As I've understood it
But I'm having difficulties with framing this definition with the general one given for stochastic processes. Given a probability space $(\Omega,\mathcal{F}, \mathrm{P})$, and a measurable space $S$, a stochastic process $X(t, \omega)$ is a set of time-dependent $S$-valued random variables. If the probability distribution of this process has density, and we assume $S=\mathbb{R}$, then $$\tag{1} \mathbb{E}[X]=\int_{\mathbb{R}} xp(x,t)\mathrm{d}x$$ Where is the $\omega$ dependence (in the screenshot above $s=\omega$, so $v(t,s)=x(t,\omega)$) ?. It's as if $\Omega\equiv S$, but is it true/possible? The sample space (the total "number" of possible paths) has nothing to deal with the measurable space which the "samples" (paths) take value on...