Hunt's hypothesis and symmetrization for general 1-dimensional diffusions

TL;DR

This paper investigates the relationship between Hunt's hypothesis and symmetrization in 1-dimensional diffusions, providing conditions under which symmetrization is possible and characterizing associated Dirichlet forms.

Contribution

It offers a new characterization of Hunt's hypothesis involving point classification and establishes criteria for symmetrization of 1D diffusions.

Findings

Symmetrization occurs if and only if Hunt's hypothesis holds and no asymmetric shunt points exist.

Provides a representation of Dirichlet forms for symmetric 1D diffusions.

Characterizes points for the diffusion related to Hunt's hypothesis.

Abstract

In this paper, we will consider the problem that how far from Hunt's hypothesis (H) to symmetrization for a general 1-dimensional diffusion. A characterization of (H) involving the classification of points for this diffusion will be first obtained. Then the main result shows that such a process is symmetrizable, if and only if (H) holds and a certain family of asymmetric shunt points is empty. Furthermore, we will also derive the representation of associated Dirichlet forms of general 1-dimensional diffusions under symmetry.