I have a differential equation $$\frac{dx(t)}{dt}=F(g(x(t),\omega)),\;\;x_0=x,$$ where $\omega$ is standard normal variable. Given that for each fixed $y\in\mathbb{R}$, $$\mathbb{E}_\omega[g(y,\omega)]=h(y),$$ here $h$ is a known function. Can we say anything about $\mathbb{E}_\omega[g(x(t),\omega)]$?
Here the things that boarders me is $x(t)$ is depends on $\omega.$ Thank you very much in advance.