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Is it true that for all continuous time Markov processes on a countable state space $S$, we have

all rows of the transition matrix $\mathbf{P}_t$ are distinct for all time $t\in[0,\infty)$ ?

Some thought: This is true when $S$ is finite. This is also true when $S$ is countably infinite and $\mathbf{P}_t=\exp{(tQ)}$, where $Q=(q_{ij}: i,j\in S)$ a bounded operator on the $\ell_1$ sequence space. The latter condition on the transition matrix is equivalent to $sup_{i}|q_{ii}|<\infty$.

How about for general continuous time Markov processes?

Is it true that for all continuous time Markov processes on a countable state space $S$, we have

all rows of the transition matrix $\mathbf{P}_t$ are distinct for all time $t\in[0,\infty)$ ?

This is true when $S$ is finite. This is also true when $S$ is countably infinite and $\mathbf{P}_t=\exp{(tQ)}$, where $Q=(q_{ij}: i,j\in S)$ a bounded operator on the $\ell_1$ sequence space. The latter condition on the transition matrix is equivalent to $sup_{i}|q_{ii}|<\infty$. Here're 2 examples for which $sup_{i}|q_{ii}|=\infty$ yet the statement is still true:

1. Birth-death chains (due to strict total positivity)
2. Branching processes (due to self-similarity in the generating function)

Any example, counter-example or reference will be appreciated.

Is it true that for all continuous time Markov process on a countable state space $S$, we have

all rows of the transition matrix $\mathbf{P}_t$ are distinct for all time $t\in[0,\infty)$ ?

Some thought: This is true when $S$ is finite. This is also true when $S$ is countably infinite and $\mathbf{P}_t=\exp{(tQ)}$, where $Q=(q_{ij}: i,j\in S)$ a bounded operator on the $\ell_1$ sequence space. The latter condition on the transition matrix is equivalent to $sup_{i}|q_{ii}|<\infty$.

How about for general continuous time Markov process?

Is it true that for all continuous time Markov processes on a countable state space $S$, we have

all rows of the transition matrix $\mathbf{P}_t$ are distinct for all time $t\in[0,\infty)$ ?

Some thought: This is true when $S$ is finite. This is also true when $S$ is countably infinite and $\mathbf{P}_t=\exp{(tQ)}$, where $Q=(q_{ij}: i,j\in S)$ a bounded operator on the $\ell_1$ sequence space. The latter condition on the transition matrix is equivalent to $sup_{i}|q_{ii}|<\infty$.

How about for general continuous time Markov processes?

Is it true that for all continuous time Markov process on a countable state space $S$, we have

all rows of the transition matrix $\mathbf{P}_t$ are distinct for all time $t\in[0,\infty)$ ?

Some thought: This is true when $S$ is finite. This is also true when $S$ is countably infinite and $\mathbf{P}_t=\exp{(tQ)}$, where $Q$ a bounded operator on the $\ell_1$ sequence space. The latter condition on the transition matrix is equivalent to $sup_{i}|q_{ii}|<\infty$.

How about for general continuous time Markov process?

Is it true that for all continuous time Markov process on a countable state space $S$, we have

all rows of the transition matrix $\mathbf{P}_t$ are distinct for all time $t\in[0,\infty)$ ?

Some thought: This is true when $S$ is finite. This is also true when $S$ is countably infinite and $\mathbf{P}_t=\exp{(tQ)}$, where $Q=(q_{ij}: i,j\in S)$ a bounded operator on the $\ell_1$ sequence space. The latter condition on the transition matrix is equivalent to $sup_{i}|q_{ii}|<\infty$.

How about for general continuous time Markov process?