A note on the equivalence of a strongly convex function and its induced contractive differential equation

Maxwell Fitzsimmons, Jun Liu

Abstract: A strongly convex function naturally induces a gradient flow that is contractive. This paper is a short investigation on when the converse to the previous statement holds. That is, given a differential equation that is contractive, does there exist a strongly convex function that induces the differential equation? We show that, if sufficient smoothness of the vector field is assumed, then the contractivity of such a differential equation with a symmetric Jacobian is equivalent to the existence of a strongly convex function which induces the differential equation as its gradient flow.