A Starter: The First-Order Master Equation

This chapter contains a preliminary analysis of the master equation in the simpler case when there is no common noise. Some of the proofs given in this chapter consist of a sketch only. One of the reasons is that some of the arguments used to investigate the mean field games (MFGs) system have been already developed in the literature. Another reason is that the chapter constitutes a starter only, specifically devoted to the simpler case without common noise. It provides details of the global Lipschitz continuity of H. The solutions of the MFG system are uniformly Lipschitz continuous, which are independently of initial conditions.

Projections of the sphere graph to the arc graph of a surface

Let [Formula: see text] be a compact surface, and [Formula: see text] be the double of a handlebody. Given a homotopy class of maps from [Formula: see text] to [Formula: see text] inducing an isomorphism of fundamental groups, we describe a canonical uniformly Lipschitz retraction of the sphere graph of [Formula: see text] to the arc graph of [Formula: see text]. We also show that this retraction is a uniformly bounded distance from the nearest point projection map.

Viscous stability of quasi-periodic tori

This paper is devoted to the study of $P$-regularity of viscosity solutions $u(x,P)$, $P\in {\Bbb R}^n$, of a smooth Tonelli Lagrangian $L:T {\Bbb T}^n \rightarrow {\Bbb R}$ characterized by the cell equation $H(x,P+D_xu(x,P))=\overline {H}(P)$, where $H: T^* {\Bbb T}^n\rightarrow {\Bbb R}$ denotes the Hamiltonian associated with $L$ and $\overline {H}$ is the effective Hamiltonian. We show that if $P_0$ corresponds to a quasi-periodic invariant torus with a non-resonant frequency, then $D_xu(x,P)$ is uniformly Hölder continuous in $P$ at $P_0$ with Hölder exponent arbitrarily close to $1$, and if both $H$ and the torus are real analytic and the frequency vector of the torus is Diophantine, then $D_xu(x,P)$ is uniformly Lipschitz continuous in $P$ at $P_0$, i.e., there is a constant $C\gt 0$ such that $\|D_xu(\cdot ,P)-D_xu(\cdot ,P_0)\|_{\infty }\le C\|P-P_0\|$ for $\|P-P_0\|\ll 1$. Similar P-regularity of the Peierls barriers associated with $L(x,v)- \langle P,v \rangle $is also obtained.

Comparison Theorem for any Solutions of Backward Stochastic Differential Equations and its Application

ƒIn this paper, we study the one-dimensional backward stochastic equations driven by continuous local martingale. We establish a generalized the comparison theorem for any solutions where the coefficient is uniformly Lipschitz continuous in z and is equi-continuous in y.