2 problems

Convexity conjecture. The Parisi functional $P_{\phi,x}$ is convex on $\mathcal{M}$ if $\phi$ is even and convex. Numerical simulation suggests this convexity, and it would imply u…

Let $\mathcal{M}$ be the collection of all nonincreasing and left-continuous functions from $[0,1]$ to $[0,1]$. For $a\in\mathcal{M}$, let $F_a$ solve … with initial condition … Th…