SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 1295-1319, January 2021.
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.

中文翻译：

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拔河类游戏的梯度和Lipschitz估计

SIAM数学分析期刊，第53卷，第2期，第1295-1319页，2021年1月。
我们定义了一个随机步长拔河游戏，并证明了几乎每个地方都存在着价值函数的梯度。我们还证明了值函数的梯度是均匀有界的，并且微弱地收敛到相应的$ p $-谐波函数的梯度。此外，当边界值接近平面时，我们建立了改进的Lipschitz估计。众所周知，这样的估计在偏微分方程的高正则性理论中起着关键作用。证明基于消除和耦合方法以及圆柱行走参数的改进版本。