Georgia Institute of Technology College of Sciences

We consider a class of dynamical systems with random switching with the following specifics: Given a finite collection of smooth vector fields on a finite-dimensional smooth manifold, we fix an initial vector field and a starting point on the manifold. We follow the solution trajectory to the corresponding initial-value problem for a random, exponentially distributed time until we switch to a new vector field chosen at random from the given collection. Again, we follow the trajectory induced by the new vector field for an exponential time until we make another switch. This procedure is iterated. The resulting two-component process whose first component records the position on the manifold, and whose second component records the driving vector field at any given time, is a Markov process. We identify sufficient conditions for its invariant measure to be unique and absolutely continuous. In the one-dimensional case, we show that the invariant densities are smooth away from critical points of the vector fields and derive asymptotics for the invariant densities at critical points.