Well-posedness and trace theory for the Kolmogorov equation on bounded domains

We establish well-posedness of the stationary Kolmogorov equation with spherical velocity on a bounded domain, subject to either inflow boundary conditions or specular reflection. For the sake of completeness, we also include the problem on the torus, which was solved already in [Albritton, Armstrong, Mourrat, Novack 2024]. We introduce a hypoelliptic space of functions whose trace is defined via the transport operator; we prove a Poincaré-type inequality with trace, which is an essential step towards the well-posedness of the inflow problem without friction. Moreover, concerning the trace problem [Albritton, Armstrong, Mourrat, Novack 2024 - Question 1.8], we obtain a partial result with the optimal weight $\mathrm{\mid }{n}_x\cdot v\mathrm{\mid }$, in which the outgoing (resp. inflow) flux is bounded by the energy inside the domain and the inflow (resp. outgoing) flux.