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

We consider the solution u of u(t) - Delta(G)(p) u = 0 in a (not necessarily bounded) domain Omega, such that u = 0 in Omega at time t = 0 and u = 1 on the boundary of Omega at all times. Here, Delta(G)(p) is the game-theoretic or normalized p-laplacian. We derive new precise asymptotic formulas for t -> 0, that generalize that of S.R.S. Varadhan [39] for large deviations and that of the second author and S. Sakaguchi [26] for the heat content of a ball touching the boundary. We also determine the behavior for t -> 0 of the q-mean of u on such a ball. Applications to time-invariant level surfaces of u are then obtained. (C) 2018 Elsevier Masson SAS. All rights reserved.