# 2-D Brownian motion with correlated noise

1. Oct 18, 2012

### Dessert

dx/dt = η(t)
dy/dt = ζ(t)

where
<η(t)>=<ζ(t)>=0
<η(t)η(t')> = 2Dδ(t-t')
<ζ(t)ζ(t')> = 2Dδ(t-t')

If <η(t)ζ(t')> = 0, we have the standard 2-D diffusion equation and the analytical solution is known.

If <η(t)ζ(t')> = 2Dδ(t-t'), or η(t) = ζ(t), we can transform it into a 1-D problem and the analytical solution is also known.

What if <η(t)ζ(t')> = 2Dcδ(t-t') where 0<c<1 which is correlation of the two noises? We can still write down a 2-D diffusion equation, but is the analytical solution known?

Last edited: Oct 18, 2012