A limit theorem on convergence of random walk functionals to a solution of the Cauchy problem for the equation $\frac{\partial u}{\partial t}=\frac{\sigma^2}2\,\Delta u$ with complex $\sigma$

The paper is devoted to 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 $\mathrm{Re}\,\sigma^2\geqslant0$. The above family includes as a particular case both the heat equation (when $\mathrm{Im}\,\sigma=0$) and the Schrödinger equation (when $\mathrm{Re}\,\sigma^2=0$).