标题： 漂移扩散方程在$\mathbb{R}^d$中的区域：本质自伴性和随机完备性 显示英文标题

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

摘要： 我们考虑在区域$ \Omega \subset \mathbb R^d$上漂移-扩散方程的量子和随机约束问题。 我们获得了关于系数在区域$\Omega$边界附近行为的各种充分条件，这些条件确保了漂移-扩散算子对称形式的本质自伴性或随机完备性，$-\frac{1}{\rho_\infty}\,\nabla\cdot \rho_\infty\mathbb D\nabla$。 证明基于在 [29] 中为有界区域$\mathbb R^d$上的量子约束发展出的方法。 特别是对于随机约束，我们将 Liouville 性质与弱解的 Agmon 类指数估计相结合。 显示英文摘要

摘要： 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.