 Solution

This is the synchronous product of simple symmetric random walks, rather than a walk that changes only one coordinate at a time. Independence of the three coordinate processes givesOdd-time return probabilities vanish. The p-series with exponent $3/2$ converges, so the sum of all return probabilities is finite. By the recurrence criterion by return probabilities, the origin is a transient state: the three-dimensional process is transient. From the origin its communicating class consists of integer triples whose coordinates have the same parity; translation gives the same classification for every other class.