Memoirs of the American Mathematical Society 2009; 77 pp; softcover Volume: 204 ISBN10: 0821847155 ISBN13: 9780821847152 List Price: US$68 Individual Members: US$40.80 Institutional Members: US$54.40 Order Code: MEMO/204/960
 The authors study singular perturbations of optimal stochastic control problems and differential games arising in the dimension reduction of system with multiple time scales. They analyze the uniform convergence of the value functions via the associated HamiltonJacobiBellmanIsaacs equations, in the framework of viscosity solutions. The crucial properties of ergodicity and stabilization to a constant that the Hamiltonian must possess are formulated as differential games with ergodic cost criteria. They are studied under various different assumptions and with PDE as well as controltheoretic methods. The authors also construct an explicit example where the convergence is not uniform. Finally they give some applications to the periodic homogenization of HamiltonJacobi equations with noncoercive Hamiltonian and of some degenerate parabolic PDEs. Table of Contents  Introduction and statement of the problem
 Abstract ergodicity, stabilization, and convergence
 Uncontrolled fast variables and averaging
 Uniformly nondegenerate fast diffusion
 Hypoelliptic diffusion of the fast variables
 Controllable fast variables
 Nonresonant fast variables
 A counterexample to uniform convergence
 Applications to homogenization
 Bibliography
