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Consider the Langevin process, described by a vector (positions and momenta)in $\mathbb{R}^{d}\times\mathbb{R}^d$. Let $\mathcal O$ be a $\mathcal{C}^2$open bounded and connected set of $\mathbb{R}^d$. Recent works showed theexistence of a unique quasi-stationary distribution (QSD) of the Langevinprocess on the domain $D:=\mathcal{O}\times\mathbb{R}^d$. In this article, westudy the overdamped limit of this QSD, i.e. when the friction coefficient goesto infinity. In particular, we show that the marginal law in position of theoverdamped limit is the QSD of the overdamped Langevin process on the domain$\mathcal{O}$.

Quasi-stationary distribution for the Langevin process in cylindrical domains, part II: overdamped limit