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### Open Access Journals

CPAA

We prove some comparison principles for viscosity solutions of fully nonlinear degenerate elliptic equations
that satisfy some conditions of partial non-degeneracy instead of the usual uniform ellipticity or
strict monotonicity. These results
are applied to the well-posedness
of the Dirichlet problem under suitable conditions at the characteristic points of the boundary.
The examples motivating the theory are operators of the form of sum of squares of vector fields
plus a nonlinear first order Hamiltonian and the Pucci operator over the Heisenberg group.

CPAA

We study Hamilton-Jacobi equations with upper semicontinuous initial data
without convexity assumptions on the Hamiltonian. We analyse the
behavior of generalized

*u.s.c*solutions at the initial time $t=0$, and find necessary and sufficient conditions on the Hamiltonian such that the solution attains the initial data along a sequence (right accessibility).
CPAA

We study the homogenization of Hamilton-Jacobi equations with oscillating initial data and non-coercive Hamiltonian, mostly of the Bellman-Isaacs form arising in optimal control and differential games. We describe classes of equations for which pointwise homogenization fails for some data. We prove locally uniform homogenization for various Hamiltonians with some partial coercivity and some related restrictions on the oscillating variables, mostly motivated by the applications to differential games, in particular of pursuit-evasion type. The effective initial data are computed under some assumptions of asymptotic controllability of the underlying control system with two competing players.

NHM

We consider $N$-person differential games involving linear systems affected by white noise, running cost quadratic in the control and in the displacement of the state from a reference position, and with
long-time-average integral cost functional.
We solve an associated system of Hamilton-Jacobi-Bellman and Kolmogorov-Fokker-Planck equations and find explicit Nash equilibria in the form of linear feedbacks. Next we compute the limit as the number $N$ of players goes to infinity, assuming they are almost identical and with suitable scalings of the parameters. This provides a
quadratic-Gaussian solution to a system of two
differential equations of the kind introduced by Lasry and Lions in the theory of Mean Field Games [22]. Under a natural normalization the uniqueness of this solution depends on the sign of a single parameter.
We also discuss some singular limits, such as vanishing noise, cheap control, vanishing discount. Finally, we compare the L-Q model with other Mean Field models of population distribution.

DCDS

In this paper we study the boundary value problem for the
Hamilton-Jacobi-Isaacs equation of pursuit-evasion
differential games with state constraints.
We prove existence of a continuous viscosity solution and
a comparison theorem that we apply to establish uniqueness
of such a solution and its
uniform approximation by solutions of discretized equations.

DCDS

We consider the short time behaviour of stochastic systems affected by a stochastic volatility evolving at a faster time scale. We study the asymptotics of a logarithmic functional of the process by methods of the theory of homogenization and singular perturbations for fully nonlinear PDEs. We point out three regimes depending on how fast the volatility oscillates relative to the horizon length. We prove a large deviation principle for each regime and apply it to
the asymptotics of option prices near maturity.

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