An Asymptotic Mean Value Characterization for a Class of Nonlinear Parabolic Equations Related to Tug-of-War Games

@article{Manfredi2010AnAM,
  title={An Asymptotic Mean Value Characterization for a Class of Nonlinear Parabolic Equations Related to Tug-of-War Games},
  author={Juan J. Manfredi and Mikko Parviainen and Julio D. Rossi},
  journal={SIAM J. Math. Anal.},
  year={2010},
  volume={42},
  pages={2058-2081}
}
We characterize solutions to the homogeneous parabolic p-Laplace equation $u_{t}=|\nabla u|^{2-p}\Delta_{p}u=(p-2)\Delta_{\infty}u+\Delta u$ in terms of an asymptotic mean value property. The results are connected with the analysis of tug-of-war games with noise in which the number of rounds is bounded. The value functions for these games approximate a solution to the PDE above when the parameter that controls the size of the possible steps goes to zero. 
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References

SHOWING 1-10 OF 28 REFERENCES
On the Equivalence of Viscosity Solutions and Weak Solutions for a Quasi-Linear Equation
TLDR
The main result states that in both the elliptic and the parabolic case, the viscosity supersolutions coincide with the potential-theoretic supersolutions.
The infinity Laplacian, Aronsson’s equation and their generalizations
The infinity Laplace equation Δ ∞ u = 0 arose originally as a sort of Euler-Lagrange equation governing the absolute minimizer for the L ∞ variational problem of minimizing the functional ess-sup U
Tug-of-war and the infinity Laplacian
We consider a class of zero-sum two-player stochastic games called tug-of-war and use them to prove that every bounded real-valued Lipschitz function F on a subset Y of a length space X admits a
An asymptotic mean value characterization for p-harmonic functions
We characterize p-harmonic functions in terms of an asymptotic mean value property. A p-harmonic function u is a viscosity solution to Δ p u = div(|∇u| p-2 ∇u) = 0 with 1 < p ≤ ∞ in a domain Ω if and
Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations
This paper treats degenerate parabolic equations of second order $$u_t + F(\nabla u,\nabla ^2 u) = 0$$ (14.1) related to differential geometry, where ∇ stands for spatial derivatives of u =
On absolutely minimizing lipschitz extensions and PDE $$\Delta_\infty (u) = 0$$
Abstract.We prove the existence of Absolutely Minimizing Lipschitz Extensions by a method which differs from those used by G. Aronsson in general metrically convex compact metric spaces and R. Jensen
Dynamic Programming Principle for tug-of-war games with noise
We consider a two-player zero-sum-game in a bounded open domain Ω described as follows: at a point x  ∈ Ω, Players I and II play an e -step tug-of-war game with probability α , and with probability β
A deterministic‐control‐based approach to fully nonlinear parabolic and elliptic equations
We show that a broad class of fully nonlinear, second‐order parabolic or elliptic PDEs can be realized as the Hamilton‐Jacobi‐Bellman equations of deterministic two‐person games. More precisely:
An Evolution Equation Involving the Normalized p-Laplacian
This paper considers an initial-boundary value problem for the evolution equation associated with the normalized $p$-Laplacian. There exists a unique viscosity solution $u,$ which is globally
ar 2 00 4 On Absolutely Minimizing Lipschitz Extensions and PDE ∆ ∞ ( u ) = 0
We prove the existence of Absolutely Minimizing Lipschitz Extensions by a method which differs from those used by G. Aronsson in general metrically convex compact metric spaces and R. Jensen in
...
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