I asked this problem on MSE some while ago, but it has stubbornly resisted any attempts at solving it. Maybe there is someone here who can either close the gap in one of the existing answers or has an independent answer.

Let the following stochastic system be given

$$dX_t=-X_t \, dt+dW_t,$$ $$dY_t=X_tY_t(1-Y_t)(dt+dV_t),$$

where $W_t$ and $V_t$ are independent Wiener processes, $X_0\sim\mathcal{N}(0,1/2)$ (the stationary measure), and $Y_0\in[0,1]$.

I would like to show that $\lim_{t\to\infty}Y_t\in\{0,1\}$ almost surely, or at least that the invariant probability measure for $Y$ is of the form $\alpha\delta_0+(1-\alpha)\delta_1$ for some $\alpha\in[0,1]$.