Analytic continuation of stochastic mechanics

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