Random extremal solutions of differential inclusions

Given a Lipschitz continuous multifunction F on \({\mathbb{R}^{n}}\), we construct a probability measure on the set of all solutions to the Cauchy problem \(\dot{x}\in F(x)\) with x(0) = 0. With probability one, the derivatives of these random solutions take values within the set extF(x) of extreme points for a.e. time t. This provides an alternative approach in the analysis of solutions to differential inclusions with non-convex right hand side.