Consider a regular continuous time Markov chain $B(t)$ with state space the non-negative integers. Suppose that
the transition probabilities of the underlying DTMC, which you may assume to be positive recurrent,
satisfy $P_{0,1} = 1, P_{i,i+1} > 0$ for $i \ge 1, P_{i,i−1} > 0$ for $i \ge 1$, and $P_{i,j} = 0$ for all $i, j$ such that $|i − j| \ge 2$.
Let $P_{i,j} (t) = P (B(t) = j|B(0) = i).$
Use C-K equations to prove that
there exists a finite positive constant $c$, depending only on the transition probabilities and exponential
rates (not time), such that
$P_{1,0}(t) = cP_{0,1}(t)$ for all $t \ge 0.$
I try to use C-K differential equations to show $(\frac{P_{1,0}(t)}{P_{0,1}(t)})'=0.$ While Forward C-K equation says $$P'_{ij} (t) = \sum_{k\neq j}P_{ik}(t)P_{kj}\gamma_k-P_{ij}(t)\gamma_j$$ and Backward C-K equation says $$ P'_{ij}(t)=\sum_{k\neq i}P_{kj}(t)P_{ik}\gamma_i-P_{ij}(t)\gamma_i$$ Since $(\frac{P_{1,0}(t)}{P_{0,1}(t)})'=\frac{P'_{10}(t)P_{01}(t)-P_{1,0}(t)P'_{0,1}(t)}{P^2_{01}(t)}$ and $P'_{10}(t)=P_{11}(t)P_{10}\gamma_1-P_{10}(t)\gamma_0$ and $P'_{01}(t)=P_{11}(t)P_{01}\gamma_0-P_{01}(t)\gamma_0.$ It seems the numerator of the derivative involves $P_{11}(t)$ and does not cancel.