On asymptotic expansions for the fractional infinity Laplacian

We propose two asymptotic expansions of two interrelated integral-type averages, in the context of the fractional $\mathrm{\infty }$-Laplacian ${\mathrm{\Delta }}_{\mathrm{\infty }}^s$ for $s\in (\frac{1}{2},1)$. This operator has been introduced and first studied in [Bjorland, C., Caffarelli, L. and Figalli, A., \textsl{Nonlocal Tug-of-War and the inifnity fractional Laplacian}, Comm. Pure Appl. Math., $\mathbf{\text{65}}$, pp. 337–380, (2012)]. Our expansions are parametrised by the radius of the removed singularity $ϵ$, and allow for the identification of ${\mathrm{\Delta }}_{\mathrm{\infty }}^s\varphi (x)$ as the ${ϵ}^{2s}$-order coefficient of the deviation of the $ϵ$-average from the value $\varphi (x)$, in the limit $ϵ\to 0+$. The averages are well posed for functions $\varphi$ that are only Borel regular and bounded.