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A unifying approach to non-minimal quasi-stationary distributions for one-dimensional diffusions

Part of: Markov processes Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  08 August 2022

Kosuke Yamato Open the ORCID record for Kosuke Yamato [Opens in a new window]
Kosuke Yamato*
Affiliation:
Kyoto University
*
*Postal address: Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, Japan. Email address: yamato.kosuke.43r@st.kyoto-u.ac.jp
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Abstract

We study convergence to non-minimal quasi-stationary distributions for one-dimensional diffusions. We give a method for reducing the convergence to the tail behavior of the lifetime via a property we call the first hitting uniqueness. We apply the results to Kummer diffusions with negative drift and give a class of initial distributions converging to each non-minimal quasi-stationary distribution.

Keywords

One-dimensional diffusion processesquasi-stationary distributionslimit theoremsKummer diffusion processes

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 60B10: Convergence of probability measures
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 1106 - 1128
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopedia of Mathematics and its Applications 27). Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion: Facts and Formulae, 2nd edn (Probability and its Applications). Birkhäuser, Basel.CrossRefGoogle Scholar
Cattiaux, P., Collet, P., Lambert, A., Martínez, S., Méléard, S. and Martín, J. S. (2009). Quasi-stationary distributions and diffusion models in population dynamics. Ann. Prob. 37, 1926–1969.CrossRefGoogle Scholar
Coddington, E. A. and Levinson, N. (1955). Theory of Ordinary Differential Equations. McGraw-Hill, New York, Toronto and London.Google Scholar
Collet, P., Martínez, S. and Martín, J. S. (1995). Asymptotic laws for one-dimensional diffusions conditioned to nonabsorption. Ann. Prob. 23, 1300–1314.CrossRefGoogle Scholar
Collet, P., Martínez, S. and Martín, J. S. (2013). Quasi-Stationary Distributions: Markov Chains, Diffusions and Dynamical Systems (Probability and its Applications). Springer, Heidelberg.CrossRefGoogle Scholar
Göing-Jaeschke, A. and Yor, M. (2003). A survey and some generalizations of Bessel processes. Bernoulli 9, 313–349.CrossRefGoogle Scholar
Hening, A. and Kolb, M. (2019). Quasistationary distributions for one-dimensional diffusions with singular boundary points. Stoch. Process. Appl. 129, 1659–1696.CrossRefGoogle Scholar
Itô, K. (2006). Essentials of Stochastic Processes (Translations of Mathematical Monographs 231). American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Kolb, M. and Steinsaltz, D. (2012). Quasilimiting behavior for one-dimensional diffusions with killing. Ann. Prob. 40, 162–212.CrossRefGoogle Scholar
Kotani, S. (2007). Krein’s strings with singular left boundary. Rep. Math. Phys. 59, 305–316.CrossRefGoogle Scholar
Kotani, S. and Watanabe, S. (1982). Kren’s spectral theory of strings and generalized diffusion processes. In Functional Analysis in Markov Processes (Katata/Kyoto 1981) (Lecture Notes Math. 923), pp. 235–259. Springer, Berlin and New York.CrossRefGoogle Scholar
Lamperti, J. (1958). An occupation time theorem for a class of stochastic processes. Trans. Amer. Math. Soc. 88, 380–387.CrossRefGoogle Scholar
Littin, J. (2012). Uniqueness of quasistationary distributions and discrete spectra when $\infty$ is an entrance boundary and 0 is singular. J. Appl. Prob. 49, 719–730.CrossRefGoogle Scholar
Lladser, M. and Martín, J. S. (2000). Domain of attraction of the quasi-stationary distributions for the Ornstein–Uhlenbeck process. J. Appl. Prob. 37, 511–520.CrossRefGoogle Scholar
Magnus, W., Oberhettinger, F. and Soni, R. P. (1966). Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd edn (Grundlehren der mathematischen Wissenschaften 52). Springer, New York.CrossRefGoogle Scholar
Mandl, P. (1961). Spectral theory of semi-groups connected with diffusion processes and its application. Czechoslovak Math. J. 11, 558–569.CrossRefGoogle Scholar
Martínez, S. and Martín, J. S. (2001). Rates of decay and h-processes for one dimensional diffusions conditioned on non-absorption. J. Theoret. Prob. 14, 199–212.CrossRefGoogle Scholar
Martínez, S., Picco, P. and Martín, J. S. (1998). Domain of attraction of quasi-stationary distributions for the Brownian motion with drift. Adv. Appl. Prob. 30, 385–408.CrossRefGoogle Scholar
McKean, H. P. Jr (1956). Elementary solutions for certain parabolic partial differential equations. Trans. Amer. Math. Soc. 82, 519–548.CrossRefGoogle Scholar
Rogers, L. C. G. (1984). A diffusion first passage problem. In Seminar on Stochastic Processes, 1983 (Gainesville, Fla., 1983) (Progress Prob. Statist. 7), pp. 151–160. Birkhäuser, Boston.CrossRefGoogle Scholar
Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales, Vol. 2, Itô Calculus (Cambridge Mathematical Library), 2nd edn. Cambridge University Press, Cambridge.Google Scholar
Takeda, M. (2019). Existence and uniqueness of quasi-stationary distributions for symmetric Markov processes with tightness property. J. Theoret. Prob. 32, 2006–2019.CrossRefGoogle Scholar
Takemura, T. and Tomisaki, M. (2012). h transform of one-dimensional generalized diffusion operators. Kyushu J. Math. 66, 171–191.CrossRefGoogle Scholar
Yamato, K. (2021). Existence of Laplace transforms of the spectral measures for one-dimensional diffusions with an exit boundary. Infinitely divisible processes and related topics (25), The Institute of Statistical Mathematics Cooperative Research Report 55–59.Google Scholar
Yano, K. (2006). Excursion measure away from an exit boundary of one-dimensional diffusion processes. Publ. Res. Inst. Math. Sci. 42, 837–878.CrossRefGoogle Scholar