In this manuscript, we propose a structural condition on non-separable Hamiltonians, which we term displacement monotonicity condition, to study second order mean field games master equations.A rate of dissipation of a bilinear form is brought to bear a global (in time) well-posedness theory, based on a-priori uniform Lipschitz estimates on the solution in the measure variable. Displacement monotonicity being sometimes in dichotomy with the widely used Lasry-Lions monotonicity condition, the novelties of this work persist even when restricted to separable Hamiltonians.

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We endow the set of probability measures on a weighted graph with a Monge-Kantorovich metric, induced by a function defined on the set of vertices. The graph is assumed to have n vertices and so, the boundary of the probability simplex is an affine (n − 2)chain. Characterizing the geodesics of minimal length which may intersect the boundary, is a challenge we overcome even when the endpoints of the geodesics don't share the same connected components. It is our hope that this work would be a preamble to the theory of Mean Field Games on graphs.

This paper is concerned with existence of viscosity solutions of non-translation invariant nonlocal fully nonlinear equations. We construct a discontinuous viscosity solution of such nonlocal equation by Perron's method. If the equation is uniformly elliptic, we prove the discontinuous viscosity solution is Hölder continuous and thus it is a viscosity solution.

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We prove comparison theorems and uniqueness of viscosity solutions for a class of nonlocal equations. This class of equations includes Bellman-Isaacs equations containing operators of Lévy type with measures depending on x and control parameters, as well as elliptic nonlocal equations that are not strictly monotone in the u variable. The proofs use the knowledge about regularity of viscosity solutions of such equations. Mathematics Subject Classification. 35R09, 35D40, 35J60, 47G20, 45K05, 93E20.

It is well known that the monotonicity condition, either in Lasry-Lions sense or in displacement sense, is crucial for the global well-posedness of mean field game master equations, as well as for the uniqueness of mean field equilibria and solutions to mean field game systems.In the literature, the monotonicity conditions are always taken in a fixed direction. In this paper we propose a new type of monotonicity condition in the opposite direction, which we call the anti-monotonicity condition, and establish the global well-posedness for mean field game master equations with nonseparable Hamiltonians. Our anti-monotonicity condition allows our data to violate both the Lasry-Lions monotonicity and the displacement monotonicity conditions.