A continuous time tug-of-war game for parabolic p(x,t)-Laplace-type equations

Heino, Joonas

doi:10.1142/s0219199718500475

Cited by 3 publications

(3 citation statements)

References 18 publications

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“…In the tug-of-war game, when one takes into account the number of plays that the players play and considering sets of possible movements that may depend on space and time, the continuum value function will be the solution of the parabolic equation involved infinity Laplacian. See the details for example [22,32] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

A weighted eigenvalue problem of the biased infinity Laplacian ^*

Liu

Yang

2021

Nonlinearity

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We study a weighted eigenvalue problem of the β-biased infinity Laplacian operator arising from the β-biased tug-of-war. We characterize the principal eigenvalue by the comparison principle and show that β-biased infinity Laplacian operator possesses two principal eigenvalues, corresponding to a positive and a negative principal eigenfunction. When a parameter is less than the principal eigenvalue, certain existence and uniqueness results of the inhomogeneous equations related to this problem are established. As an application, we obtain the decay estimates for viscosity solutions of the parabolic problem associated to the β-biased infinity Laplacian. In the process, we also establish the Lipschitz regularity and Harnack inequality by barrier method.

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Section: Introductionmentioning

confidence: 99%

A weighted eigenvalue problem of the biased infinity Laplacian ^*

Liu

Yang

2021

Nonlinearity

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“…As interpreted in [15,29], parabolic equations of the type considered in (1.1) arise naturally from a two-player zero-sum stochastic differential game (SDG) with probabilities depending on space and time. It is defined in terms of an n-dimensional state process, and is driven by a 2n-dimensional Brownian motion for n ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, Parviainen-Ruosteenoja [29] proved the Hölder and Harnack estimates for a more general game that was called p(x, t)-game without using the PDE techniques and showed that the value functions of the game converge to the unique viscosity solution of the Dirichlet problem to the normalized p(x, t)-parabolic equation (n + p(x, t))u t (x, t) = ∆ N p(x,t) u(x, t). In addition, Heino [15] formulated a stochastic differential game in continuous time and obtained that the viscosity solution to a terminal value problem involving the parabolic normalized p(x, t)-Laplace operator is unique under suitable assumptions. However, whether or not the spatial gradient ∇u of (1.1) is Hölder continuous was still unknown.…”

Section: Introductionmentioning

confidence: 99%