First Eigenvalue of One-dimensional Diffusion Processes

Jian Wang (School of Mathematics and Computer Science, Fujian Normal University)

Abstract


We consider the first Dirichlet eigenvalue of diffusion operators on the half line. A criterion for the equivalence of the first Dirichlet eigenvalue with respect to the maximum domain and that to the minimum domain is presented. We also describle the relationships between the first Dirichlet eigenvalue of transient diffusion operators and the standard Muckenhoupt's conditions for the dual weighted Hardy inequality. Pinsky's result [17] and Chen's variational formulas [8] are reviewed, and both provide the original motivation for this research.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 232-244

Publication Date: May 24, 2009

DOI: 10.1214/ECP.v14-1464

References

  1. C. Albanese and A. Kuznetsov. Transformations of Markov processes and classification scheme for solvable driftless diffusions. http://arxiv.org/0710.1596, 2007
  2. S. G. Bobkov and F. Goetze. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1999), no.1, 1-28. Math. Review 1682772
  3. S. G. Bobkov and F. Goetze. Hardy type inequalities via Riccati and Sturm-Liouville equations. Sobolev Spaces in Mathematics I: Sobolev Type Inequalities, Inter. Math. Ser. 8 (2009), 69-86.
  4. M. F. Chen. Analystic proof of dual variational formula for the first eigenvalue in dimension one. Sci. Chin. Ser. A 42 (1999), no. 8, 805-815. Math. Review 1738551
  5. M. F. Chen. Explicit bounds of the first eigenvalue. Sci. Chin. Ser. A 43 (2000), no. 10, 1051-1059. Math. Review 1802148
  6. M. F. Chen. Variational formulas and approximation theorems for the first eigenvalue in dimension one. Sci. Chin. Ser. A 44 (2001), no. 4, 409-418. Math. Review 1831443
  7. M. F. Chen. Exponential decay of birth-death processes.Preprint, 2008.
  8. M. F. Chen. Eigenvalues, Inequalities and Ergodic Theory. Springer, London, 2005. Math. Review 2105651
  9. M. F. Chen and F. Y. Wang. Estimation of spectral gap for elliptic operators.Trans. Amer. Math.Soc. 349 (1997), no. 3, 1239-1267. Math. Review 1401516
  10. M. Fukushima, Y. Oshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Stud. Math., 19, Berlin,1994. Math. Review 1303354
  11. K. Ito and Jr. H. P. McKean.. Diffusion Processes and Their Sample Paths. Springer-Verlag, Berlin, Heidelberg and New York, 1965. Math. Review 0199891
  12. Y. H. Mao. Nash inequalities for Markov processes in dimension one. Acta Math. Sin. Eng. Ser. 18 (2002), no. 1, 147-156. Math. Review 1894847
  13. P. Mandl. Analytical Treatment of One-dimensional Markov Processes. Springer-Verlag, Berlin, Heidelberg and New York, 1968. Math. Review 0247667
  14. L. Miclo. An example of application of discrete Hardy's inequalities. Markov Processes Relat. Fields 5 (1999), no.3, 319-313. Math. Review 1710983
  15. B. Muckenhoupt. Hardy inequality with weights. Studia. Math. 44 (1972), 31-38. Math. Review 0311856
  16. R. D. Nussbaum and Y. Pinchover. On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications. Journal d'Analyse Mathematique 59 (1992),162-177.Math. Review 1226957
  17. R. G. Pinsky.Explicit and almost explicit spectral calculations for diffusion operators. J. Funct. Anal. 256 (2009), 3279-3312.
  18. M. Reed. and B. Simon. Methods of Modern Mathematical Physics. Analysis of Operators, Academic Press, New York,1978.Math. Review 0493421
  19. J. Wang. First Dirichlet eigenvalue of transient birth-death processes.Preprint, 2008 .
  20. F. Y. Wang. Application of coupling methods to the Neumann eigenvalue problem.Probab.Theory Relat. Fields 98 (1994), no. 3, 299-306. Math. Review 1262968
  21. F. Y. Wang. Functional Inequalities, Markov Semigroups and Spectral Theory. Science Press, Beijing/New York, 2004.
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.