Short-time behavior for game-theoretic p-caloric functions

We consider the solution of $u_t-De^G_p u=0$ in a (not necessarily bounded) domain, satisfying $u=0$ initially and $u=1$ on the boundary at all times. Here, $De^G_p u$ is the {it game-theoretic} or {it normalized} $p$-laplacian. We derive new precise asymptotic formulas for short times, that generalize the work of S. R. S. Varadhan cite{Va} for large deviations and that of the second author and S. Sakaguchi cite{MS-AM} for the heat content of a ball touching the boundary. We also compute the short-time behavior of the $q$-mean of $u$ on such a ball. Applications to time-invariant level surfaces of $u$ are then derived.