A formulation of Wardrop vector equilibrium principle in probabilistic Lebesgue spaces

In this note we first review the concept of a Wardrop equilibrium in the finite dimensional scalar and vector cases, and its relationship with scalar and vector variational inequalities. We then extend the concept of a vector equilibrium to the Lebesgue space of square integrable functions, with respect to a given probability measure, so as to model a vector network equilibrium problem with random data.