A probabilistic approximation of the evolution operator $\exp(t(S\nabla,\nabla))$ with a complex matrix $S$

We consider some problems concerning a probabilistic interpretation of the Cauchy problem solution for the equation $\frac{\partial u}{\partial t}=\frac12(S\nabla,\nabla)u$, where $S$ is a symmetric complex matrix such that $\operatorname{Re}S\ge0$.