2024/12/02 by Zhizhang Wang, Backhoff, Julio, Xin Zhang +2 · 1 citation
Mathematics · Physics and Astronomy · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Black Holes and Theoretical Physics
paper · pdf · doi:10.48550/arxiv.2412.01995
We consider a competition between d+1 players, and aim to identify the "most exciting game'' of this kind. This is translated, mathematically, into a stochastic optimization problem over martingales that live on the d-dimensional subprobability simplex Δ and terminate on the vertices of Δ (so-called win-martingales), with a cost function related to a scaling limit of Shannon entropies. We uncover a surprising connection between this problem and the seemingly unrelated field of Monge-Ampère equations: If g solves \begincases g(x)=log det((1)/(2)∇2 g(x)), x ∈ Δ,
g(x)=∞, x∈ ∂ Δ, \endcases then the winning-probability of the players in the most exciting game is described by dMs=√\frac2 (∇2 g(Ms))-11-s dBs. To formalize this, a detailed quantitative analysis of the Monge-Ampère equation for g is crucial. This is then leveraged to prove that M is indeed an optimal win-martingale.