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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Space-time processes, parabolic functions and one-dimensional diffusions


Author: Tze Leung Lai
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
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0334337-X
Keywords: Space-time processes, parabolic functions, characteristic operators, semigroups, infinitesimal generators, martingales, diffusions, inacessible boundaries, entrance boundaries, natural boundaries, scale, positive increasing (decreasing) solutions, boundary crossing probabilities
Article copyright: © Copyright 1973 American Mathematical Society