Maximum principles and the method of moving planes for the uniformly elliptic nonlocal Bellman operator and applications

In this paper, we establish various maximum principles and develop the method of moving planes and the sliding method (on general unbounded domains) for equations involving the uniformly elliptic nonlocal Bellman operator. As a consequence, we derive multiple applications of these maximum principles and the moving planes method. For instance, we prove symmetry, monotonicity and uniqueness results and asymptotic properties for solutions to various equations involving the uniformly elliptic nonlocal Bellman operator in bounded domains, unbounded domains, epigraph or ${\mathbb{R}}^n$. In particular, the uniformly elliptic nonlocal Monge-Amp`{e}re operator introduced by Caffarelli and Charro in \cite{CC} is a typical example of the uniformly elliptic nonlocal Bellman operator.