If we assume a discrete set of events, our initial probability distribution function can be given by
$$ w_i(0) = \delta_{i,0}, $$and its time-development after a given time step \( \Delta t=\epsilon \) is
$$ w_i(t) = \sum_{j}W(j\rightarrow i)w_j(t=0). $$The continuous analog to \( w_i(0) \) is
$$ w(\mathbf{x})\rightarrow \delta(\mathbf{x}), $$where we now have generalized the one-dimensional position \( x \) to a generic-dimensional vector \( \mathbf{x} \). The Kroenecker \( \delta \) function is replaced by the \( \delta \) distribution function \( \delta(\mathbf{x}) \) at \( t=0 \).