3. Two-Sided Classification of Isolated Singular Points

Abstract

In this chapter, we investigate the behaviour of a solution of (1) in the two-sided neighbourhood of an isolated singular point. Many properties related to the “two-sided” behaviour follow from the results of Section 2.3. However, there are some properties that involve both the right type and the left type of a point. The corresponding statements are formulated in Section 3.1.

Section 3.2 contains an informal description of the behaviour of a solution for various types of isolated singular points.

The statements formulated in Section 3.1 are proved in Sect 3.3.

The results of Section 3.1 show that the isolated singular points of only 4 types can disturb uniqueness. These points are called here the branch points. Disturbing uniqueness, the branch points give rise to a variety of “bad” solutions. In particular, one can easily construct a non-Markov solution in the neighbourhood of a branch point. This is the topic of Section 3.4.

However, it turns out that all the strong Markov solutions in the neighbourhood of a branch point admit a simple description. It is given in Section 3.5.

Throughout this chapter, we assume that $\sigma(x)\neq0$ for all $x\in\mathbb{R}$.