Let $x_t$ be a zero mean, time homogeneous Markovian process over time $t$ starting from $x_0=0$. Is it necessary that, in continuous time setting, the variance of $x_t$ does not decrease over $t$?

What are the examples of $x_t$ where the variance at $t$ decreases over some interval of $t$? The following are my successful ( 1) ) and failed ( 2) ) attempts in constructing the examples.

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. I first looked at the following jump diffusion process.

$$dx_t = -\alpha x_t dt+dz_t+ y\eta dN_t,\, x_0 = 0,$$ 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 $y\gg 1$.

It does not seem 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 may have to make the jumps negatively correlated to $z_t$.

Another setup I thought of 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 still has its variance increasing over time.

Therefore, I am still without a valid example in this setup.

3. What are the examples for continuous path? I suspect they do not exist. Can anyone prove this if the variance has to increase over $t$?

Let $x_t$ be a zero mean, time homogeneous Markovian process (chiefly look at the case where the value is in $1$ dimension) over time $t$ starting from $x_0=0$. Is it necessary that, in continuous time setting, the variance of $x_t$ does not decrease over $t$?

What are the examples of $x_t$ where the variance at $t$ decreases over some interval of $t$? The following are my successful ( 1) ) and failed ( 2) ) attempts in constructing the examples.

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. I first looked at the following jump diffusion process.

$$dx_t = -\alpha x_t dt+dz_t+ y\eta dN_t,\, x_0 = 0,$$ 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 $y\gg 1$.

It does not seem 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 may have to make the jumps negatively correlated to $z_t$.

Another setup I thought of 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 still has its variance increasing over time.

Therefore, I am still without a valid example in this setup.

3. What are the examples for continuous path? As Martin Hairer shows in his solution below, this can be achieved in higher-than-1 dimensional value space. This circumvents the difficulty of having to revisit the same point going away from $0$ and come closer to $0$. Now if we tackle the difficulty of revisiting the same value, and restrict the process to $1$ dimension. What is the answer? I suspect the satisfactory process does not exist. Can anyone prove this if the variance has to increase over $t$?

Let $x_t$ be a zero mean, time homogeneous Markovian process over time $t$ starting from $x_0=0$. Is it necessary that, in continuous time setting, the variance of $x_t$ increases over $t$?

What are the examples of $x_t$ where the variance at $t$ does not increase over $t$? The following are my successful ( 1) ) and failed ( 2) ) attempts in constructing the examples.

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. I first looked at the following jump diffusion process.

$$dx_t = -\alpha x_t dt+dz_t+ y\eta dN_t,\, x_0 = 0,$$ 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 $y\gg 1$.

It does not seem 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 may have to make the jumps negatively correlated to $z_t$.

Another setup I thought of 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 still has its variance increasing over time.

Therefore, I am still without a valid example in this setup.

3. What are the examples for continuous path? I suspect they do not exist. Can anyone prove this if the variance has to increase over $t$?

Let $x_t$ be a zero mean, time homogeneous Markovian process over time $t$ starting from $x_0=0$. Is it necessary that, in continuous time setting, the variance of $x_t$ does not decrease over $t$?

What are the examples of $x_t$ where the variance at $t$ decreases over some interval of $t$? The following are my successful ( 1) ) and failed ( 2) ) attempts in constructing the examples.

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. I first looked at the following jump diffusion process.

$$dx_t = -\alpha x_t dt+dz_t+ y\eta dN_t,\, x_0 = 0,$$ 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 $y\gg 1$.

It does not seem 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 may have to make the jumps negatively correlated to $z_t$.

Another setup I thought of 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 still has its variance increasing over time.

Therefore, I am still without a valid example in this setup.

3. What are the examples for continuous path? I suspect they do not exist. Can anyone prove this if the variance has to increase over $t$?