On near optimal trajectories for a game associated with the \infty-Laplacian

A two-player stochastic differential game representation has recently been obtained for solutions of the equation -\Delta_\infty u=h in a \calC^2 domain with Dirichlet boundary condition, where h is continuous and takes values in \RR\setminus\{0\}. Under appropriate assumptions, including smoothness of u, the vanishing \delta limit law of the state process, when both players play \delta-optimally, is identified as a diffusion process with coefficients given explicitly in terms of derivatives of the function u.