  We unify two paradigmatic mesoscopic mechanisms for the emergence of
nonextensive statistics, namely the multiplicative noise mechanism leading to a
{\it linear} Fokker-Planck (FP) equation with {\it inhomogenous} diffusion
coefficient, and the non-Markovian process leading to the {\it nonlinear} FP
equation with {\it homogeneous} diffusion coefficient. More precisely, we
consider the equation $\frac{\partial p(x,t)}{\partial
t}=-\frac{\partial}{\partial x}[F(x) p(x,t)] + 1/2D \frac{\partial^2}{\partial
x^2} [\phi(x,p)p(x,t)]$, where $D \in {\cal R}$ and $F(x)=-\partial V(x)
/\partial x$, $V(x)$ being the potential under which diffusion occurs. Our aim
is to find whether $\phi(x,p)$ exists such that the inhomogeneous linear and
the homogeneous nonlinear FP equations become unified in such a way that the
(ubiquitously observed) $q$-exponentials remain as stationary solutions. It
turns out that such solutions indeed exist for a wide class of systems, namely
when $\phi(x,p)=[A+BV(x)]^\theta [p(x,t)]^{\eta}$, where $A$, $B$, $\theta$ and
$\eta$ are (real) constants.
  Our main result can be sumarized as follows: For $\theta \neq 1$ and
arbitrary confining potential $V(x)$, $p(x,\infty) \propto \lbrace
1-\beta(1-q)V(x)\rbrace ^{1/(1-q)} \equiv e_q^{-\beta V(x)}$, where $q= 1+
\eta/(\theta-1)$. The present approach unifies into a single mechanism,
essentially {\it long memory}, results currently discussed and applied in the
literature.
