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We consider the solution of $u_t-\Delta^G_p u=0$ in a (not necessarilybounded) domain, satisfying $u=0$ initially and $u=1$ on the boundary at alltimes. Here, $\Delta^G_p u$ is the game-theoretic or normalized $p$-laplacian.We derive new precise asymptotic formulas for short times, that generalize thework of S. R. S. Varadhan for large deviations and that of the second authorand S. Sakaguchi for the heat content of a ball touching the boundary. We alsocompute 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.

Short time behaviour for game-theoretic $p$-caloric functions