The main type of noise I know other than white noise is a colored noise (Ornstein-Uhlenbeck) of the form:
$$d\eta = \lambda \eta dt + \alpha dW_t$$
with exponential correlation. I'm interested in noise in which the distribution is heavy tailed so that events far from the mean are more common, and have much longer tails in the correlation function so that correlations last much longer than the Ornstein-Uhlenbeck process (I'm unsure if this correlation and heavy tailed noise coincide or not). I believe this is also called "pink noise". Is there a known stochastic differential equation of the form:
$$dP(t) = f(P(t))dt + g(P(t))dW_t$$
describing such a process? (I'm using $P$ for pink noise).