I am currently working on the understanding of the stochastic nature of the Schroedinger equation. This has a notable history dating back to Nelson's works and relative criticisms. But one can take a different path and, starting from a random walk process with probability

\begin{equation} P(k;N) = \binom{N}{k}\left(\frac{1}{2}\right)^N, \end{equation}

one can assume the existence of a "square root" of this process with a complex amplitude

\begin{equation} A(k;N) = \binom{N}{k}^\frac{1}{2}\left(\frac{1}{2}\right)^{N/2}e^{i\phi(k,N)} \end{equation}

such that $P(k;N)=|A(k;N)|^2$ and $\phi(k,N)$ are some phases exactly determined. These are quantum amplitudes. Is this anything making sense? Does any literature exist about?