The paper puts forward a new method of construction of a probabilistic representation of solutions to initial-boundary value problems for a number of evolution equations (in particular, for the Schrödinger equation) in a bounded subdomain of ℝ² with smooth boundary. Our method is based on the construction of a special extension of the initial function from the domain to the entire plane. For problems with Neumann boundary condition, this method produces a new approach to the construction of a Wiener process “reflected from the boundary,” which was first introduced by A. V. Skorokhod.