You can always specify $x_0$ to be drawn from a given distribution (a constant is a special case). In that case, a solution is just given by iterating forward according to the model.

However, if you want a covariance stationary solution $\{ x_t \}$, then $x_0$ (or any other $x_t$ for that matter) is necessarily not deterministic (except for the trivial solution $x_t = 0$ when $\rho = 0$).

When $\{ \epsilon_t \}$ is i.i.d., then covariance stationary solutions are also strictly stationary and all $x_t$'s have the same distribution, including $x_0$.

You can always specify $x_0$ to be drawn from a given distribution (a constant is a special case). In that case, a solution is just given by iterating forward according to the model.

However, if you want a covariance stationary solution $\{ x_t \}$, then $x_0$ (or any other $x_t$ for that matter) necessarily cannot be deterministic.

When $\{ \epsilon_t \}$ is i.i.d., then covariance stationary solutions are also strictly stationary and all $x_t$'s have the same distribution, including $x_0$.