Processes that converge to the Pareto distribution

Question

Do any stochastic processes generate the Pareto distribution as the steady-state statistic of the ensemble?

For example,

$$ dS_t = f(t, S_t, W_t) $$

where in the Ito sense the p.d.f. of $ g(S_t) $ is Pareto and $ g(\cdot) $ is one-to-one.

Answer 1

Not exactly what I was after, but moving closer.

Let $ R $ be a random variable, then the p.d.f of $ v = R^t $ is given by,

$$\begin{align} \rho_v &= \int dR ~ \rho_R ~ \delta(v - R^t) \\ &= \frac{\rho_R(v^{1/t})}{t v^{(t-1)/t}} ~~ \forall t > 0 \end{align}$$

which is a Power-law probability distribution.

For $ t \rightarrow \infty $,

$$ \rho_v \propto \frac{1}{v} $$