Limit Theorems for Discontinuous Random Evolutinos with Applications to Initial Value Problems and to Markov Processes on $N$ Lines

Abstract

Let $X(t); t \geqq 0$ be a stationary continuous-time Markov chain with state space $\{1,2,\cdots, N\}$ and jump times $t_1, t_2,\cdots$. Let $T_\alpha(t); t \geqq 0, 1 \leqq \alpha \leqq N$, be semi-groups and $\Pi_{jk} (u); u \geqq 0, 1 \leqq j \neq k \leqq N$, operators defined on Banach space $B$. Under suitable conditions on these operators, including commutativity, and an appropriate time change in $\varepsilon > 0$ on $X(t)$, we give limiting behavior for the discontinuous random evolutions $T_{X(0)}(t_1^\varepsilon) \Pi_{X(0)X(t_1)} (\varepsilon)T_{X(t_1)}(t_2^\varepsilon - t_1^\varepsilon)\cdots T_{X(t_\nu)}(t - t_\nu^\varepsilon)$ as $\varepsilon \rightarrow 0$. By considering the `expectation semi-group' of the discontinuous random evolutions, we prove a type of singular perturbation theorem and give formulas for the asymptotic solution. These results rely on a limit theorem for the joint distribution of the occupation-time and number-of-jump random variables of the chain $X(\bullet)$. We prove this theorem and with `random evolution' techniques use it to give new proofs of limit theorems for Markov processes on $N$ lines. Analogous results are obtained when the controlling process is a discrete-time finite-state Markov chain.