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

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We consider the (viscosity) solution uε of the elliptic equation ε²Δₚᵍ u= u in a domain (not necessarily bounded), satisfying u=1 on its boundary. Here, Δₚᵍ is the game-theoretic or normalized p-laplacian. We derive asymptotic formulas for ε→ 0+ involving the values of uε, in the spirit of Varadhan's work Va, and its q-mean on balls touching the boundary, thus generalizing that obtained in MS-AM for p=q=2. As in a related parabolic problem, investigated in BM, we link the relevant asymptotic behavior to the geometry of the domain.