Drift-diffusion equations on domains in ${\mathbb{R}}^d$: essential self-adjointness and stochastic completeness

Abstract

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.

v2: minor corrections and two additional references

References in corpus (1)

• Self-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces