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 Omega subset of R-d. We obtain various sufficient conditions on the behavior of the co-efficients near the boundary of Omega which ensure the essential self-adjointness or stochastic completeness of the symmetric form of the drift-diffusion operator, -1/rho(infinity) del . rho D-infinity del. 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. (C) 2017 Elsevier Inc. All rights reserved.