  We study the dynamics of a particle in a space that is non-differentiable.
Non-smooth geometrical objects have an inherently probabilistic nature and,
consequently, introduce stochasticity in the motion of a body that lives in
their realm. We use the mathematical concept of fiber bundle to characterize
the multivalued nature of geodesic trajectories going through a point that is
non-differentiable. Then, we generalize our concepts to everywhere non-smooth
structures. The resulting theoretical framework can be considered a
hybridization of the theory of surfaces and the theory of stochastic processes.
We keep the concepts as general as possible, in order to allow for the
introduction of other fundamental processes capable of modeling the fractality
or the fluctuations of any conceivable continuous, but non-differentiable
space.
