Uniform measure density condition and game regularity for tug-of-war games

Heino, Joonas

doi:10.3150/16-bej882

Cited by 2 publications

(2 citation statements)

References 17 publications

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“…We provide an existence proof using an approximation based on game theory (this approach will be very useful since it allows us to gain some intuition that will be used when dealing with the asymptotic behaviour of the solutions). For references concerning games (tug-of-war games) and fully nonlinear PDEs we refer to [2,8,13,14,16,17,[19][20][21][24][25][26] and to [10,18] for parabolic versions. Here we propose a parabolic version of the game introduced in [6] in order to show existence of a viscosity solution to (1.1).…”

Section: Introductionmentioning

confidence: 99%

The evolution problem associated with eigenvalues of the Hessian

Blanc

Esteve

Rossi

2020

J. London Math. Soc.

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In this paper, we study the evolution problem ⎧ ⎪ ⎨ ⎪ ⎩ ut(x, t) − λj(D 2 u(x, t)) = 0, in Ω × (0, +∞), u(x, t) = g(x, t), on ∂Ω × (0, +∞), u(x, 0) = u0(x), in Ω, where Ω is a bounded domain in R N (which verifies a suitable geometric condition on its boundary) and λj(D 2 u) stands for the jth eigenvalue of the Hessian matrix D 2 u. We assume that u0 and g are continuous functions with the compatibility condition u0(x) = g(x, 0), x ∈ ∂Ω. We show that the (unique) solution to this problem exists in the viscosity sense and can be approximated by the value function of a two-player zero-sum game as the parameter measuring the size of the step that we move in each round of the game goes to zero. In addition, when the boundary datum is independent of time, g(x, t) = g(x), we show that viscosity solutions to this evolution problem stabilize and converge exponentially fast to the unique stationary solution as t → ∞. For j = 1, the limit profile is just the convex envelope inside Ω of the boundary datum g, while for j = N , it is the concave envelope. We obtain this result with two different techniques: with partial differential equations (PDE) tools and with game-theoretical arguments. Moreover, in some special cases (for affine boundary data), we can show that solutions coincide with the stationary solution in finite time (which depends only on Ω and not on the initial condition u0).

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Section: Introductionmentioning

confidence: 99%

The evolution problem associated with eigenvalues of the Hessian

Blanc

Esteve

Rossi

2020

J. London Math. Soc.

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“…We provide an existence proof using an approximation based on game theory (this approach will be very useful since it allows us to gain some intuition that will be used when dealing with the asymptotic behaviour of the solutions). For references concerning games (Tug-of-War games) and fully nonlinear PDEs we refer to [3,9,14,15,17,18,20,21,22,25,26,27] and to [11,19] for parabolic versions. Here we propose a parabolic version of the game introduced in [7] in order to show existence of a viscosity solution to (1.1).…”

Section: Introductionmentioning

confidence: 99%

The evolution problem associated with eigenvalues of the Hessian

Blanc,

Esteve,

Rossi

2019

Preprint

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In this paper we study the evolution problemwhere Ω is a bounded domain in R N (that verifies a suitable geometric condition on its boundary) and λ j (D 2 u) stands for the j−st eigenvalue of the Hessian matrix D 2 u. We assume that u 0 and g are continuous functions with the compatibility condition u 0 (x) = g(x, 0), x ∈ ∂Ω.We show that the (unique) solution to this problem exists in the viscosity sense and can be approximated by the value function of a two-player zero-sum game as the parameter measuring the size of the step that we move in each round of the game goes to zero.In addition, when the boundary datum is independent of time, g(x, t) = g(x), we show that viscosity solutions to this evolution problem stabilize and converge exponentially fast to the unique stationary solution as t → ∞. For j = 1 the limit profile is just the convex envelope inside Ω of the boundary datum g, while for j = N it is the concave envelope. We obtain this result with two different techniques: with PDE tools and and with game theoretical arguments. Moreover, in some special cases (for affine boundary data) we can show that solutions coincide with the stationary solution in finite time (that depends only on Ω and not on the initial condition u 0 ).

show abstract

Uniform measure density condition and game regularity for tug-of-war games

Cited by 2 publications

References 17 publications

The evolution problem associated with eigenvalues of the Hessian

The evolution problem associated with eigenvalues of the Hessian

The evolution problem associated with eigenvalues of the Hessian

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