Monotonicity, Gibbons’ conjecture and Liouville theorem for the nonlocal Monge–Ampère problem with the Marchaud time derivative

In this paper, we primarily establish maximum principles for the nonlocal Monge–Ampère problem with the Marchaud time derivative, and apply these principles to study the monotonicity, Gibbons’ conjecture and Liouville theorem for a class of nonlocal Monge–Ampère parabolic equations with the Marchaud time derivative. Specifically, to overcome the challenges posed by the space-time nonlocality of the operator [Formula: see text], we first develop various maximum principles for the operator [Formula: see text]. On this basis, we derive the monotonicity of the solution to this class of equation. Secondly, combining the sliding method, we investigate the Gibbons’ conjecture related to this class of equation. Finally, the Liouville theorem is established by applying the maximum principle of the antisymmetric function.