Drift-diffusion equations on domains in $\mathbb{R}^d$: essential   self-adjointness and stochastic completeness
  We consider the problem of quantum and stochastic confinement for drift-diffusion equations on domains $ \Omega \subset \mathbb R^d$. We obtain various sufficient conditions on the behavior of the coefficients near the boundary of $\Omega$ which ensure the essential self-adjointness or stochastic completeness of the symmetric form of the drift-diffusion operator, $-\frac{1}{\rho_\infty}\,\nabla\cdot \rho_\infty\mathbb D\nabla$. The proofs are based on the method developed in [29] for quantum confinement on bounded domains in $\mathbb R^d$. In particular for stochastic confinement we combine the Liouville property with Agmon type exponential estimates for weak solutions. 
Gheorghe Nenciu and Irina Nenciu
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