Importance sampling, Fokker-Planck and Langevin equations

A stochastic process is simply a function of two variables, one is the time, the other is a stochastic variable \( X \), defined by specifying

  1. the set \( \left\{x\right\} \) of possible values for \( X \);
  2. the probability distribution, \( w_X(x) \), over this set, or briefly \( w(x) \)

The set of values \( \left\{x\right\} \) for \( X \) may be discrete, or continuous. If the set of values is continuous, then \( w_X (x) \) is a probability density so that \( w_X (x)dx \) is the probability that one finds the stochastic variable \( X \) to have values in the range \( [x, x +dx] \) .