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.
References
Bañuelos, R., Kulczycki, T.: The Cauchy process and the Steklov problem. Preprint, 2003
Bañuelos, R., Latała, R., Méndez–Hernández, P.: A Brascamp–Lieb–Luttinger-type inequality and applications to symmetric stable processes. Proc. Amer. Math. Soc. 129, 2997–3008 (2001)
Bass, R.F., Cranston, M.: Exit times for symmetric stable processes in ℝn, Ann. Probab. 11, 578–588 (1983)
Bertoin, J.: Lévy Processes, Cambridge University Press, Cambridge (1996)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)
Blumenthal, R.M., Getoor, R.K.: Some theorems on stable processes. Trans. Amer. Math. Soc. 95, 263–273 (1960)
Burkholder, D.L.: Exit times of Brownian motion, harmonic majorization, and Hardy spaces. Adv. in Math. 26, 182–205 (1977)
Chavel, I.: Eigenvalues in Riemannian Geometry. Academic, New York (1984)
Chen, Z.Q., Song, R.: Intrinsic ultracontractivity, conditional lifetimes and conditional gauge for symmetric stable processes on rough domains. Illinois J. Math. 44, 138–160 (2000)
Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge (1989)
DeBlassie, R.D.: Exit times from cones in ℝn of Brownian motion. Probab. Theory Relat. Fields 74, 1–29 (1987)
DeBlassie, R.D.: The first exit time of a two-dimensional symmetric stable process from a wedge. Ann. Probab. 18, 1034–1070 (1990)
Feller, W.: An Introduction to Probability Theory and its Applications, Wiley, New York (1971)
Kulczycki, T.: Intrinsic ultracontractivity for symmetric stable processes. Bull. Polish Acad. Sci. Math. 46, 325–334 (1998)
Méndez–Hernández, P.J.:Exit times from cones in ℝn of symmetric stable processes. Illinois J. Math. 46, 155–163 (2002)
Port, S.C., Stone, C.J.: Brownian Motion and Classical Potential Theory. Academic, New York (1978)
Zolotarev, V.M.: One-Dimensional Stable Distributions. Amer. Math. Soc., Providence, R.I. (1986)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-004-0347-x
Keywords
- Differential Equation
- Partial Differential Equation
- Brownian Motion
- Rational Index
- Stable Process