Non-local tug-of-war with noise for the geometric fractional $p$-Laplacian

Abstract

This paper concerns the fractional $p$-Laplace operator $\Delta_p^s$ in non-divergence form, which has been introduced in [Bjorland, Caffarelli, Figalli (2012)]. For any $p\in [2,\infty)$ and $s\in (\frac{1}{2},1)$ we first define two families of non-local, non-linear averaging operators, parametrised by $\epsilon$ and defined for all bounded, Borel functions $u:\mathbb{R}^N\to \mathbb{R}$. We prove that $\Delta_p^s u(x)$ emerges as the $\epsilon^{2s}$-order coefficient in the expansion of the deviation of each $\epsilon$-average from the value $u(x)$, in the limit of the domain of averaging exhausting an appropriate cone in $\mathbb{R}^N$ at the rate $\epsilon\to 0$. Second, we consider the $\epsilon$-dynamic programming principles modeled on the first average, and show that their solutions converge uniformly as $\epsilon\to 0$, to viscosity solutions of the homogeneous non-local Dirichlet problem for $\Delta_p^s$, when posed in a domain $\mathcal{D}$ that satisfies the external cone condition and subject to bounded, uniformly continuous data on $\mathbb{R}^N\setminus \mathcal{D}$. Finally, we interpret such $\epsilon$-approximating solutions as values to the non-local Tug-of-War game with noise. In this game, players choose directions while the game position is updated randomly within the infinite cone that aligns with the specified direction, whose aperture angle depends on $p$ and $N$, and whose $\epsilon$-tip has been removed.