Let $X_t, t\geq -1$ be an ergodic process and let $\mu$ be its invariant measure. For example, let us consider $X_t$ solution of an overdamped Langevin equation $$ dX_t = -V'(X_t)\, dt + dW_t, $$ where $W_t$ is a Brownian motion. If $V$ is a dissipative potential, we have that the invariant measure $\mu$ of $X_t$ has density $\rho$ $$ \rho(x)= \frac1Z \exp\left(-V(x)\right), $$ with $Z$ the normalization constant. The latter can be derived by solving the stationary Fokker-Planck equation associated to the SDE above.
For simplicity, let $X_{-1}\sim \mu$ so that $X_t\sim \mu$ for all $t \geq -1$. Let now for $t \geq 0$ $$ Y_t=\int_{t-1}^tX_s \, ds, $$ be a "moving average" of $X_t$.
Is there an argument why $Y_t$ is itself ergodic? What will be the relationship between the invariant measures of $X_t$ and $Y_t$?