  The Transition Path Theory (TPT) of complex systems has proven a robust means
for statistically characterizing the ensemble of trajectories that connect any
two preset flow regions, say $\mathcal A$ and $\mathcal B$, directly. More
specifically, transition paths are such that they start in $\mathcal A$ and
then go to $\mathcal B$ without detouring back to $\mathcal A$ or $\mathcal B$.
This way, they make an effective contribution to the transport from $\mathcal
A$ to $\mathcal B$. Here, we explore its use for building a scheme that enables
predicting the evolution of an oil spill in the ocean. This involves
appropriately adapting TPT such that it includes a reservoir that pumps oil
into a typically open domain. Additionally, we lift up the restriction of the
oil not to return to the spill site en route to a region that there is interest
to be protected. TPT is applied on oil trajectories available up to the
present, e.g., as integrated using velocities produced by a data assimilative
system or as inferred from high-frequency radars, to make a prediction of
transition oil paths beyond, without relying on forecasted oil trajectories. As
a proof of concept we consider a hypothetical oil spill in the Trion oil field,
under development within the Perdido Foldbelt in the northwestern Gulf of
Mexico, and the \emph{Deepwater Horizon} oil spill. This is done using
trajectories integrated from climatological and hindcast surface velocity and
winds as well as produced by satellite-tracked surface drifting buoys, in each
case discretized into a Markov chain that provides a framework for the
TPT-based prediction.
