# Can a continuous markov chain explode with probability strictly between 0 and 1

A continuous markov process $$(X_{t})_{t\geq_{0}}$$ with corresponding Q matrix Q defined on a countable space $$I$$ is said to be explosive if $$\mathbb{P}_{i}(\xi < \infty) >0$$ for some $$i \in I$$ where $$\xi = \underset{ n }{ \text{sup} }J_{n}$$, where $$J_{n}$$ is the jumping time.

My question is if we fix the initial state, does there exist a continuous Markov chain with a corresponding Q matrix that explodes with probability p, s.t $$0 < p < 1$$

Note here that we have conditioned on the initial state of the Markov chain, if we furthermore fix the trajectory, the Markov chain either explodes with probability 0 or 1 by some properties of exponential random variables.

I asked this question to my professor, and he responded by saying that we can consider a 3-dimensional random walk, which starts at some states that are not the origin. He said that we can kill the process once it reaches zero, and since the probability for the process to reach zero is between 0 and 1, we are done. However, I'm really confused about what he meant by that killing this process would result in an explosion, isn't $$\xi$$, the explosion time defined as $$\underset{ n }{ \text{sup } } J_{n}$$?

• Take two MCs with state spaces $X_1$ and $X_2$, one of them being explosive with probability $1$ and another being explosive with probability $0$. Then consider an MC with a state space $X = \{\Delta\} \cup X_1 \cup X_2$ by leaving the dynamics on each $X_i$ the same and starting at $\Delta$ in such a way that you jump to $X_1$ from it with probability $p$. Would that work?
– SBF
Apr 29, 2023 at 15:39
• @Ilya Thank you for your comment. I am a little confused what you mean. Can you elaborate on "leaving the dynamics on each $X_i$ the same"? Also do you mean once we get to $\triangle$ the probability of going to $X_1$ is p, and $X_2$ is $1-p$? May 13, 2023 at 17:25

Recall that in a pure-birth process, explosion occurs if and only if $$\sum_{n=0}^\infty\lambda_n^{-1}<\infty$$. Consider a CTMC on the integers with $$\lambda_n := Q_{n,n+1}$$ for $$n\geqslant 0$$ and $$\mu_n:= Q_{n,n-1}$$ for $$n\leqslant 0$$ with $$\sum_{n=0}^\infty \lambda_n^{-1}<\infty$$ but $$\sum_{n=0}^\infty \mu_n^{-1}=+\infty$$.
Conditioned on the initial state being $$0$$, then explosion occurs with probability $$\frac{\lambda_0}{\lambda_0+\mu_0}$$, which lies in $$(0,1)$$ assuming $$\lambda_0,\mu_0>0$$.
• Thank you for your answer. I have given you an upvote, but I am really confused about what you mean. Do you mean this birth-death process takes place on $\mathbb{Z}$or the non-negative integers? And why is the explosion probability $\frac{\lambda_0}{\lambda_0+\mu_0}$? May 13, 2023 at 17:35
• I meant that $\lambda_n$ is defined for nonnegative $n$ and $\mu_n$ for nonpositive $n$, that should make it more clear. May 14, 2023 at 7:43