for vanishing $x$. Therefore, for $p\in(-1,0)$, the PDF presents dominant exponential-like behavior both for low and large values, without any restriction on the value of $s$. Outside that interval, the [*power law*]{} $x^{-2s}$ in (\[ss0\]) asymptotically dominates, for either small (if $p>0$) or large (if $p<-1$) argument. Then, normalization in $[0,\infty)$ restricts the possible values of $s$ according to: $s<1/2$ (if $p>0$), $x>1/2$ (if $p<-1$).

In the marginal cases, Eq. (\[ss0\]) explicitly is:

 I: For $p\equiv r-2s=-1,$ $$\label{caseI}
\rho(x)=\frac{\rho_o}{x^{2s+\gamma_\mu}}\;
\exp( -\gamma_\mu\theta/x ) \;,$$ with $2s>1-\gamma_\mu$ for nomalizability.

II: For $p\equiv r-2s=0$, $$\label{caseII}
\rho(x)=\rho_o x^{\gamma_\mu\theta-2s}
\exp( -\gamma_\mu x ) \;,$$ with $2s<\gamma_\mu\theta+1$ for normalization, but $2s<\gamma_\mu\theta$ to avoid the divergence at the origin.

Fig. \[fig:m\] displays the possible PDF asymptotic shapes in $(s,r)$ space. Notice that the $s=0$ axis gives the solution for mean-reverting models with purely additive fluctuations. Let us analyze some special cases. In the trivial case $(s,r)=(0,1)$, corresponding to the Ornstein-Ulhenbeck process [@books] $$\label{OU}
{\rm d} x \;=\;-\gamma[x-\theta]{\rm d}t + \mu {\rm d}z \;,$$ the noisy contribution becomes additive and the stationary PDF is Gaussian (truncated at $x=0$).

Although we are dealing with $\theta>0$, it is worth of mention the case $(\theta,s,r)=(0,1,1)$, corresponding to the geometric Brownian process, 