This is an edit to the previous question, on stationary process, which was answered by Richard below.
Let $x_t$ be a zero mean, time homogeneous Markovian process over time $t$ starting from $x_0=0$. What are the examples of $x_t$ where the variance at $t$ does not increase over $t$?
1) In discrete time and discrete state, the followig is a very simple example where the variance periodically oscillates over time.
$$x_{t+1} = \eta(1-|x_t|),\, x_0=0;\, \eta\in\{-1,1\},\mbox{ with probability of } \frac{1}{2} \mbox{ on each value of }\eta.$$
2) In continuous time, but discontinuous path setting, is the following jump diffusion process a correct example?
$$dx_t = -\alpha x_t dt+dz_t+ y\eta dN_t,\, x_0 = 0,$$ where $\alpha\gg 0$, $z_t$ is the standard brownian motion with mean $0$ and standard deviation $t$, $N_t$ is the Poisson process with frequency $0<\lambda\ll 1$, $\eta$ takes on values $-1$ or $1$ with $0.5$ probability each, $z_{t_1}$, $N_{t_2}$ and $\eta$ are independent of each other at arbitrary $t_1$ and $t_2$, and constant $y\gg 1$.
On second thought, this is not a correct example. One can solve this equation and one will find the variance of this process is the sum of the variance from $dz_t$ and that from $dN_t$ due their independence. We will have to make the jumps negatively correlated to $z_t$.
A better setup is to shift $x_t$ beyond a barrier directly back to the $x=0$ line. So the process resides on the topology of two cylinders touched along a longitude. However, it seems to me, even this set up with $x_t$ being either a standard Browniam motion or mean reverting one without any jump process still has its variance increasing with time.
Therefore, I am still without a valid example in this setup.
3) What are the examples for continuous path? I suspect it is not possible. Can anyone prove this if it is indeed impossible?
