6533b838fe1ef96bd12a46ad
RESEARCH PRODUCT
Gradient and Lipschitz Estimates for Tug-of-War Type Games
Mikko ParviainenHannes LuiroAmal Attouchisubject
osittaisdifferentiaaliyhtälöt91A15 35B65 35J92gradient regularityApplied MathematicsTug of warMathematical analysisstochastic two player zero-sum gameType (model theory)Lipschitz continuityComputational MathematicsMathematics - Analysis of PDEsLipschitz estimateBellman equationtug-of-war with noiseFOS: MathematicsUniform boundednesspeliteoriaAlmost everywherep-LaplaceValue (mathematics)AnalysisAnalysis of PDEs (math.AP)Mathematicsstokastiset prosessitdescription
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
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2021-01-01 |