# Limit theorems for symmetric random walks and probabilistic approximation of the Cauchy problem solution for Schrödinger type evolution equations

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

Research output

2 Citations (Scopus)

### Abstract

In the present paper we discuss a possibility to construct both a probabilistic representation and a probabilistic approximation of the Cauchy problem solution for an equation $\frac{\partial u}{\partial t}=\frac{\sigma^2}{2}\,\Delta u+V(x)u,$ where $\sigma$ is a complex parameter such that $\mathrm{Re}\,\sigma^2\geqslant 0$. This equation coincides with the heat equation when $\mathrm{Im}\,\sigma=0$ and with the Schr\"odinger equation when $\sigma^2=iS$ where $S$ is a positive number.
Original language English 4455 --- 4472 Stochastic Processes and their Applications 125 12 https://doi.org/doi:10.1016/j.spa.2015.07.005 Published - 2015 Yes