Rough dynamics on smooth manifolds

We define a notion of irregular path with finite p-th variation on a manifold, for arbitrarily large integer p, and develop an  intrinsic calculus for functionals of such paths.
Our construction   may be viewed as a higher-order analogue of L. Schwartz's "second order differential geometry".
But, unlike the latter, our framework is purely geometric, without any probabilistic ingredient, and only relies on the existence of a connection on the manifold. 

-- A Probability and Stochastic Analysis Seminar --