  We embed Nelson's stochastic quantization in the Schwartz-Meyer second order
geometry framework. The result is a non-perturbative theory of quantum
mechanics on (pseudo)-Riemannian manifolds. Within this approach, we derive
stochastic differential equations for massive spin-0 test particles charged
under scalar potentials, vector potentials and gravity. Furthermore, we derive
the associated Schr\"odinger equation. The resulting equations show that
massive scalar particles must be conformally coupled to gravity in a theory of
quantum gravity. We conclude with a discussion of some prospects of the
stochastic framework.
