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We lay the theoretical and mathematical foundations of the square root ofBrowniam motion and we prove the existence of such a process. In doing so, weconsider Brownian motion on quantized noncommutative Riemannian manifolds andshow how a set of stochastic processes on sets of complex numbers can bedevised. This class of stochastic processes are shown to yield at the outset aChapman-Kolmogorov equation with a complex diffusion coefficient that can bestraightforwardly reduced to the Schr\"odinger equation. The existence of theseprocesses has been recently shown numerically. In this work we provide ananalogous support for the existence of the Chapman-Kolmogorov-Schr\"odingerequation for them, performing a Monte Carlo study. It is numerically seen as aWick rotation can turn the heat kernel into the Schr\"odinger one, mapping suchkernels through the corresponding stochastic processes. In this way, weintroduce a new kind of improper complex stochastic process. This permits areformulation of quantum mechanics using purely geometrical concepts that arestrongly linked to stochastic processes. Applications to economics are alsoentailed.

Quantized Noncommutative Riemann Manifolds and Stochastic Processes: The theoretical foundations of the square root of Brownian motion