= Solution
{c}
Suppose a <stationary distribution> $\pi$ exists. Its <stationarity equations> are
$$
\pi_j=p\pi_{j-a}+q\pi_{j+b}\qquad(j\in\mathbb Z).
$$
Since $\sum_j\pi_j=1$, its masses tend to zero as $|j|\to\infty$, and a positive maximum $M=\pi_{j_0}$ is attained. Both terms on the right are at most $M$. Since $p,q>0$ and $p+q=1$, equality forces $\pi_{j_0-a}=\pi_{j_0+b}=M$. Applying the same argument repeatedly at $j_0-a,j_0-2a,\ldots$ gives
$$
\pi_{j_0-ka}=M>0\qquad(k\geq0).
$$
These are infinitely many distinct states, contradicting $\sum_j\pi_j=1$. Thus
$$
\boxed{\text{There is no stationary distribution.}}
$$
This <maximum principle> proof uses neither a finite <expected value> of the state nor the <coprimality> assumption.
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