Drift-diffusion equations on domains in Rd: Essential self-adjointness and stochastic completeness

We consider the problem of quantum and stochastic confinement for drift-diffusion equations on domains Ω⊂Rd. We obtain various sufficient conditions on the behavior of the coefficients near the boundary of Ω which ensure the essential self-adjointness or stochastic completeness of the symmetric form of the drift-diffusion operator, −1ρ∞∇⋅ρ∞D∇. The proofs are based on the method developed in [31] for quantum confinement on bounded domains in Rd. In particular for stochastic confinement we combine the Liouville property with Agmon type exponential estimates for weak solutions.