Resolvent approach to diffusions with discontinuous scale

TL;DR

This paper extends the resolvent approach to diffusions with discontinuous scale functions, showing that the associated reproducing kernel can generate a Markov process similar to classical quasidiffusions.

Contribution

It generalizes the resolvent method to handle diffusions with non-decreasing, possibly discontinuous scale functions, linking solutions of the {lpha}-harmonic equation to Markov processes.

Findings

Reproducing kernel generated by two positive monotone solutions.

Induces a Markov process consistent with previous semigroup and Dirichlet form approaches.

Extends diffusion theory to discontinuous scale functions.

Abstract

Quasidiffusion is an extension of regular diffusion which can be described as a Feller process on $\mathbb{R}$ with infinitesimal operator $L=\frac{1}{2}D_mD_s$. Here, $s(x) = x$ and $m$ refers to the (not necessarily fully supported) speed measure. In this paper, we will examine an analogous operator where the scale function s is general and only assumed to be non-decreasing. We find that, like regular diffusion or quasidiffusion, the reproducing kernel can still be generated by two specific positive monotone solutions of the {\alpha}-harmonic equation $Lf = \alpha f$ for each $\alpha>0$. Our main result shows that this reproducing kernel is able to induce a Markov process, which is identical to that obtained in [25] using a semigroup approach or in [17] through Dirichlet forms. Further investigations into the properties of this process will be presented.