Department of Mathematics - MATH/IEDA Joint Seminar - On the most uncertain match

One of Aldous’ open problems is to identify the max-entropy win-probability martingale. Namely, given two players of equal strength, such that the win-probability is a martingale diffusion, which of these processes has maximum entropy and hence gives the most uncertain match. We study a terminal-boundary value problem for the nonlinear parabolic PDE 2e_t(t,x)=log(-e_xx(t,x)) derived by Aldous and prove its well-posedness and regularity of its solution by combining PDE analysis and probabilistic tools. We establish key qualitative properties of the solution including concavity, monotonicity, convergence to a steady state for long remaining time and the asymptotic behaviour shortly before the terminal time. This talk is based on the joint-work with Howison, Possamaï and Reisinger.