Mathematics Colloquia and Seminars

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Tug-of-War is a zero sum, two player game played by moving a token in a domain D in R^d until it hits the boundary of D. At each step Player II pays Player I a certain value determined by the current position of the token. The order of moves is determined by fair coin tosses. In a work of Peres, Schramm, Sheffield and Wilson these games were used to prove the existence and uniqueness of solutions for certain Infinity Laplace equations with Dirichlet boundary conditions. Their method was later generalized to remove certain assumptions, study mixed boundary conditions and some variations of the equation. In this talk we will look at the limiting behavior of Tug-of-War of prescribed horizon and use it to prove existence results for the Infinity Laplace equation with vanishing Neumann boundary conditions. This is a joint work with Yuval Peres, Scott Sheffield and Stephanie Somersille.