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46
On the definition and properties of p-harmonious 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) =
A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs
, 2009
"... We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in [10], and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. Our first main result p ..."
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Cited by 22 (2 self)
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We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in [10], and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. Our first main result provides the convergence of the discrete-time approximation and derives a bound on the discretization error in terms of the time step. An explicit implementable scheme requires to approximate the conditional expectation operators involved in the discretization. This induces a further Monte Carlo error. Our second main result is to prove the convergence of the latter approximation scheme, and to derive an upper bound on the approximation error. Numerical experiments are performed for the approximation of the solution of the mean curvature flow equation in dimensions two and three, and for two and five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman equations arising in the theory of portfolio optimization in financial mathematics.
AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR A CLASS OF NONLINEAR PARABOLIC EQUATIONS RELATED TO Tug-of-war Games
"... We characterize solutions to the homogeneous parabolic p-Laplace equation ut = |∇u|2−p ∆pu = (p − 2)∆∞u + ∆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 t ..."
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Cited by 18 (9 self)
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We characterize solutions to the homogeneous parabolic p-Laplace equation ut = |∇u|2−p ∆pu = (p − 2)∆∞u + ∆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 game approximate a solution to the PDE above when the parameter that controls the size of the possible steps goes to zero.
A MIXED PROBLEM FOR THE INFINITY LAPLACIAN VIA Tug-of-war Games
, 2009
"... In this paper we prove that a function u ∈ C(Ω) is the continuous value of the Tug-of-War game described in [19] if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary conditions ..."
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Cited by 16 (8 self)
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In this paper we prove that a function u ∈ C(Ω) is the continuous value of the Tug-of-War game described in [19] if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary conditions
Level set approach for fractional mean curvature flows
, 2009
"... Abstract. This paper is concerned with the study of a geometric flow whose law involves a singular integral operator. This operator is used to define a non-local mean curvature of a set. Moreover the associated flow appears in two important applications: dislocation dynamics and phasefield theory fo ..."
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Cited by 13 (2 self)
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Abstract. This paper is concerned with the study of a geometric flow whose law involves a singular integral operator. This operator is used to define a non-local mean curvature of a set. Moreover the associated flow appears in two important applications: dislocation dynamics and phasefield theory for fractional reaction-diffusion equations. It is defined by using the level set method. The main results of this paper are: on one hand, the proper level set formulation of the geometric flow; on the other hand, stability and comparison results for the geometric equation associated with the flow.
A PDE perspective of the normalized infinity Laplacian, preprint
"... The inhomogeneous normalized infinity Laplace equation was derived from the tug-of-war game in [PSSW] with the positive right-hand-side as a running payoff. The existence, uniqueness and comparison with polar quadratic functions were proved in [PSSW] by the game theory. In this paper, the normalized ..."
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Cited by 12 (2 self)
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The inhomogeneous normalized infinity Laplace equation was derived from the tug-of-war game in [PSSW] with the positive right-hand-side as a running payoff. The existence, uniqueness and comparison with polar quadratic functions were proved in [PSSW] by the game theory. In this paper, the normalized infinity Laplacian, formally written as △N ∞u = | ▽ u | −2 �n i,j=1 ∂xiu∂xju∂2xixju, is defined in a canonical way with the second derivatives in the local maximum and minimum directions, and understood analytically by a dichotomy. A comparison with polar quadratic polynomials property, the counterpart of the comparison with cones property, is proved to characterize the viscosity solutions of the inhomogeneous normalized infinity Laplace equation. We also prove that there is exactly one viscosity solution of the boundary value problem for the infinity Laplace equation △ N ∞u = f with positivef in a bounded open subset of R n. The stability of the inhomogeneous infinity Laplace equation △ N ∞u = f with strictly positive f and of the homogeneous equation △ N ∞u = 0 by small perturbation of the right-hand-side and the boundary data is established in the last part of the work. Our PDE method approach is quite different from those in [PSSW]. The stability result in this paper appears to be new.
A stochastic differential game for the inhomogeneous
, 2008
"... ∞-Laplace equation ..."
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On billiards for a game interpretation of the Neumann problems for curvature flows
, 2008
"... This paper constructs a family of discrete games, whose value func-tions converge to the unique viscosity solution of the Neumann bound-ary problem of the curve shortening flow equation. To derive the boundary condition, a billiard semiflow is introduced and its basic properties near the boundary ar ..."
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Cited by 6 (3 self)
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This paper constructs a family of discrete games, whose value func-tions converge to the unique viscosity solution of the Neumann bound-ary problem of the curve shortening flow equation. To derive the boundary condition, a billiard semiflow is introduced and its basic properties near the boundary are studied for convex and more gen-eral domains. It turns out that Neumann boundary problems of mean curvature flow equations can be intimately connected with purely de-
Tug of war with noise: a game theoretic view of the p-Laplacian
- Duke Math. J
"... Fix a bounded domain Ω ⊂ R d, a continuous function F: ∂Ω → R, and constants ǫ> 0 and 1 < p,q < ∞ with p −1 + q −1 = 1. For each x ∈ Ω, let u ǫ (x) be the value for player I of the following two-player, zero-sum game. The initial game position is x. At each stage, a fair coin is tossed and ..."
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Cited by 5 (0 self)
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Fix a bounded domain Ω ⊂ R d, a continuous function F: ∂Ω → R, and constants ǫ> 0 and 1 < p,q < ∞ with p −1 + q −1 = 1. For each x ∈ Ω, let u ǫ (x) be the value for player I of the following two-player, zero-sum game. The initial game position is x. At each stage, a fair coin is tossed and the player who wins the toss chooses a vector v ∈ B(0,ǫ) to add to the game position, after which a random “noise vector ” with mean zero and variance q p |v|2 in each orthogonal direction is also added. The game ends when the game position reaches some y ∈ ∂Ω, and player I’s payoff is F(y). We show that (for sufficiently regular Ω) as ǫ tends to zero the functions u ǫ converge uniformly to the unique p-harmonic extension of F. Using a modified game (in which ǫ gets smaller as the game position approaches ∂Ω), we prove similar statements for general bounded domains Ω and resolutive functions F. These games and their variants interpolate between the tug of war games studied by Peres, Schramm, Sheffield, and Wilson (p = ∞) and the motion-by-curvature games introduced by Spencer and studied by Kohn and Serfaty (p = 1). They generalize the relationship between Brownian motion and the ordinary Laplacian and yield new results about p-capacity and p-harmonic measure. 1
A remark on the discrete deterministic game approach for curvature flow equations
- Hokkaido University
"... Abstract. This paper constructs a family of discrete two-person games, whose values converge to the unique viscosity solution of a general curva-ture flow equation in dimension two. We summarize all of the techniques needed for such second-order games. We introduce barrier games, which can be consid ..."
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Cited by 5 (3 self)
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Abstract. This paper constructs a family of discrete two-person games, whose values converge to the unique viscosity solution of a general curva-ture flow equation in dimension two. We summarize all of the techniques needed for such second-order games. We introduce barrier games, which can be considered as a combination of the classical barrier argument and game perspectives.
