On the approximation of the solutions of some evolution equations by the expectations of functionals of random walks

We consider some problems associated with a probabilistic representation and a probabilistic approximation of the Cauchy problem solution for the family of equations $\frac{\partial u}{\partial t}=\frac{\sigma^2}2\Delta u$ with a complex parameter $\sigma$ such that $\operatorname{Re}\sigma^2\geqslant0$. This equation coincides with the heat equation when $\operatorname{Im}\sigma=0$ and with the Schrödinger equation when $\operatorname{Re}\sigma^2=0$.