  We study a (relativistic) Wiener process on a complexified
(pseudo-)Riemannian manifold. Using Nelson's stochastic quantization procedure,
we derive three equivalent descriptions for this problem. If the process has a
purely real quadratic variation, we obtain the one-sided Wiener process that is
encountered in the theory of Brownian motion. In this case, the result
coincides with the Feyman-Kac formula. On the other hand, for a purely
imaginary quadratic variation, we obtain the two-sided Wiener process that is
encountered in stochastic mechanics, which provides a stochastic description of
a quantum particle on a curved spacetime.
