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

TL;DR

This paper derives precise short-time asymptotic formulas for solutions of game-theoretic p-caloric functions with boundary conditions, generalizing classical results and analyzing level surfaces.

Contribution

It introduces new asymptotic formulas for game-theoretic p-caloric functions, extending Varadhan's large deviations and heat content results to this nonlinear setting.

Findings

Derived asymptotic formulas for short-time behavior of solutions.

Analyzed the short-time behavior of the q-mean of solutions.

Applications to time-invariant level surfaces of solutions.

Abstract

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.