Geodesic
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~$\sigma$
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 20, Tome 420 (2013), pp. 88-102

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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 $\mathrm{Re}\,\sigma^2=0$). 
@article{ZNSL_2013_420_a4,
     author = {I. A. Ibragimov and N. V. Smorodina and M. M. Faddeev},
     title = {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~$\sigma$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {88--102},
     publisher = {mathdoc},
     volume = {420},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_420_a4/}
}
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I. A. Ibragimov; N. V. Smorodina; M. M. Faddeev. 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~$\sigma$. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 20, Tome 420 (2013), pp. 88-102. http://geodesic.mathdoc.fr/item/ZNSL_2013_420_a4/