A1 Journal article (refereed)
Gradient and Lipschitz Estimates for Tug-of-War Type Games (2021)

Attouchi, A., Luiro, H., & Parviainen, M. (2021). Gradient and Lipschitz Estimates for Tug-of-War Type Games. SIAM Journal on Mathematical Analysis, 53(2), 1295-1319. https://doi.org/10.1137/19M1256816

JYU authors or editors

Publication details

All authors or editors: Attouchi, Amal; Luiro, Hannes; Parviainen, Mikko

Journal or series: SIAM Journal on Mathematical Analysis

ISSN: 0036-1410

eISSN: 1095-7154

Publication year: 2021

Volume: 53

Issue number: 2

Pages range: 1295-1319

Publisher: Society for Industrial and Applied Mathematics

Publication country: United States

Publication language: English

DOI: https://doi.org/10.1137/19M1256816

Publication open access: Not open

Publication channel open access:

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/77975

Web address of parallel published publication (pre-print): https://arxiv.org/abs/1904.05147


We define a random step size tug-of-war game and show that the gradient of a value function exists almost everywhere. We also prove that the gradients of value functions are uniformly bounded and converge weakly to the gradient of the corresponding $p$-harmonic function. Moreover, we establish an improved Lipschitz estimate when boundary values are close to a plane. Such estimates are known to play a key role in the higher regularity theory of partial differential equations. The proofs are based on cancellation and coupling methods as well as an improved version of the cylinder walk argument.

Keywords: game theory; stochastic processes; partial differential equations

Free keywords: gradient regularity; Lipschitz estimate; p-Laplace; stochastic two player zero-sum game; tug-of-war with noise

Contributing organizations

Ministry reporting: Yes

Preliminary JUFO rating: 2

Last updated on 2022-17-06 at 11:04