Abstract
In this paper, we prove some limit theorems for killed Brownian motion during its life time. The emphases are on quasi-stationarity and quasi-ergodicity and related problems. On one hand, using an eigenfunction expansion for the transition density, we prove the existence and uniqueness of both quasi-stationary distribution (qsd) and mean ratio quasi-stationary distribution (mrqsd). The later is shown to be closely related to laws of large numbers (LLN) and to quasi-ergodicity. We further show that the mrqsd is the unique stationary distribution of a certain limiting ergodic diffusion process of the BM conditioned on not having been killed. We also show that a phase transition occurs from mrqsd to qsd. On the other hand, we study the large deviation behavior related to the above problems. A key observation is that the mrqsd is the unique minimum of certain large deviation rate function. We further prove that the limiting diffusion process also satisfies a large deviation principle with the rate function attaining its unique minimum at the mrqsd. These give interpretations of the mrqsd from different points of view, and establish some intrinsic connections among the above topics. Some general results concerning Yaglom limit, moment convergence and LLN are also obtained.
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References
Breyer L A, Roberts G O. A quasi-ergodic theorem for evanescent processes. Stochastic Process Appl, 1999, 84: 177–186
Chung K L, Zhao Z X. From Brownian Motion to Schrodinger’s Equation. Berlin: Springer, 1995
Darroch J N, Seneta E. On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J Appl Probab, 1967, 4: 192–196
Dembo A, Zeitouni O. Large Deviations Techniques and Applications, 2nd ed. New York: Springer, 1998
Donsker M D, Varadhan S R S. On the principle eigenvalue of second-order elliptic differential operators. Comm Pure Appl Math, 1976, 29: 595–621
Van Doorn E A, Pollett P K. Quasi-stationary distributions. http://eprints.eemcs.utwente.nl/20245/, 2011
Evans L C. Partial Differential Equations. Providence: American Mathematical Society, 1998
Flaspohler D C. Quasi-stationary distributions for absorbing continuous-time denumerable markov chains. Ann Inst Statist Math, 1973, 26: 351–356
Fukushima M, Oshima O, Takeda M. Dirichlet Forms and Symmetric Markov Processes. Berlin-New York: Walter de Gruyter, 2011
Pinsky R G. On the convergence of diffusion processes conditioned to remain in a bounded for large time to limiting positive recurrent diffusion processes. Ann Probab, 1985, 18: 363–378
Port S C, Stone C J. Brownian Motion and Classical Potential Theory. New York: Academic Press, 1978
Stroock D W. An Introduction to the Theory of Large Deviations. Berlin: Springer, 1984
Vere-Jones D. Some limit theorems for evanescent processes. Australian J Statist, 1969, 11: 67–78
Yang Y. Brownian motion with boundary conditions (in Chinese). Undergraduate Thesis. Beijing: Tsinghua University, 2011
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Chen, J., Jian, S. Some limit theorems of killed Brownian motion. Sci. China Math. 56, 497–514 (2013). https://doi.org/10.1007/s11425-012-4430-y
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DOI: https://doi.org/10.1007/s11425-012-4430-y
Keywords
- killed Brownian motion
- quasi-stationary distribution (qsd)
- mean ratio quasi stationary distribution (mrqsd)
- large deviation principle (LDP)
- phase transition