  Mean value formulas are of great importance in the theory of partial
differential equations: many very useful results are drawn, for instance, from
the well known equivalence between harmonic functions and mean value
properties. In the nonlocal setting of fractional harmonic functions, such an
equivalence still holds, and many applications are now-days available. The
nonlinear case, corresponding to the $p$-Laplace operator, has also been
recently investigated, whereas the validity of a nonlocal, nonlinear,
counterpart remains an open problem. In this paper, we propose a formula for
the \emph{nonlocal, nonlinear mean value kernel}, by means of which we obtain
an asymptotic representation formula for harmonic functions in the viscosity
sense, with respect to the fractional (variational) $p$-Laplacian (for $p\geq
2$) and to other gradient dependent nonlocal operators.
