Spectral gap and irreducibility for collisional transport semigroups with general vector fields

Motivated by kinetic theory, this paper investigates the spectral theory and long-time asymptotics of semigroups $ \left( W(t)\right) _{t\geq 0} $ associated with integro-differential equations $ \frac{\partial f}{\partial t} +a.\frac{\partial f}{\partial x}+b.\frac{\partial f}{\partial v}+\sigma f = Kf\ $ in $ L^{p}(\Omega \times \mathbb{R}^{d}) $ $ (1\leq p<+\infty) $ under non incoming boundary condition. We consider a broad classes of vector fields $ \left( a, b\right) $ satisfying appropriate transversality conditions, together with a general class of scattering operators $ K $. The analysis focuses on the existence of a spectral gap for $ \left( W(t)\right) _{t\geq 0} $ and on its irreducibility. A systematic construction is provided, and several open questions are highlighted.