A Stochastic Interpretation of the Cauchy Problem Solution for the Equation $\boldsymbol{\partial_tu= (\sigma^2 /2) \Delta u+V(x)u}$ with Complex $\boldsymbol{\sigma}$

#### M.M. Faddeev, I.A. Ibragimov, N.V. Smorodina

2016, v.22, Issue 2, 203-226

ABSTRACT

Using classical probabilistic methods we construct
a probabilistic approximation in $L_2$ of the Cauchy problem solution
for an equation
$\partial u / \partial t= (\sigma^2/2) \,\Delta u+V(x)u,$
where $\mathrm{Re}\,V\leqslant 0$ and $\sigma$ is a complex parameter
with $\mathrm{Re}\,\sigma^2\geqslant 0$.
This equation coincides with the heat equation when $\mathrm{Im}\,\sigma=0$ and with the
Schr\"odinger equation when $\mathrm{Re}\,\sigma^2=0$.

Keywords: limit theorem, Schr\"odinger equation, Feynman measure, random walk, evolution equation