The paper is devoted to some problems associated with a probabilistic representation and probabilistic approximation of the Cauchy problem solution for the family of equations ∂u∂t=σ22Δu with a complex parameter σ such that Re σ2 ≥ 0. The above family includes as a particular case both the heat equation (Im σ = 0) and the Schrödinger equation (Re σ2 = 0).