is it all fun and games?

upon the death of my thesis problem i devoted myself to being (mathematically) unproductive and avoided entangling alliances of all sorts.

it worked well until this afternoon.

as you may have guessed, i started entertaining crackpot theories, mostly about how to raise the problem from the dead. i'll mention something the moment that i see something rigorous in the works, but don't hold your breath; i won't.

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as for the title, it seems like the p-harmonic folks are spending more time at games, especially tug-of-war!

• a friend of mine told me once about a game-theoretic formulation of mean curvature flow via pde,
  
  
• earlier this year, Peres, Schramm, Sheffield, and Wilson gave a formulation of the infinity-Laplacian in terms of "tug of war,"
  
  
• then earlier this summer, i heard a talk of Jan Malý about a game-theoretic version of the eikonal equation,
  

and now it seems that Peres and Sheffield have gone further and interpolated some of their earlier joint work:


Tug of war with noise: a game theoretic view of the p-Laplacian

Fix a bounded domain Ω in R^d, a continuous function F on the boundary of Ω, and constants ε > 0, p > 1, and q > 1 with p^-1 + q^-1 = 1. For each x in Ω, let u^ε(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 of length at most epsilon 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 on the boundary of Ω, and player I's payoff is F(y).

We show that (for sufficiently regular Ω) as ε tends to zero the functions u^ε converge uniformly to the unique p-harmonic extension of F. Using a modified game (in which ε gets smaller as the game position approaches the boundary), we prove similar statements for general bounded domains Ω and resolutive functions F.