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Eigentime Identity for One-Dimensional Diffusion Processes

Part of: Markov processes General theory of linear operators

Published online by Cambridge University Press:  30 January 2018

Li-Juan Cheng  and
Yong-Hua Mao
Li-Juan Cheng*
Affiliation:
Zhejiang University of Technology and Beijing Normal University
Yong-Hua Mao*
Affiliation:
Beijing Normal University
*
∗ Postal address: Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China.
∗ Postal address: Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China.
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Abstract

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The eigentime identity for one-dimensional diffusion processes on the halfline with an entrance boundary at ∞ is obtained by using the trace of the deviation kernel. For the case of an exit boundary at ∞, a similar eigentime identity is presented with the aid of the Green function. Explicit equivalent statements are also listed in terms of the strong ergodicity or the uniform decay for diffusion processes.

Keywords

Hitting timelifetimeeigentime identitystrong ergodicityuniform decayeigenvalue

MSC classification

Primary: 60J60: Diffusion processes 47A75: Eigenvalue problems
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 224 - 237
Copyright
© Applied Probability Trust 

References

Aldous, D. J. and Fill, J. A. (2002). Reversible Markov chains and random walks on graphs. Unpublished manuscript. Available at http://www.stat.berkeley.edu/≃aldous/RWG/book.html.Google Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.CrossRefGoogle Scholar
Cheng, L.-J. and Mao, Y.-H. (2013). Passage time distribution for one-dimensional diffusion processes. Preprint.Google Scholar
Douglas, R. G. (1998). Banach Algebra Techniques in Operator Theory (Graduate Texts Math. 179). Springer, New York.CrossRefGoogle Scholar
Feller, W. (1952). The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 55, 468–519.CrossRefGoogle Scholar
Feller, W. (1954). Diffusion processes in one dimension. Trans. Amer. Math. Soc. 77, 1–31.CrossRefGoogle Scholar
Itô, K. and Mckean, H. P. Jr. (1965). Diffusion Processes and Their Sample Paths. Spinger, Berlin.Google Scholar
Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Liggett, T. M. (1989). Exponential L2 convergence of attractive reversible nearest particle systems. Ann. Prob. 17, 403–432.CrossRefGoogle Scholar
Mandl, P. (1968). Analytical Treatmeat of One-Dimensional Markov Processes. Springer, Berlin.Google Scholar
Mao, Y.-H. (2002). Deviation kernels for one-dimensional diffusion processes. Preprint. Available at http://math.bnu.edu.cn/probab/Workshop2002/Preprints/MaoYH.pdf.Google Scholar
Mao, Y.-H. (2002). Strong ergodicity for Markov processes by coupling methods. J. Appl. Prob. 39, 839–852.CrossRefGoogle Scholar
Mao, Y.-H. (2004). The eigentime identity for continuous-time ergodic Markov chains. J. Appl. Prob. 41, 1071–1080.CrossRefGoogle Scholar
Mao, Y. H. (2006). Eigentime identity for trainsient Markov chains. J. Math. Anal. Appl. 315, 415–424.CrossRefGoogle Scholar
Mao, Y. H. (2006). Some new results on strong ergodicity. Front. Math. China 1, 105–109.CrossRefGoogle Scholar
Mao, Y.-H. and Ouyang, S.-X. (2006). Strong ergodicity and uniform decay for Markov processes. Math. Appl. 19, 580–586.Google Scholar
Pitman, J. and Yor, M. (2003). Hitting, occupation and inverse local times of one-dimensional diffusions: martingale and excursion approaches. Bernoulli 9, 1–24.CrossRefGoogle Scholar
Reed, M. and Simon, B. (1972). Methods of Modern Mathematical Physics. I. Functional Analysis. Academic Press, New York.Google Scholar
Reed, M. and Simon, B. (1975). Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. Academic Press, New York.Google Scholar
Wu, L. (2004). Essential spectral radius for Markov semigroups. I. Discrete time case. Prob. Theory Relat. Fields 128, 255–321.CrossRefGoogle Scholar