Asymptotics for the resolvent equation associated to the game-theoretic p-laplacian

We consider the (viscosity) solution $u^e$ of the elliptic equation $e^2De_p^G u= u$ in a domain (not necessarily bounded), satisfying $u=1$ on its boundary. Here, $De_p^G$ is the {it game-theoretic or normalized $p$-laplacian}. We derive asymptotic formulas for $e o 0^+$ involving the values of $u^e$, in the spirit of Varadhan's work cite{Va}, and its $q$-mean on balls touching the boundary, thus generalizing that obtained in cite{MS-AM} for $p=q=2$. As in a related parabolic problem, investigated in cite{BM}, we link the relevant asymptotic behavior to the geometry of the domain.