Solution:

For $n \in \mathbb{Z}$, let $a_n$ be the fraction of the time Mr. Fat spends at $n$. By symmetry, $a_n = a_{-n}$ for all $n$.

For $n > 0$, we have
$$a_n = \frac{2}{5} a_{n-1} + \frac{2}{5} a_{n+1},$$
or
$$a_{n+1} = \frac{5}{2} a_n - a_{n-1}.$$
This Fibonacci-like recurrence can be solved explicitly to obtain
$$a_n = \alpha \cdot 2^{|n|} + \beta \cdot 2^{-|n|}$$
for all $n \in \mathbb{Z}$. Now we also have
$$\sum_{n \in \mathbb{Z}} a_n = 1,$$
so we better have $\alpha = 0$, so that $a_0 = \beta$ and $a_{\pm 1} = \frac{\beta}{2}$.

Now we also have
$$a_0 = \frac{2}{5} a_{-1} + \frac{2}{5} a_1 + \frac{1}{5},$$
so $\beta = \frac{1}{3}$.

This matches perfectly with $\sum_{n \in \mathbb{Z}} a_n = 1$.