Trajectories are sampled to obtain histograms of the path history defined as probability density \(Q({\mathbf{x}}, t | {\mathbf{x}}_{f}, t_{f}; {{\mathbf{x}}_{A}}, t_{0})\), for \(t_{0} < t < t_{f}\). We can express the path history as \[\label{eq:23} Q({\mathbf{x}}, t) = \frac{{\mathrm{P}}({\mathbf{x}}_{f}, t_{f} | {\mathbf{x}}, t){\mathrm{P}}({\mathbf{x}}, t | {{\mathbf{x}}_{A}}, t_{0})}{{\mathrm{P}}({\mathbf{x}}_{f}, t_{f} | {{\mathbf{x}}_{A}}, t_{0})},\] where \(v_{f} = 0.6\), \(t_{f} = 0\) and (effectively) \(t_{0} = - \infty\) (trajectories take a long time to reach \(v_{f}\)). Each pane in Fig. \ref{fig:mc_type2} represents a discrete approximation from \(10^{3}\) simulation trials of \(Q\) at a different point in time. The most probable path (dashed red curve) coincides with the peak of the histogram as a function of time.