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January/February 2022 Non-local tug-of-war with noise for the geometric
Marta Lewicka
Adv. Differential Equations 27(1/2): 31-76 (January/February 2022).

Abstract

This paper concerns the fractional ${\mathrm{\bf p}}$-Laplace operator $\Delta_{\mathrm{\bf p}}^s$ in non-divergence form, which has been introduced in [2]. For any ${\mathrm{\bf p}}\in [2,\infty)$ and $s\in (\frac{1}{2},1)$, we first define two families of non-local, non-linear averaging operators, parametrized by $\epsilon$ and defined for all bounded, Borel functions $u:\mathbb R^N\to \mathbb R$. We prove that $\Delta_{\mathrm{\bf p}}^s u(x)$ emerges as the $\epsilon^{2s}$-order coefficient in the expansion of the deviation of each $\epsilon$-average from the value $u(x)$, in the limit of the domain of averaging exhausting an appropriate cone in $\mathbb R^N$ at the rate $\epsilon\to 0$. Second, we consider the $\epsilon$-dynamic programming principles modeled on the first average, and show that their solutions converge uniformly as $\epsilon\to 0$, to viscosity solutions of the homogeneous non-local Dirichlet problem for $\Delta_{\mathrm{\bf p}}^s$, posed in a domain ${\mathcal{D}}$ satisfying the external cone condition and subject to bounded, uniformly continuous data on $\mathbb R^N\setminus {\mathcal{D}}$. Finally, we interpret such $\epsilon$-approximating solutions as values to the non-local Tug-of-War game with noise. In this game, players choose directions while the game position is updated randomly within the infinite cone that aligns with the specified direction, whose aperture angle depends on ${\mathrm{\bf p}}$ and $N$, and whose $\epsilon$-tip has been removed.

Citation

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Marta Lewicka. "Non-local tug-of-war with noise for the geometric." Adv. Differential Equations 27 (1/2) 31 - 76, January/February 2022.

Information

Published: January/February 2022
First available in Project Euclid: 6 January 2022

Subjects:
Primary: 26A33 , 35Q91 , 35R11 , 35R99 , 91A23

Rights: Copyright © 2022 Khayyam Publishing, Inc.

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Vol.27 • No. 1/2 • January/February 2022
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