Suppose I have an infinite and reversible Markov chain $X_n$ with transition kernel $P_1(x,y)$ and stationary measure $\pi(x)$.
For concreteness, suppose my state space is on a graph and my edges have weights $a_{xy}$, so that $P_1(x,y)=\frac {a_{xy}}{\sum_{z\sim x}a_{xz}}$ I'm interested in references which deal with the following:
Suppose $a_{x^*y^*}$ (the weight between vertices $x^*$ and $y^*$) is changed to $b_{x^*y^*}$. In particular, by changing a single weight, I get a new transition kernel $P_2(x,y)$.
What can be said about the behavior of $P^n_2(x,x)$ compared to $P^n_1(x,x)$ as $n$ gets large?
If I tell you that every state in this Markov chain is transient, can something be said about the Radon Nikodym derivative $dP_1/dP_2$, in the sense that $\displaystyle P_1^n(x,x)=E_x^{(2)}\left(\frac{dP_1}{dP_2} 1_{X_n=x}\right)$.
I would like to say that $P_1^n(x,x)$ and $P_2^n(x,x)$ become commensurate but, this likely needs strong conditions on what $b_{x^*y^*}$ can be.