Maximum principles and Aleksandrov-Bakelman-Pucci type estimates for non-local Schr\"odinger equations with exterior conditions

TL;DR

This paper develops maximum principles and ABP-type estimates for non-local Schrödinger equations with exterior conditions, using probabilistic methods related to subordinate Brownian motion, and establishes existence, uniqueness, and Liouville theorems.

Contribution

It introduces new maximum principles and ABP estimates for non-local Schrödinger operators with exterior conditions, extending classical results to a probabilistic framework.

Findings

Proved elliptic and parabolic ABP-type estimates.

Established maximum and anti-maximum principles for non-local Schrödinger equations.

Derived Liouville-type theorems for harmonic and semi-linear solutions.

Abstract

We consider Dirichlet exterior value problems related to a class of non-local Schr\"odinger operators, whose kinetic terms are given in terms of Bernstein functions of the Laplacian. We prove elliptic and parabolic Aleksandrov-Bakelman-Pucci type estimates, and as an application obtain existence and uniqueness of weak solutions. Next we prove a refined maximum principle in the sense of Berestycki-Nirenberg-Varadhan, and a converse. Also, we prove a weak anti-maximum principle in the sense of Cl\'ement-Peletier, valid on compact subsets of the domain, and a full anti-maximum principle by restricting to fractional Schr\"odinger operators. Furthermore, we show a maximum principle for narrow domains, and a refined elliptic ABP-type estimate. Finally, we obtain Liouville-type theorems for harmonic solutions and for a class of semi-linear equations. Our approach is probabilistic, making use of the properties of subordinate Brownian motion.