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 editorsAttouchi, Amal; Luiro, Hannes; Parviainen, Mikko

Journal or seriesSIAM Journal on Mathematical Analysis



Publication year2021


Issue number2

Pages range1295-1319

PublisherSociety for Industrial and Applied Mathematics

Publication countryUnited States

Publication languageEnglish


Publication open accessNot 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.

Keywordsgame theorystochastic processespartial differential equations

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

Contributing organizations

Ministry reportingYes

Reporting Year2021

JUFO rating2

Last updated on 2024-03-04 at 20:16