Let $S_n$ be a transient, irreducible random walk starting from $0$. Then I want to prove that $$\sum_{n\geq0}{p_n(0,x)}=P_0[S_n=x\text{ for some }n\geq0]\sum_{n\geq0}{p_n(0,0)}$$
where $p_n$ is the n-th step transition probability and $P_0$ the conditional probability of the chain starting at $0$. As a hint is given to use the strong Markov property but I always get something not useful. I tried to use the SMP on the first return time to $0$. Does someone has a better intuition? Thanks
I kept trying to prove this and I found a similar formulation of this, which is $$\sum_{n\geq0}{p_n(0,x)}=P_0[S_n=x\text{ for some }n\geq0]\sum_{n\geq0}{p_n(x,x)}.$$ Are both formulation equivalent?
Since a random walk is spatially homogeneous, we have $p_n(0,0)=p_n(x,x)$ for all $n\geq 0$. Therefore your two equations are equivalent.