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

We consider the solution of $u_t-\Delta^G_p u=0$ in a (not necessarily bounded) domain, satisfying $u=0$ initially and $u=1$ on the boundary at all times. Here, $\Delta^G_p u$ is the game-theoretic or normalized $p$-laplacian. We derive new precise asymptotic formulas for short times, that generalize the work of S. R. S. Varadhan for large deviations and that of the second author and S. Sakaguchi 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.