Q过程的不变分布(Ⅱ)

INVARIANT DISTRIBUTION OF Q-PROCESS (Ⅱ)

设E为一个可数集,Q=（qi,j;i,j∈E）为E×E上的矩阵,满足m为E上的概率分布满足何时存在Q过程,使得m是它的不变分布? 这个问题由Williams（1979）作为一个开问题提出.文[15]对全稳定情形,解决了这个问题;本文对单瞬时情形,完整地解决了该问题.

Let E be a countable set, Q =(qi,j;i,j ∈E) be a matrix defined on E× E such that qi,j≥0 (i≠j),∑k≠i qi,k=-qii≤∞, i∈E, m = (mi;i∈E) is a set of strictly positive∑i≠jmiqi,j=-mjqj,j, j∈Eprobability distribution such thatin what condition does there exist Q-process such that m is a invariant distribution of its?The question was given by Williams (1979) as an open problem. The paper [15] solves the problem when Q = (qi,j;i,j∈E) is total stable.In this paper the anthors completely solve the problem when Q = (qi,j;i,j ∈E) has a single instantaneous state.