So I am looking at the case of adding multiplicative noise to an amplitude equation and I am following a paper. It states that in cylindrical polar coordinates the probability density is
$\rho(r)=Nr\exp \bigg(\frac{2\mu r^2 -r^4 }{2 \sigma_2^2}\bigg) $
and
$p(x,y) =K \exp \bigg(\frac{2\mu (x^2+y^2) - (x^2 + y^2)^2 }{2\sigma_2^2} \bigg)$.
However, when you simply put $r^2=x^2 + y^2$ into one you do not get the other. Does any one have any thoughts to why this would be?
Kindest regards,
Catherine