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Higher order PDEs and symmetric stable processes
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  • Published: 29 April 2004

Higher order PDEs and symmetric stable processes

  • R. Dante DeBlassie1 

Probability Theory and Related Fields volume 129, pages 495–536 (2004)Cite this article

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Abstract.

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 Banuelos 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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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA

    R. Dante DeBlassie

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  1. R. Dante DeBlassie
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Correspondence to R. Dante DeBlassie.

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Dante DeBlassie, R. Higher order PDEs and symmetric stable processes. Probab. Theory Relat. Fields 129, 495–536 (2004). https://doi.org/10.1007/s00440-004-0347-x

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  • Received: 30 October 2003

  • Revised: 24 January 2004

  • Published: 29 April 2004

  • Issue Date: August 2004

  • DOI: https://doi.org/10.1007/s00440-004-0347-x

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Brownian Motion
  • Rational Index
  • Stable Process
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