Wiener-Hopf-Faktorisierung f�r substochastische �bergangsfunktionen in angeordneten R�umen

Abstract

If (E, \(\mathfrak{B}\) is a measurable space with a reflexive and transitive ordering such that for every pair x, y e E, x≦y or y≦x holds, then it is intuitive to define increasing, decreasing and equal-height transition functions. P(x, dy) is increasing for instance if $$P(x,\{ y:y \leqq x\} ) = 0{\text{ for all }}x\varepsilon E.$$ If P is an arbitrary substochatsic transition function, then it is shown, that $$(I - sP) = \left( {I - \sum\limits_1^\infty {s^k P_k^ - } } \right) \circ \left( {I - \sum\limits_1^\infty {s^k P_k^ \circ } } \right) \circ \left( {I - \sum\limits_1^\infty {s^k P_k^ + } } \right)$$ holds for 0<=s<1 if the Pk*are certain substochastic transition functions. Here the notation P+ indicates that these transition functions are increasing; similarly P0 and P− is to be interpreted. The Pk*are unique. Their probability interpretation is given. The relation to Wiener-Hopf-Factorization and to Spitzer's Identity is explained. The result is applied to the problem of first exit from a set in a Markov Chain and to the factorization of an infinite matrix into an upper and a lower triangular matrix.