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arXiv · 1904.05147

Gradient and Lipschitz estimates for tug-of-war type games

Amal Attouchi·Hannes Luiro·Mikko Parviainen
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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 higher regularity theory of partial differential equations. The proofs are based on cancellation and coupling methods as well as improved version of the cylinder walk argument.

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Amal Attouchi, Hannes Luiro, Mikko Parviainen. 2019-04-10. Gradient and Lipschitz estimates for tug-of-war type games. https://arxiv.org/abs/1904.05147

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