Given a continuous Markov chain that is a birth-death process on countable space $I$, suppose it starts at state $ i_{0}$, what do we know about the distribution of $X_{t}$ specifically, $\mathbb{E}_{i_{0}}(X_{t})$.
I know for the Poisson process when we start at 0, each $X_{t}$ follows a Poisson distribution. On the other hand for the birth process, it seems to be missing such a characterization, so does there exist anything to say about its distribution in general?
Now for the general birth-death process assume for simplicity, it does not explode and has an invariant distribution, which is equivalent to being positive recurrent. Can we say anything about $\mathbb{E}_{i_{0}}(X_{t})$? I think it has something to do with the expected return time and the time spent in state i. However, I'm not really sure if time spent in i is just proportion to t in a linear way.
I know some parts of this question might be difficult compare to others so essentially, I'm looking for the following.
- Distribution at t for a general birth-death chain, and $\mathbb{E}_{i_{0}}(X_{t})$
- The above if we assume positive recurrent
- The case when it is a birth process.
I appreciate comments or answers to any part of the question.
I forgot to add, of course, we can solve for the minimal nonnegative solution of the forward or backward equation, but that doesn't look anyhow easy.