A Survey of Max-Type Recursive Distributional Equations

In certain problems in a variety of applied probability settingss (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form $X = g((\xi_i, X_i), i \geq 1)$, where equality means in distribution. Here $(\xi_i)$ and $g(\cdot)$ are given and the $(X_i)$ are inddependent copies of the unknown distribution $X$. We survey this area, emphasizing examples where the function $g$ is essentially a "maximum" or "minimum" function. We draw attention to the theoretical question of endogeny : in the associated recursive tree process $X_{{\bf i}}$, are the $X_{{\bf i}}$ measurable functions of the innovation process $(\xi_{{\bf i}})$ ?