Langevin’S Equation in Geometric Form

Abstract

At the beginning of this Section we use the so-called white noise w(t), which is defined as the ordinary derivative of a Wiener process w(t). From Statement 6.23 it follows that w(t) does not exist in the standard sense, but one can make its definition sensible in terms of distributions. White noise is useful for deriving clearly a well-posed differential equation for a physical process. Below, having introduced the Langevin’s equation in differential form by means of white noise, we avoid dealing with it by using the integral form of the equation and taking into account that, by definition,

$$ \int_0^t {A(t)w(t)dt = } \int_0^t {A(t)dw(t)} $$

(cf. the standard procedure of distribution theory).