English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We propose a new finite difference approximation to the Dirichlet problem forthe homogeneous $\mathbf{p}$-Laplace equation posed on an $N$-dimensionaldomain, in connection with the Tug of War games with noise. Our game and therelated mean-value expansion that we develop, superposes the ``deterministicaverages'' ``$\frac{1}{2}(\inf +\sup)$'' taken over balls, with the``stochastic averages'' ``$\fint$'', taken over $N$-dimensional ellipsoidswhose aspect ratio depends on $N,\mathbf{p}$ and whose orientations span alldirections while determining $\inf / \sup$. We show that the unique solutions$u_\epsilon$ of the related dynamic programming principle are automaticallycontinuous for continuous boundary data, and coincide with the well-definedgame values. Our game has thus the min-max property: the order of supremizingthe outcomes over strategies of one player and infimizing over strategies oftheir opponent, is immaterial. We further show that domains satisfying theexterior corkscrew condition are game regular in this context, i.e. the family$\{u_\epsilon\}_{\epsilon\to 0}$ converges uniformly to the unique viscositysolution of the Dirichlet problem.

Random Tug of War games for the ${\mathbf p}$-Laplacian: ${\mathbf{1