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
Abstract
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
Reporting Year: 2021
JUFO rating: 2