Random Motions

Abstract

In my two earlier papers [1], [2], I developed a general theory of continuous random processes. Under very general conditions, it was proved that if the state of a physical system at every given moment is fully determined by n parameters x ₁,x ₂,…,x _n , and if these n parameters continuously¹ change with timet, then the corresponding distribution functions satisfy the Fokker-Planck differential equation. In the general case of such random processes the increments △x _i of the parameter x _i ,- are of the same order as (△t)^1/2. This implies that in the general case △x _i : △t → ∞ as △t → 0, so that we cannot speak about a definite rate of variation of x _i . We will now show how to apply this general theory to random motions, for which we assume that not only the system’s coordinates, but also their derivatives with respect to time vary continuously.

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‘Zufällige Bewegungen (Zur Theorie der Brownschen Bewegung)’, Ann. of Math. 35 (1934), 116-117.