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Importance sampling, Fokker-Planck and Langevin equations

An arbitrary number of other stochastic variables may be derived from X . For example, any Y given by a mapping of X , is also a stochastic variable. The mapping may also be time-dependent, that is, the mapping depends on an additional variable t

Y_X (t) = f(X, t).

The quantity Y_X (t) is called a random function, or, since t often is time, a stochastic process. A stochastic process is a function of two variables, one is the time, the other is a stochastic variable X . Let x be one of the possible values of X then

y(t) = f (x, t),

is a function of t , called a sample function or realization of the process. In physics one considers the stochastic process to be an ensemble of such sample functions.