I have two very simple continuous-time Markov chains $\displaystyle M^a$ and $\displaystyle M^b$ defined on states $\displaystyle (i,j)\in\{(0,0),(1,0),(0,1)\}$ by the following transition rates: For $\displaystyle M^a$

$\displaystyle

p_{10,01}=\mu, ~~

p_{01,10}=\mu (1-z),~~

p_{01,00}=\mu z,

$

and for $\displaystyle M^b$

$\displaystyle

p_{10,01}=\mu z, ~~

p_{01,00}=\mu z

$

where 0<z<1 and the omitted transition rates are zero.

Assume that both chains start in state (0,1). Now, consider the random variable $\displaystyle X^a(t)=X_1^a(t)+X_2^a(t) $, i.e., the sum of the state indices i and j over time, for $\displaystyle M^a$, and $\displaystyle X^b(t)=X_1^b(t)+X_2^b(t) $ analogously.

Is it true that $\displaystyle X^a(t)$ has the same distribution of $\displaystyle X^b(t)$ for all t?

Is it true that $\displaystyle X^a(t)\le_{st}X^b(t)$? (or $\displaystyle X^a(t)\ge_{st}X^b(t)$?)

Why?

Thanks