【摘要】 <正> 1. 一般空间上的纯间断的时齐马可夫过程的概率转移函数,其定义述之如下。 设{X,F}是一个可测空间——这就是说,X是不论怎样一个集合,F是X的某些子集所集成的一个σ-体,包括着X为其成员。我们假定:凡独点集皆属于F。那么,更多还原

【Abstract】 Let X be a Euclidean space and bo the cless of Borel sets in X. We consider x (probabilily transition) function P(t,x,E) satisfying; the following conditions:(Ⅰ) the domains of the arguments are: t≥0, E, and x∈Y, where Y is a Borel seb in X;(Ⅱ) P(t,x,E) is a mensurable function of a; and a probability function of E;(Ⅲ) P(t,x,Y) = 1;(Ⅳ) P(s+t,x,E) = fP(s,x,dy)P(t,y,E);(Ⅴ) P(0,x, {x}) = 1:(Ⅵ) lim P(t,x,{x})=1We putThe following results are arrived at.(A) Suppse that for a certain point x we have q(x)<∞. Thend/dtrP(t,x,E)=P (t,x,E) exisfs at every t>0 for every E. P’(t,x,E) is a continuous function of t and a completely additive function of E. Further, wo have|P’(t,x,E)| ≤q(x),(B) If for a certain point x and a certain set EY we have xzY-E, q(x)< ∞, and lim P(t,y, {y}) = l uniformly on E, then,no(C) If for a certain point x and a certain got EY we have q(x)<∞ and that q(y) is bounded on E, then P’(t,x,E) is a function of boundel variation of t.(D) Suppore that q(x) is finite for every x∈Y. Lot EY be a get such that q(x) is bounded on E. Then更多还原