Spectral analysis of stable processes on the positive half-line

2015/09/22 by Kuznetsov, Alexey, Kwasnicki, Mateusz
#60G52 #60J35 #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1509.06435

Abstract

We study the spectral expansion of the semigroup of a general stable process killed on the first exit from the positive half-line. Starting with the Wiener-Hopf factorization we obtain the q-resolvent density for the killed process, from which we derive the spectral expansion of the semigroup via the inverse Laplace transform. The eigenfunctions and co-eigenfunctions are given rather explicitly in terms of the double sine function and they give rise to a pair of integral transforms which generalize the classical Fourier sine transform. Our results provide the first explicit example of a spectral expansion of the semigroup of a non-symmetric Levy process killed on the first exit form the positive half-line.

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