ИСТИНА

The talk is devoted to the general one-dimensional diffusion. We discuss the definition and characterization of such processes: scale, speed measure, killing measure. The generating operator is considered on an extended space of functions (as compared with a classical approach). We give a local characterization potential functions and excessive functions. For the general one-dimensional diffusion we give a necessary and sufficient conditions that the optimal strategy in the optimal stopping problem has a threshold or an island character.