A Discrete Stochastic Interpretation of the Dominative $p$-Laplacian

Abstract

We build a discrete stochastic process adapted to the (nonlinear) dominative $p$-Laplacian $$ \mathcal{D}_p u(x):=\Delta u + (p-2)\lambda_{N} , $$ where $\lambda_{N}$ is the largest eigenvalue of $D^2 u$ and $p > 2$. We show that the discrete solutions of the Dirichlet problems at scale $\varepsilon$ tend to the solution of the Dirichlet problem for $\mathcal{D}_p$ as $\varepsilon\to 0$. We assume that the domain and the boundary values are both Lipschitz.