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Spectral analysis of subordinate Brownian motions on the half-line

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Kwaśnicki M.

Studia Mathematica

2011

tom 206

nr 3

211-271

We study one-dimensional Lévy processes with Lévy-Khintchine exponent ψ(ξ²), where ψ is a complete Bernstein function. These processes are subordinate Brownian motions corresponding to subordinators whose Lévy measure has completely monotone density; or, equivalently, symmetric Lévy processes whose Lévy measure has completely monotone density on (0,∞). Examples include symmetric stable processes and relativistic processes. The main result is a formula for the generalized eigenfunctions of transition operators of the process killed after exiting the half-line. A generalized eigenfunction expansion of the transition operators is derived. As an application, a formula for the distribution of the first passage time (or the supremum functional) is obtained.

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1 Studia Mathematica

1 Kwaśnicki M.

1 2011

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Spectral analysis of subordinate Brownian motions on the half-line