I am looking for some references for the following problem.
Consider a graph $G$ and a simple continuous time random walk $(X_t)_{t\geqslant 0}$ on this graph. Consider the family of events $(e_t)_{t\geqslant 0}$ where $e_t$ is the event $\{X_t$ has jumped an even number of times$\}$ and the family $(o_t)_{t\geqslant 0}$ the event for odd number of jumps. What is a sharp bound for the following quantity:
$$G(x,t)=\mathbb{P}_0(X_t=x\cap e_t)-\mathbb{P}_0(X_t=x\cap o_t)$$
Where $\mathbb{P}_0$ means the probability measure given that the random walk is starting from 0.
Trivial examples:
On $\mathbb{Z}$, if $x$ is at even distance of the origin then the walker had to make an even number of steps to be at that position at time $t$ and we have: $$G(x,t)=\mathbb{P}_0(X_t=x\cap e_t)=\mathbb{P}_0(X_t=x)\sim \frac{1}{\sqrt t}$$
If we add a loop with rate 1 on each site and $x$ at even distance of the origin then: $$G(x,t)= e^{-t} [\cosh(t)\mathbb{P}_0(X_t=x)-\sinh(t)\mathbb{P}_0(X_t=x)] = e^{-2t}\mathbb{P}_0(X_t=x)$$
Less Trivial
I am looking for references for this type of question for less trivial graph such as the triangular graph. For various reasons, in the triangular graph, it should look like: $$G(x,t) \sim e^{-4t} \mathbb{P}_0(X_t=x)$$
If you can prove it or provide me with some references, it would be much appreciated.
PS: This question has been asked on Math StackExchange and it was advised to ask this here.