## rules for gaussian random variables - ornstein uhlenbeck process

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

The Langevin equation for the Ornstein-Uhlenbeck process is

$$\dot{x} = -\kappa x(t) + \eta (t)$$

where the noise $$\eta$$ has azero mean and variance $$<\eta (t)\eta (t')> = 2D(t-t')\delta$$ with $$D \equiv kT/M\gamma$$. Assume the process was started at $$t0 = - \infty$$. Using the fact that $$\eta (t)$$ 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
 PhysOrg.com science news on PhysOrg.com >> King Richard III found in 'untidy lozenge-shaped grave'>> Google Drive sports new view and scan enhancements>> Researcher admits mistakes in stem cell study

 Tags gaussian, langevin, ornstein-uhlenbeck, statistical physics