  Motivated by marginals-mimicking results for It\^o processes via SDEs and by
their applications to volatility modeling in finance, we discuss the weak
convergence of the law of a hypoelliptic diffusions conditioned to belong to a
target affine subspace at final time, namely $\mathcal{L}(Z_t|Y_t = y)$ if
$X_{\cdot}=(Y_\cdot,Z_{\cdot})$. To do so, we revisit Varadhan-type estimates
in a small-noise regime (as opposed to small-time), studying the density of the
lower-dimensional component $Y$. The application to stochastic volatility
models include the small-time and, for certain models, the large-strike
asymptotics of the Gyongy-Dupire's local volatility function. The final product
are asymptotic formulae that can (i) motivate parameterizations of the local
volatility surface and (ii) be used to extrapolate local volatilities in a
given model.
