Iyer, Sarvesh Ravichandran2025-01-142025-01-142024-07122p.http://hdl.handle.net/10263/7487This thesis is under the supervision of Prof. Siva AthreyaRecently, 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.enJump processesHarnack inequalitiesMartingale problemGeometric stable processThesis