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

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 ∂u∂t=σ22Δu+V(x)u, where σ is a complex parameter such that Reσ2⩾0. This equation coincides with the heat equation when Imσ2=0 and with the Schrödinger equation when σ2=iS where S is a positive number.