Space-time processes, parabolic functions and one-dimensional diffusions
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- Trans. Amer. Math. Soc. 175 (1973), 409-438 Request permission
Abstract:
In this paper, we study the properties of the space-time process and of parabolic functions associated with a Markov process. Making use of these properties and the asymptotic behavior of the first passage probabilities near the boundary points, we prove certain theorems concerning when $u(X(t),t)$ is a martingale, where $X(t)$ is a conservative regular one-dimensional diffusion with inaccessible boundaries. A characterization of the class of parabolic functions associated with classical diffusions is also obtained.References
- Michael A. Arbib, Hitting and Martingale characterizations of one-dimensional diffusions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 (1965), 232–247 (1965). MR 192551, DOI 10.1007/BF00533754
- J. L. Doob, Martingales and one-dimensional diffusion, Trans. Amer. Math. Soc. 78 (1955), 168–208. MR 70885, DOI 10.1090/S0002-9947-1955-0070885-7
- J. L. Doob, A probability approach to the heat equation, Trans. Amer. Math. Soc. 80 (1955), 216–280. MR 79376, DOI 10.1090/S0002-9947-1955-0079376-0
- E. B. Dynkin, Theory of Markov processes, Prentice-Hall, Inc., Englewood Cliffs, N.J.; Pergamon Press, Oxford-London-Paris, 1961. Translated from the Russian by D. E. Brown; edited by T. Köváry. MR 0131900
- E. B. Dynkin, Markovskie protsessy, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR 0193670
- Kiyoshi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der mathematischen Wissenschaften, Band 125, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-New York, 1965. MR 0199891
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
- Tze Leung Lai, Martingales and boundary crossing probabilities for Markov processes, Ann. Probability 2 (1974), 1152–1167. MR 436336, DOI 10.1214/aop/1176996503
- Petr Mandl, Analytical treatment of one-dimensional Markov processes, Die Grundlehren der mathematischen Wissenschaften, Band 151, Academia [Publishing House of the Czechoslovak Academy of Sciences], Prague; Springer-Verlag New York Inc., New York, 1968. MR 0247667
- Herbert Robbins and David Siegmund, Boundary crossing probabilities for the Wiener process and sample sums, Ann. Math. Statist. 41 (1970), 1410–1429. MR 277059, DOI 10.1214/aoms/1177696787
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 175 (1973), 409-438
- MSC: Primary 60J60
- DOI: https://doi.org/10.1090/S0002-9947-1973-0334337-X
- MathSciNet review: 0334337