A Stochastic Differential Game for the Inhomogeneous ∞-laplace Equation by Rami Atar

A stochastic differential game for the inhomogeneous -Laplace equation

Given a bounded C domain G ⊂ R and functions g ∈ C(∂G,R) and h ∈ C(Ḡ,R \ {0}), let u denote the unique viscosity solution to the equation −2∆∞u = h in G with boundary data g. We provide a representation for u as the value of a two-player zero-sum stochastic differential game. AMS 2000 subject classifications: 91A15, 91A23, 35J70, 49L20

On near optimal trajectories for a gameassociated with the -Laplacian

A two-player stochastic differential game representation has recently been obtained for solutions of the equation −∆∞u = h in a C 2 domain with Dirichlet boundary condition, where h is continuous and takes values in R \ {0}. Under appropriate assumptions, including smoothness of u, the vanishing δ limit law of the state process, when both players play δ-optimally, is identified as a diffusion p...

A di erential game with onstrained dynami sand vis osity solutions of a related HJB equation

A PDE Perspective of The Normalized Infinity Laplacian

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 4∞u = | 5 u|−2 ∑n i,j=1 ∂xiu∂xju∂ xixju, is defined in a...

Modified Laplace decomposition method for fractional Volterra-Fredholm integro-differential equations

This paper successfully applies the Adomian decomposition  and the modified Laplace Adomian decomposition methods to find  the approximate solution of a nonlinear fractional Volterra-Fredholm integro-differential equation. The reliability of the methods and reduction in the size of the computational work give these methods a wider applicability. Also, the behavior of the solution can be formall...