Resolvent approach to diffusions with discontinuous scale

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 α-harmonic equation $Lf = αf$ for each $α>0$. 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.