Other Examples and Additional Notes
A solution of a Langevin equation for a particular realization of the fluctuating force is of no interest by itself, what is of interest are correlation functions of the slow variables after averaging over the fluctuating force.
One method of solution makes use of the Fokker–Planck equation, which provides a deterministic equation satisfied by the time dependent probability density. Alternatively numerical solutions can be obtained by Monte Carlo simulation. Other techniques, such as path integrals have also been used, drawing on the analogy between statistical physics and quantum mechanics (the Fokker-Planck equation is formally equivalent to the Schrödinger equation).
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