G-capacity under degenerate case and its application

TL;DR

This paper investigates the G-capacity in degenerate cases of the G-heat equation, establishing viscosity solutions and analyzing the properties of associated indicator functions, with implications for stochastic analysis.

Contribution

It introduces a new type of viscosity solution for the degenerate G-heat equation and characterizes the G-capacity for Borel sets, advancing understanding of stochastic capacities.

Findings

Derived a viscosity solution for degenerate G-heat equations

Calculated G-capacity for Borel sets in this context

Showed indicator functions lack quasi-continuous versions unless constant

Abstract

In this paper, we first find a type of viscosity solution of $G$-heat equation under degenerate case, and then obtain the related $G$-capacity $c(\{B_{T}\in A\})$ for any Borel set $A$. Furthermore, we prove that $I_{A}(B_{T})$ has no quasi-continuous version when it is not a constant function.