  We study the effects of IID random perturbations of amplitude $\epsilon > 0$
on the asymptotic dynamics of one-parameter families $\{f_a : S^1 \to S^1, a
\in [0,1]\}$ of smooth multimodal maps which "predominantly expanding", i.e.,
$|f'_a| \gg 1$ away from small neighborhoods of the critical set $\{ f'_a = 0
\}$. We obtain, for any $\epsilon > 0$, a \emph{checkable, finite-time}
criterion on the parameter $a$ for random perturbations of the map $f_a$ to
exhibit (i) a unique stationary measure, and (ii) a positive Lyapunov exponent
comparable to $\int_{S^1} \log |f_a'| \, dx$. This stands in contrast with the
situation for the deterministic dynamics of $f_a$, the chaotic regimes of which
are determined by typically uncheckable, infinite-time conditions. Moreover,
our finite-time criterion depends on only $k \sim \log (\epsilon^{-1})$
iterates of the deterministic dynamics of $f_a$, which grows quite slowly as
$\epsilon \to 0$.

Dynamical Systems