  We unify Brownian motion and quantum mechanics in a single mathematical
framework. In particular, we show that non-relativistic quantum mechanics of a
single spinless particle on a flat space can be described by a Wiener process
that is rotated in the complex plane. We then extend this theory to
relativistic stochastic theories on manifolds using the framework of second
order geometry. As a byproduct, our results suggest that a consistent path
integral based formulation of a quantum theory on a Lorentzian (Riemannian)
manifold requires an Ito deformation of the Poincare (Galilean) symmetry,
arising due to the coupling of the quadratic variation to the affine
connection.
