The aim of the course is to explain how tools developed in spectral theory and semi-classical analysis are used to study the long time behaviour  of  processes conditioned to remain a  region of the phase space as well as the concentration of  quasi-stationary distributions in the small temperature regime.

This course is composed of two parts. The first part is dedicated to the study of the long time behaviour of elliptic diffusion processes conditioned to remain in a bounded region of the phase space. For that purpose, the semi group of the absorbed process is investigated using techniques developed for PDE and it will be proved that such conditioned processes converge in time to a quasi-stationary distribution.  The second part of the course is concerned with the study of the concentration of this quasi-stationary distribution in the small temperature regime. To this end, tools from semi-classical analysis will be introduced.