Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle

Han, Jeongmin

doi:10.3934/cpaa.2020114

Cited by 7 publications

(4 citation statements)

References 17 publications

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“…The method has been significantly generalized and applied to parabolic and elliptic equations [39,40,50,64,65]. Coupling methods in the discrete setting have also been used to establish Hölder and Lipschitz regularity in nonlinear potential theory, and in particular, for the p-Laplacian via the connection to stochastic two player tug-of-war games [2,3,34,44,47]. There are also recent application to Hölder regularity for the Robin problem [41,42].…”

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confidence: 99%

Lipschitz regularity of graph Laplacians on random data clouds

Calder¹,

Trillos²,

Lewicka³

2020

Preprint

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In this paper we study Lipschitz regularity of elliptic PDEs on geometric graphs, constructed from random data points. The data points are sampled from a distribution supported on a smooth manifold. The family of equations that we study arises in data analysis in the context of graph-based learning and contains, as important examples, the equations satisfied by graph Laplacian eigenvectors. In particular, we prove high probability interior and global Lipschitz estimates for solutions of graph Poisson equations. Our results can be used to show that graph Laplacian eigenvectors are, with high probability, essentially Lipschitz regular with constants depending explicitly on their corresponding eigenvalues. Our analysis relies on a probabilistic coupling argument of suitable random walks at the continuum level, and an interpolation method for extending functions on random point clouds to the continuum manifold. As a byproduct of our general regularity results, we obtain high probability L ∞ and approximate C 0,1 convergence rates for the convergence of graph Laplacian eigenvectors towards eigenfunctions of the corresponding weighted Laplace-Beltrami operators. The convergence rates we obtain scale like the L 2 -convergence rates established in [13].

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confidence: 99%

Lipschitz regularity of graph Laplacians on random data clouds

Calder¹,

Trillos²,

Lewicka³

2020

Preprint

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“…The motivation to study parabolic equations involving the normalized p-Laplacian stems partially from connections to time-dependent tug-of-war games [29,31,20] and image processing [18]. For regularity results concerning this equation, we refer to [6,3,8,21,19].…”

Section: Introductionmentioning

confidence: 99%

Gradient regularity for a singular parabolic equation in non-divergence form

Attouchi¹,

Ruosteenoja²

2019

Preprint

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“…The coupling approach to the regularity of different variants of tug-of-war with noise was first developed in [LP18] and applied in [PR16,AHP17,ALPR,Han]. As it turned out, for the continuous time diffusion processes and the Laplacian, the coupling method was utilized in connection to the regularity already at the beginning of 90's by Cranston [Cra91], utilizing the tools developed in [LR86].…”

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confidence: 99%

Asymptotic Hölder regularity for the ellipsoid process

Arroyo

Parviainen

2020

ESAIM: COCV

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We obtain an asymptotic Hölder estimate for functions satisfying a dynamic programming principle arising from a so-called ellipsoid process. By the ellipsoid process we mean a generalization of the random walk where the next step in the process is taken inside a given space dependent ellipsoid. This stochastic process is related to elliptic equations in non-divergence form with bounded and measurable coefficients, and the regularity estimate is stable as the step size of the process converges to zero. The proof, which requires certain control on the distortion and the measure of the ellipsoids but not continuity assumption, is based on the coupling method.

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Local Lipschitz regularity for functions satisfying a time-dependent dynamic programming principle

Cited by 7 publications

References 17 publications

Lipschitz regularity of graph Laplacians on random data clouds

Lipschitz regularity of graph Laplacians on random data clouds

Gradient regularity for a singular parabolic equation in non-divergence form

Asymptotic Hölder regularity for the ellipsoid process

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