The First Exit Time of a Two-Dimensional Symmetric Stable Process from a Wedge

DeBlassie, R. Dante

doi:10.1214/aop/1176990735

Cited by 24 publications

(26 citation statements)

References 15 publications

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“…For α < 2, this result is due to DeBlassie (1990) for d = 2 and Méndez-Hernández (2002) for d ≥ 3. When α = 2, the result is due to Burkholder (1977).…”

Section: Proof Of Remark 14 (Ii)mentioning

confidence: 87%

Higher order PDEs and symmetric stable processes

DeBlassie

2004

Probab. Theory Relat. Fields

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We describe a connection between the semigroup of a symmetric stable process with rational index and higher order partial differential equations. As an application, we obtain a variational formula for the eigenvalues associated with the process killed upon leaving a bounded open set D. The variational formula is more "user friendly" than the classical Rayleigh-Ritz formula. We illustrate this by obtaining upper bounds on the eigenvalues in terms of Dirichlet eigenvalues of the Laplacian on D. These results generalize some work of Bañuelos and Kulczycki on the Cauchy process. Along the way we prove an operator inequality for the operators associated with the transition densities of Brownian motion and the Brownian motion killed upon leaving D.

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“…For α < 2, this result is due to DeBlassie (1990) for d = 2 and Méndez-Hernández (2002) for d ≥ 3. When α = 2, the result is due to Burkholder (1977).…”

Section: Proof Of Remark 14 (Ii)mentioning

confidence: 87%

Higher order PDEs and symmetric stable processes

DeBlassie

2004

Probab. Theory Relat. Fields

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“…The purpose of the present paper is to obtain an analogue of the results in [DZ] and [BS] for conditioned symmetric stable processes in generalized cones and to extend the results in [De2] and [M] for the unconditioned processes to more general cones. Our method is motivated by, but different then, the method in [DZ].…”

Section: Introductionmentioning

confidence: 87%

“…In particular, the results in [DZ] and [BS] provide the analogue of Burkholder's result for conditioned Brownian motion. In [De2], DeBlassie obtained a counterpart of Burkholder's result for symmetric stable processes in R 2 . DeBlassie's result was extended to all dimensions by P. Méndez-Hernández [M].…”

Section: Introductionmentioning

confidence: 94%

Symmetric Stable Processes in Cones

Bañuelos

Bogdan

2004

Potential Analysis

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“…Extension of some of the abovementioned results pertaining to cones to the case 0 < α < 2 (that is, for {X t }) were given in [12], [13], [16] and [1]; see also [7]. It should be emphasized that while there are many similarities between {X t } and {B 2t }, there also exist essential differences in their respective properties and their proofs.…”

Section: Theorem 1 Let P ≥mentioning

confidence: 99%