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

TL;DR

This paper establishes conditions under which drift-diffusion operators on domains in Euclidean space are essentially self-adjoint or stochastically complete, using boundary behavior and advanced analytical methods.

Contribution

It provides new sufficient conditions for quantum and stochastic confinement of drift-diffusion equations based on boundary coefficient behavior.

Findings

Conditions for essential self-adjointness derived

Criteria for stochastic completeness established

Methods extend previous quantum confinement techniques

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.