### 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 |
---|---|

Pages (from-to) | 4455 --- 4472 |

Journal | Stochastic Processes and their Applications |

Volume | 125 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2015 |

Externally published | Yes |

## Fingerprint Dive into the research topics of 'Limit theorems for symmetric random walks and probabilistic approximation of the Cauchy problem solution for Schrödinger type evolution equations'. Together they form a unique fingerprint.

## Cite this

Ibragimov, I. A., Smorodina, N. V., & Faddeev, M. M. (2015). Limit theorems for symmetric random walks and probabilistic approximation of the Cauchy problem solution for Schrödinger type evolution equations.

*Stochastic Processes and their Applications*,*125*(12), 4455 --- 4472. https://doi.org/doi:10.1016/j.spa.2015.07.005