The study of partial differential equations with random data forms the basis of theories of turbulence and phase transitions. However, there are very few examples, where we have any real understanding of such ensembles.

This talk is a description of the rich structure of one such problem: the analysis of perhaps a basic class of nonlinear PDE with random data (scalar conservation laws and Hamilton- Jacobi equations). I will rephrase our results on kinetic theory and PDE in terms of path transformations in order to connect with topics of current interest to probabilists and motivate a conjecture on path transformations that commute in law.