On the notion of weak stability and related issues of hybrid diffusion systems

This work is concerned with the weak stability of hybrid diffusion systems, which consist of a number of diffusions modulated by a jump process. First, we show that when the jump component is nearly completely decomposable into a number of ergodic jump processes, the overall system is still weakly stable (or positive recurrent). Next we show that even if the process contains transient states, the positive recurrence can still be preserved. Then, we examine the asymptotic distribution when only one ergodic group of states is involved in the jump process and the state space of the continuous state belongs to a compact set. Our attention is devoted to asymptotic distribution in this case. The distribution is obtained by utilizing a spectrum gap property of the underlying process.