The Non-Decreasing Additive Functionals And The Non-Decreasing Multiplicative Functionals Of The Semi-Dynamic System

Piecewise deterministic markov processes is a kind of the widely used markov processes, which is in accordance with the decisive system evolution between the two adjacent jump times. This paper investigate the additive functionals of the semi-dynamic system which is closely related to Piecewise deterministic markov processes.This paper is mainly divided into two parts. The first part is de-voted to the definition and properties of the additive functionals and the multiplicative functionals of the semi-dynamic system. In the second part we shall investigate the definition and properties of the non-descending additive functionals and non-descending multiplica-tive functionals and some examples. Finally, the relationship between non-descending additive functionals and non-decreasing multiplicative functionals is given and proved:If F (x,t) is a non-decreasing multiplicative functionals, then A=-slog (F) is a non-decreasing additive functionals. The mapping from a non-decreasing multiplicative functionals to a non-decreasing addi-tive functionals which is Fâ†'SlogF is a bijective mapping,and A= SlogM(?)M= SexpA.