Probability Theory and Related Fields

, Volume 129, Issue 4, pp 495–536 | Cite as

Higher order PDEs and symmetric stable processes



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.


Differential Equation Partial Differential Equation Brownian Motion Rational Index Stable Process 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

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