Construction of the Local Time and Skorokhod Decomposition

Abstract

In this chapter we further analyze the boundary behavior of the diffusion process ${{{(}}{{\text{X}}_t}{\text{)}}_{t \ge {\text{0}}}}$ constructed in Chapter 4. We keep the notion of Chapter 4, in particular ${{\rm{\Gamma }}_{\rm{2}}}$ denotes an open C ²-smooth boundary part of $\partial \Omega$. We construct the local time at the boundary part ${{\rm{\Gamma }}_2} \cap {\{ \varrho}\,{\rm{> }}\,{\rm{0}}\}$ as an additive functional of the process ${{{(}}{{\text{X}}_t}{\text{)}}_{t \ge {\text{0}}}}$. For this we first need to refine a construction result for additive functionals of Fukushima, Oshima and Takeda ([FOT11]) to our setting, see Theorem 6.1.11. The construction of the local time is based on boundedness properties of the corresponding potential that are proven using our elliptic regularity result of Chapter 3.1, see Theorem 6.2.1.