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

Abstract We consider the problem of quantum and stochastic confinement for drift-diffusion equations on domains Ω ⊂ R d . 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, − 1 ρ ∞ ∇ ⋅ ρ ∞ D ∇ . The proofs are based on the method developed in [31] for quantum confinement on bounded domains in R d . In particular for stochastic confinement we combine the Liouville property with Agmon type exponential estimates for weak solutions.