Maximum Principles and Aleksandrov-Bakelman-Pucci Type Estimates for Nonlocal Schro Odinger Equations with Exterior Conditions

Abstract

We consider Dirichlet exterior value problems related to a class of nonlocal Schrödinger operators, whose kinetic terms are given in terms of Bernstein functions of the Laplacian. We prove elliptic and parabolic Aleksandrov--Bakelman--Pucci (ABP) 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 antimaximum principle in the sense of Clément--Peletier, valid on compact subsets of the domain, and a full antimaximum principle by restricting to fractional Schrödinger 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 semilinear equations. Our approach is probabilistic, making use of the properties of subordinate Brownian motion.