On the approximation of the solutions of some evolution equations by the expectations of functionals of random walks

S. V. Tsykin

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 21, Tome 431 (2014), pp. 242-252

Résumé

We consider 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 $\operatorname{Re}\sigma^2\geqslant0$. This equation coincides with the heat equation when $\operatorname{Im}\sigma=0$ and with the Schrödinger equation when $\operatorname{Re}\sigma^2=0$.