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On the definition and properties of pharmonious functions
, 2009
"... We consider functions that satisfy the identity uε(x) = ..."
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Cited by 24 (11 self)
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We consider functions that satisfy the identity uε(x) =
AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR A CLASS OF NONLINEAR PARABOLIC EQUATIONS RELATED TO Tugofwar Games
"... We characterize solutions to the homogeneous parabolic pLaplace equation ut = ∇u2−p ∆pu = (p − 2)∆∞u + ∆u in terms of an asymptotic mean value property. The results are connected with the analysis of tugofwar games with noise in which the number of rounds is bounded. The value functions for t ..."
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Cited by 18 (9 self)
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We characterize solutions to the homogeneous parabolic pLaplace equation ut = ∇u2−p ∆pu = (p − 2)∆∞u + ∆u in terms of an asymptotic mean value property. The results are connected with the analysis of tugofwar games with noise in which the number of rounds is bounded. The value functions for these game approximate a solution to the PDE above when the parameter that controls the size of the possible steps goes to zero.
A new proof of the equivalence of weak and viscosity solutions for the pLaplace equation. —Manuscript 2011
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Harnack’s inequality for pharmonic functions via stochastic games
 Comm. Partial Differential Equations
, 1985
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Asymptotic statistical characterizations of pharmonic functions of two variables
 Rocky Mountain J. Math
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Finite Difference Methods for the infinity Laplace and pLaplace equations
"... Abstract. We build convergent discretizations and semiimplicit solvers for the Infinity Laplacian and the game theoretical pLaplacian. The discretizations simplify and generalize earlier ones. We prove convergence of the solution of the Wide Stencil finite difference schemes to the unique viscosi ..."
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Abstract. We build convergent discretizations and semiimplicit solvers for the Infinity Laplacian and the game theoretical pLaplacian. The discretizations simplify and generalize earlier ones. We prove convergence of the solution of the Wide Stencil finite difference schemes to the unique viscosity solution of the underlying equation. We build a semiimplicit solver, which solves the Laplace equation as each step. It is fast in the sense that the number of iterations is independent of the problem size. This is an improvement over previous explicit solvers, which are slow due to the CFLcondition. 1.
A GameTree approach to discrete infinity Laplacian with running costs
, 2013
"... We give a selfcontained and elementary proof for boundedness, existence, and uniqueness of solutions to dynamic programming principles (DPP) for biased tugofwar games with running costs. The domain we work in is very general, and as a special case contains metric spaces. Technically, we introduc ..."
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We give a selfcontained and elementary proof for boundedness, existence, and uniqueness of solutions to dynamic programming principles (DPP) for biased tugofwar games with running costs. The domain we work in is very general, and as a special case contains metric spaces. Technically, we introduce gametrees and show that a discretized flow converges uniformly, from which we obtain not only the existence, but also the uniqueness. Our arguments are entirely deterministic, and also do not rely on (semi)continuity in any way; in particular, we do not need to mollify the DPP at the boundary for wellposedness.