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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 $f_a$ toexhibit (i) a unique stationary measure, and (ii) a positive Lyapunov exponentcomparable to $\int_{S^1} \log |f_a'| \, dx$. This stands in contrast with thesituation for the deterministic dynamics of $f_a$, the chaotic regimes of whichare 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$\epsilon \to 0$.

Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps