Brownian motion in a harmonic potential

In this section we study a simple system, the one-dimensional Brownian motion of a peuticle in a harmonic potential  

Although this is a very simple system, it is a prototype of the problems which we shall discuss later on. 

Let us first calculate the time correlation function (a (I)x(0)) using the Smoluchowski equation. In principle this is obtained from eqn (3.47) 

The right-hand side is rewritten by integral by parts as 

The initial condition for the differential equation is obtained from 

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