On Some Conditions for Diffusion Processes to Stay on the Boundary of a Domain

In population genetics, it is frequently done to approximate discrete models by diffusion models. From a mathematical point of view, this diffusion approximation problem is usually treated as the problem of convergence of discrete models (suitably normalized and interpolated) to a diffusion process on a bounded closed domain. In many cases it is easy to get from the discrete models an explicit form of the limiting differential operator. But, in order to prove the convergence of the stochastic processes, it is necessary to prove the uniqueness of the diffusion process associated with the differential operator. This uniqueness problem is often a hard mathematical problem due to degeneracy of the operator on the boundary. Sometimes it is helpful to prove certain regularity properties for diffusion processes (not assumed to be unique) associated with the given operator. Especially in the problem of convergence of discrete models without mutation and migration, it is important to show that sample paths of the diffusion processes cannot enter the interior from the boundary.In this paper we consider this property for diffusion processes on a general bounded closed domain whose behavior in the interior is determined by a given diffusion operator. As an application of our results, we get this property for the diffusion process corresponding to the discrete models without mutation and migration. Consequently its uniqueness follows easily.