  A large variety of microscopic or mesoscopic models lead to generic results
that accommodate naturally within Boltzmann-Gibbs statistical mechanics (based
on $S_1\equiv -k \int du p(u) \ln p(u)$). Similarly, other classes of models
point toward nonextensive statistical mechanics (based on $S_q \equiv k [1-\int
du [p(u)]^q]/[q-1]$, where the value of the entropic index $q\in\Re$ depends on
the specific model). We show here a family of models, with multiplicative
noise, which belongs to the nonextensive class. More specifically, we consider
Langevin equations of the type $\dot{u}=f(u)+g(u)\xi(t)+\eta(t)$, where
$\xi(t)$ and $\eta(t)$ are independent zero-mean Gaussian white noises with
respective amplitudes $M$ and $A$. This leads to the Fokker-Planck equation
$\partial_t P(u,t) = -\partial_u[f(u) P(u,t)] +
M\partial_u\{g(u)\partial_u[g(u)P(u,t)]\} + A\partial_{uu}P(u,t)$. Whenever the
deterministic drift is proportional to the noise induced one, i.e., $f(u)
=-\tau g(u) g'(u)$, the stationary solution is shown to be $P(u, \infty)
\propto \bigl\{1-(1-q) \beta [g(u)]^2 \bigr\}^{\frac{1}{1-q}} $ (with $q \equiv
\frac{\tau + 3M}{\tau+M}$ and $\beta=\frac{\tau+M}{2A}$). This distribution is
precisely the one optimizing $S_q$ with the constraint $<[g(u)]^2 >_q \equiv
\{\int du [g(u)]^2[P(u)]^q \}/ \{\int du [P(u)]^q \}= $constant. We also
introduce and discuss various characterizations of the width of the
distributions.
