A survey of max-type recursive distributional equations

In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X=ᵈ\,g((ξᵢ,Xᵢ),i≥ 1). Here (ξi) and g(⋅) are given and the Xi are independent 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 Xi, are the Xi measurable functions of the innovations process (ξi)?