Dirichlet eigenvalue problems of irreversible Langevin diffusion

The basic objective of this research work is to investigate the asymptotic behavior of the first eigenvalue of an irreversible Langevin diffusion with zero boundary values. In particular, a reversible diffusion is perturbed by adding an antisymmetric drift which preserves the invariant measure. Then, a necessary and sufficient condition is provided for the boundness and the limiting behavior of the first eigenvalue, under Dirichlet boundary conditions and with respect to the invariant measure. In other words, we prove that the first eigenvalue is bounded if and only if the associated stochastic dynamical system has a first integral. Furthermore, we demonstrate that the limiting eigenvalue is the minimum of the Dirichlet functional over all first integrals of the divergence-free vector field. An extension of this model with a time parameter in the boundary conditions is studied, where we give another characterization to achieve the same main result.