rules for gaussian random variables - ornstein uhlenbeck process

yaboidjaf

1. The problem statement, all variables and given/known data

The Langevin equation for the Ornstein-Uhlenbeck process is

[tex]
\dot{x} = -\kappa x(t) + \eta (t)
[/tex]

where the noise [tex]\eta[/tex] has azero mean and variance [tex]<\eta (t)\eta (t')> = 2D(t-t')\delta[/tex] with [tex]D \equiv kT/M\gamma[/tex]. Assume the process was started at [tex]t0 = - \infty[/tex]. Using the fact that [tex]\eta (t)[/tex] is a Gaussian random variable, show that the moments of x(t) are:

<[x(t)]^2n> = (2n - 1)!!(D/k)^n, <[x(t)]^2n+1> = 0


3. The attempt at a solution

I've worked through solving the O-U process and can show the first two cases, where n = 0, 1 and 2 and there is a rule that the average of an odd number of gaussian variables gives 0 so i can show the 2n+1 case. But I struggle when it comes to the general case for 2n. I know that when you have an even number of these variables that you get a set number of combinations and work out the individual integrands, but I don't know how to show this for n. I tried induction and that doesn't work.

Any ideas?? any help woud be appreciated. Thanks