Let $X_t$, $t=0,1,2...$ be an ergodic Markov chain on $S=\{1,...,n\}$ with transition matrix $P=\left(P_{ij}\right)_{i,j\in S}$. Let $T^i=\inf\{t\geq1:X_t=i\}$ and $h_j^i=\mathbb{E}\left(T^i\right\vert X_0=j)$. I want to show that $\left(\frac{1}{h_i^i}\right)_{i\in S}$ is the stationary distribution of the Markov chain.
I have shown that $h_i^i=1+\sum_{k\neq i}h_k^iP_{ik}$. For $k\neq i$, $h_k^i$ is the standard expected hitting time of $i$ starting at $k$ with $h_k^i=1+\sum_{m\neq k}h_m^i P_{im}$.
I have tried verifying $\sum_i \frac{1}{h_i^i}P_{ij}=\frac{1}{h_j^j}$ while keeping in mind the above formulae but this has not been successful.
Let $\{\pi_i: i=1,2,\ldots,n\}$ denote the stationary distribution. Multiply both sides of your identity (or rather, its general form $h_i^j=1+\sum_{k\not=j}P_{ik}h_k^j$) by $\pi_i$ and sum over all $i\in S$, then use the fact that $\sum_i\pi_iP_{ik}=\pi_k$ for each $k$.