The natural analogue of Hamilton's least action principle in presence of randomness is the generalized Schrödinger problem where the Lagrangian action is replaced by the relative entropy with respect to some reference path measure. The role of the action minimizing paths between two prescribed endpoints is played by the bridges of the reference path measure and the solutions of Schrödinger problem are mixtures of these bridges. Therefore, the family of all the bridges of the reference measure encodes its whole "Lagrangian dynamics" and searching for a criterion for two path measures to solve the same Schrödinger problem, i.e. to be driven by the same source of randomness and the same force field, amounts to answer the question of our title. The answer is given in the special case of random walks.

This is a joint work with Giovanni Conforti.