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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 15 (9 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. #
Heat kernel of fractional Laplacian in cones
 Colloq. Math
"... 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 ..."
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Cited by 5 (1 self)
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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
αCONTINUITY PROPERTIES OF THE SYMMETRIC αSTABLE PROCESS
, 2004
"... Abstract. Let D be a domain of finite Lebesgue measure in Rd and let XD t be the symmetric αstable process killed upon exiting D. Each element of the set {λα i}∞i=1 of eigenvalues associated to XD t, regarded as a function of α ∈ (0, 2), is right continuous. In addition, if D is Lipschitz and bound ..."
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Cited by 4 (0 self)
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Abstract. Let D be a domain of finite Lebesgue measure in Rd and let XD t be the symmetric αstable process killed upon exiting D. Each element of the set {λα i}∞i=1 of eigenvalues associated to XD t, regarded as a function of α ∈ (0, 2), is right continuous. In addition, if D is Lipschitz and bounded, then each λα i is continuous in α and the set of associated eigenfunctions is precompact. We also prove that if D is a domain of finite Lebesgue measure, then for all 0 < α < β ≤ 2 and i ≥ 1, λ α i ≤ λ β] α/β i Previously, this bound had been known only for β = 2 and α rational. 1.
SYMMETRIC STABLE PROCESSES IN PARABOLA–SHAPED REGIONS
, 2004
"... Abstract. We identify the critical exponent of integrability of the first exit time of rotation invariant stable Lévy process from parabola–shaped region. 1. ..."
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Abstract. We identify the critical exponent of integrability of the first exit time of rotation invariant stable Lévy process from parabola–shaped region. 1.
SPECTRAL PROPERTIES OF THE CAUCHY PROCESS
, 906
"... This note is an announcement of the results. The full version of this paper, including full proofs, will be available soon. 1. ..."
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This note is an announcement of the results. The full version of this paper, including full proofs, will be available soon. 1.
SPECTRAL PROPERTIES OF THE CAUCHY PROCESS ON HALFLINE AND INTERVAL
, 906
"... Abstract. We study the spectral properties of the transition semigroup of the killed onedimensional Cauchy process on the halfline (0, ∞) and the interval (−1, 1). This process is related to the square root of onedimensional Laplacian A = − − d2 dx2 with a Dirichlet exterior condition (on a comp ..."
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Abstract. We study the spectral properties of the transition semigroup of the killed onedimensional Cauchy process on the halfline (0, ∞) and the interval (−1, 1). This process is related to the square root of onedimensional Laplacian A = − − d2 dx2 with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the halfplane. For the halfline, an explicit formula for generalized eigenfunctions ψλ of A is derived, and then used to construct spectral representation of A. Explicit formulas for the transition density of the killed Cauchy process in the halfline (or the heat kernel of A in (0, ∞)), and for the distribution of the first exit time from the halfline follow. The formula for ψλ is also used to construct approximations to eigenfunctions of A in the interval. For the eigenvalues λn of A in the interval the asymptotic formula λn = nπ π 1 − + O ( ) is 2 8 n derived, and all eigenvalues λn are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues λn are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to 9th decimal point. 1.