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

Lai, Tze Leung

doi:10.1090/S0002-9947-1973-0334337-X

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.

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