Autoregressive process with random walk perturbation.

Question

Suppose we have an autoregressive process, $$y_t=\phi y_{t-1} +u_t$$ where $|\phi|<1$. If $u_t$ is an i.i.d random variable this process is stationarity. What if $$u_t=u_{t-1}+g+\epsilon_t$$ where $\epsilon_t$ is white noise and $g$ is a constant? I tried to use the lag operator to show that it's still stationary, but I am having some issues. Thanks in advance.