MAXIMUM PRINCIPLES AND ALEKSANDROV-BAKELMAN-PUCCI TYPE ESTIMATES FOR NONLOCAL SCHRO ODINGER EQUATIONS WITH EXTERIOR CONDITIONS

摘要

We consider Dirichlet exterior value problems related to a class of nonlocal Schrodinger 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 Clement-Peletier, valid on compact subsets of the domain, and a full antimaximum principle by restricting to fractional Schrodinger 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.