A note on generalized flows

The problem of constructing flows associated with first order ordinary differential equations (ODEs) with spatially non-Lipschitz right-hand side is considered. Due to the lack of uniqueness of the solutions, standard flows cannot be defined in this case. For these situations, it is natural to introduce generalized flows which are random fields constructed by assigning a probability measure on the set of maps associated with the solutions of the ODEs. Some properties of the generalized flows are discussed here, in particular in terms of transport, via simple one-dimensional examples. These simple examples display a wide variety of behaviors and indicate that a general theory of generalized flows is likely to be inaccessible because they typically lack desirable properties such as stability with respect to perturbations or Markovianity in time.