 Solution

No weakly stationary process solves the model with nondegenerate noise. Iteration would giveFor a weakly stationary process with finite variance $V$, the Cauchy-Schwarz inequality instead gives $\mathrm{Var}(X_t-X_{t-m})=2V-2\gamma (m)\le 4V$. These two bounds contradict each other for large $m$. This argument does not assume that $X_{t-m}$ is independent of the intervening noise, so it excludes noncausal solutions too. The corresponding unit-root autoregressive process has persistent random walk behavior rather than stationary fluctuations.