Markov Semigroups Associated with One-Dimensional Lévy Operators — Regularity and Convergence —

Many diffusion operators appearing in the diffusion approximation to discrete models are degenerate. For example, the diffusion operator ½ x(1-x)(d/dx)2 + [u-(u+v)x](d/dx) (0 < x < 1) arises as a diffusion approximation for the Wright-Fisher model with mutation and migration. To obtain an error estimate for such diffusion approximation, it is useful to show the smoothness of solutions of the diffusion equations. Ethier has obtained important results for the smoothness problem, especially in the one-dimensional case. Motivated by his work, we will study the smoothness problem for certain degenerate integro-differential operators appearing in the theory of Markov processes. These operators include diffusion operators and are called Lévy operators (in this note, we treat the case without boundary). Our result can be used to get some limit theorem for stochastic processes with discontinuous paths. We will discuss the generation of the semigroups by one-dimensional Lévy operators and the differentiability preserving properties of the semigroups. Convergence problems for the semigroups are also treated.