A Probabilistic Approximation of the Evolution Operator exp (t (S∇, ∇)) with a Complex Matrix S

Abstract

We consider some problems concerning a probabilistic interpretation of the Cauchy problem solution for the equation $$ \frac{\partial u}{\partial t}=\frac{1}{2}\left(S\nabla, \kern0.33em \nabla \right)u, $$ where S is a symmetric complex matrix such that Re S ≥ 0.