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

We study the effects of IID random perturbations of amplitude $ϵ>0$ on the asymptotic dynamics of one-parameter families ${f_a : S^1 \to S^1, a \in [0,1]}$ofsmoothmultimodalmapswhich\mathrm{"}predominantlyexpanding\mathrm{"},i\mathrm{.}e\mathrm{.},$|f’_a| \gg 1$awayfromsmallneighborhoodsofthecriticalset${ f’_a = 0 }$\mathrm{.}Weobtain,forany$\epsilon > 0$, 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 ${\int }_{S^1}\log \mathrm{\mid }{f}_a^{\mathrm{\prime }}\mathrm{\mid }dx$. 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 $k\sim \log ({ϵ}^{-1})$ iterates of the deterministic dynamics of $f_a$, which grows quite slowly as $ϵ\to 0$.