Elliptic Harnack Inequality, Conformal Walk Dimension and Martingale Problem for Geometric Stable Processes

Abstract

Recently, Murugan and Kajino introduced the notion of conformal walk dimension as a bridge between parabolic and elliptic Harnack inequalities. They showed that a symmetric diffusion process satisfies the elliptic Harnack inequality if and only if its conformal walk dimension equals 2, raising the question of whether a similar characterization holds for jump processes. Using the geometric stable process, we provide a counterexample: it satisfies the elliptic Harnack inequality but has infinite conformal walk dimension. Additionally, we establish the existence and uniqueness of solutions to the martingale problem associated with geometric stable processes.