Search results for "Lipschitz estimate"
showing 2 items of 2 documents
Gradient and Lipschitz Estimates for Tug-of-War Type Games
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. peerReviewed
Asymptotic Lipschitz regularity for tug-of-war games with varying probabilities
2018
We prove an asymptotic Lipschitz estimate for value functions of tug-of-war games with varying probabilities defined in $\Omega\subset \mathbb R^n$. The method of the proof is based on a game-theoretic idea to estimate the value of a related game defined in $\Omega\times \Omega$ via couplings.