Analytic diffusion processes: definition, properties, limit theorems

This paper introduces the notion of an analytic diffusion process. Every process of this type is the limit of some sequence of random walks; however, the limit is understood not in the sense of convergence of measures but in the sense of convergence of generalized functions. Using the analytic diffusion processes it is possible to obtain a probabilistic approximation of solutions to Schrödinger evolution equations, whose right-hand side contains the elliptic operator with variable coefficient.