
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Abstract. We study exponents of integrability of the first exit time from generalized cones for conditioned rotation invariant stable Lévy processes. Along the way, we introduce the “spherical fractional Laplacian ” and derive some of its spectral properties. 1.
Eigenvalue gaps for the Cauchy process and a Poincare inequality
 J. Funct. Anal
, 2006
"... A connection between the semigroup of the Cauchy process killed upon exiting a domain D and a mixed boundary value problem for the Laplacian in one dimension higher known as the mixed Steklov problem, was established in [6]. From this, a variational characterization for the eigenvalues λn, n ≥ 1, of ..."
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Cited by 11 (4 self)
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A connection between the semigroup of the Cauchy process killed upon exiting a domain D and a mixed boundary value problem for the Laplacian in one dimension higher known as the mixed Steklov problem, was established in [6]. From this, a variational characterization for the eigenvalues λn, n ≥ 1, of the Cauchy process in D was obtained. In this paper we obtain a variational characterization of the difference between λn and λ1. We study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for λ ∗ −λ1 where λ ∗ is the eigenvalue corresponding to the “first ” antisymmetric eigenfunction for D. The proof is based on a variational characterization of λ ∗ − λ1 and on a weighted Poincaré–type inequality. The Poincaré inequality is valid for all α symmetric stable processes, 0 < α ≤ 2, and any other process obtained from Brownian motion by subordination. We also prove upper bound estimates for the spectral gap λ2 − λ1 in bounded convex domains.
Spectral gap for the Cauchy process on convex, symmetric domains
 Comm. Partial Differential Equations
"... Let D ⊂ R2 be a bounded convex domain which is symmetric relative to both coordinate axes. Assume that [−a, a] × [−b, b], a ≥ b> 0 is the smallest rectangle (with sides parallel to the coordinate axes) containing D. Let {λn}∞n=1 be the eigenvalues corresponding to the semigroup of the Cauchy pro ..."
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Cited by 9 (5 self)
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Let D ⊂ R2 be a bounded convex domain which is symmetric relative to both coordinate axes. Assume that [−a, a] × [−b, b], a ≥ b> 0 is the smallest rectangle (with sides parallel to the coordinate axes) containing D. Let {λn}∞n=1 be the eigenvalues corresponding to the semigroup of the Cauchy process killed upon exiting D. We obtain the following estimate on the spectral gap: λ2 − λ1 ≥ Cb a2 where C is an absolute constant. The estimate is obtained by proving new weighted Poincare ́ inequalities and appealing to the connection between the eigenvalue problem for the Cauchy process and a mixed boundary value problem for the Laplacian in one dimension higher