The general definition of the $\phi$-mixing coefficient is as follows: \begin{equation} \phi(\sigma_{1},\sigma_{2}) = \underset{B\in\sigma_{1},A\in\sigma_{2}}{\sup} P(A|B)-P(A) \end{equation}
Then for the process $\{X_{t}\}$, we define: \begin{equation} \phi(n) = \phi(\mathcal{F}_{-\infty}^{0}, \mathcal{F}_{n}^{\infty}) \end{equation} where $\mathcal{F}_{-\infty}^{0}$ denotes the sigma algebra generated by $X_{0}, X_{-1},\ldots$ and $\mathcal{F}_{n}^{\infty}$ denotes the sigma algebra generated by $X_{n}, X_{n+1},\ldots$
Now suppose that the process of $X_{t}$ follows a stationary AR(1) process \begin{equation} X_{t} = \rho X_{t-1} + u_{t} \end{equation} where $|\rho|<1$, and $u_{t}$ is a white noise with absolutely continuous density.
My question : is $X_{t}$ $\phi$-mixing? If so, can we determine the rate of the $\phi$-mixing coefficient? Or the coefficient diminishes at what rate?