A discrete stochastic interpretation of the Dominative $p$-Laplacian

Abstract

The Dominative $p$-Laplacian is the operator defined for $2\le p < \infty$ as follows: \begin{equation}\label{dominativep} \mathcal{L}_{p}u(x)=\frac{1}{p}\left(\lambda_{1}+\ldots+\lambda_{N-1}\right)+\frac{(p-1)}{p}\lambda_{N}, \end{equation} where we have ordered the eigenvalues of $D^{2}u(x)$ as $\lambda_{1}\le \lambda_{2}\ldots\le\lambda_{N}$. The operator $\mathcal{L}_{p}u(x)$ was introduced by Brustand to give a natural explanation of the superposition principle for the $p$-Laplace equation. In this paper, we present a discrete stochastic approximation to the unique viscosity solution of the Dirichlet problem for the Dominative $p$-Laplace Equation.