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12
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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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. #
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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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.
ON THE SHAPE OF THE GROUND STATE EIGENFUNCTION FOR STABLE PROCESSES
, 2004
"... MÉNDEZHERNÁNDEZ Abstract. We prove that the ground state eigenfunction for symmetric stable processes of order α ∈ (0, 2) killed upon leaving the interval (−1, 1) is concave on (−1 1 ..."
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MÉNDEZHERNÁNDEZ Abstract. We prove that the ground state eigenfunction for symmetric stable processes of order α ∈ (0, 2) killed upon leaving the interval (−1, 1) is concave on (−1 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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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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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
Trace Estimates for Stable Processes
, 2008
"... In this paper we study the behaviour in time of the trace (the partition function) of the heat semigroup associated with symmetric stable processes in domains of Rd. In particular, we show that for domains with the so called Rsmoothness property the second terms in the asymptotic as t → 0 involves ..."
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In this paper we study the behaviour in time of the trace (the partition function) of the heat semigroup associated with symmetric stable processes in domains of Rd. In particular, we show that for domains with the so called Rsmoothness property the second terms in the asymptotic as t → 0 involves the surface area of the domain, just as in the case of Brownian motion. Contents §1. Introduction and statement of main result
ON THE TRACES OF SYMMETRIC STABLE PROCESSES ON LIPSCHITZ DOMAINS
, 903
"... Abstract. It is shown that the second term in the asymptotic expansion as t→0of the trace of the semigroup of symmetric stable processes (fractional powers of the Laplacian) of order α, for any 0 < α < 2, in Lipschitz domains is given by the surface area of the boundary of the domain. This bri ..."
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Abstract. It is shown that the second term in the asymptotic expansion as t→0of the trace of the semigroup of symmetric stable processes (fractional powers of the Laplacian) of order α, for any 0 < α < 2, in Lipschitz domains is given by the surface area of the boundary of the domain. This brings the asymptotics for the trace of stable processes in domains of Euclidean space on par with those of Brownian motion (the Laplacian), as far as boundary smoothness is concerned.
SPECTRAL PROPERTIES OF THE CAUCHY PROCESS
, 2009
"... This note is an announcement of the results. The full version of this paper, including full proofs, will be available soon. ..."
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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.
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.