Diffusions from infinity

Bansaye, Vincent; Collet, Pierre; Martinez, Servet; Méléard, Sylvie; San Martin, Jaime

doi:10.1090/tran/7841

Diffusions from infinity
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by Vincent Bansaye, Pierre Collet, Servet Martinez, Sylvie Méléard and Jaime San Martin PDF

Trans. Amer. Math. Soc. 372 (2019), 5781-5823 Request permission

Abstract:

In this paper we consider diffusions on the half line $(0,\infty )$ such that the expectation of the arrival time at the origin is uniformly bounded in the initial point. This implies that there is a well defined diffusion process starting from infinity which takes finite values at positive times. We study the behavior of hitting times of large barriers and, in a dual way, the behavior of the process starting at infinity for small time. In particular, we prove that the process coming down from infinity is in small time governed by a specific deterministic function. Suitably normalized fluctuations of the hitting times are asymptotically Gaussian. We also derive the tail of the distribution of the hitting time of the origin and a Yaglom limit for the diffusion starting from infinity. We finally prove that the distribution of this process killed at the origin is absolutely continuous with respect to the speed measure. The density is expressed in terms of the eigenvalues and eigenfunctions of the generator of the killed diffusion.

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