From the Lagrangian point of view, it is \(\vec{x}_{in}=\vec{x}_{in}\left(t_{in}\right)\). The determination of the domain \(\partial\ \Omega^+\ =\ \partial\ \Omega^+\left(t_{in}\right)\), together with the knowledge of the Lagrangian trajectories \(\vec{x}_{in}=\vec{x}_{in}\left(t_{in}\right)\) determines the a switch to the Eulerian perspective over the parcels movements. Willing to eliminate the the dependence on space from the probability, to obtain the bulk probabilities, we have to perform and Integration over the domain \(\Omega\), excluding the boundaries, and one over the entering boundary \(\partial\Omega^+\) to get all the input parcels. We obtain therefore what was called in \cite{Rigon_2016} residence time distribution:
\(\)