Importance sampling, Fokker-Planck and Langevin equations

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 $t=0$.

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