Consider an infection process starting from the origin of a d-dimensional hyper-cubical lattice. The process evolves as follows: If a site is infected, it infects its neighbours with rate $r$ i.e. in time $dt$ the infinitesimal probability of infection is $rdt$. At time $t=0$, only the origin is infected. Suppose we look at all possible trajectories of this process that start at the origin and end at some specified point at time $t$. How can we go about assigning weights (probabilities) to these trajectories? From what I understand, in the evolution of this infection, some trajectories should be more likely than others (one can always think of pathological examples).
Any help would be greatly appreciated. If there are any clarifications needed, I'll be happy to provide them :)