FLUCTUATIONS OF LEVY PROCESSES AND SCATTERING THEORY

Initial work by Spitzer was extended to show that the behavior of the bivariate processes (X-t, inf(0 <= s <= t) X-s) or (X-t, sup(0 <= s <= t) X-s), where X is a Levy process, can be entirely reconstructed on the basis of the Wiener-Hopf factorization of the Levy exponent of X. This paper is meant to establish that a similar device can be used to investigate the trivariate Markov process (X-t, inf(0 <= s <= t) X-s, sup(0 <= s <= t) X-s). This involves substituting (2,2)-matrices for the scalar functions involved in the Spitzer-type factorization. The computation of this matrix from the Levy exponent of X is a Rieman n-Hilbert problem, which is the same as the one appearing in the inverse scattering problem.