Elliptic Harnack inequality, conformal walk dimension and Martingale problem for geometric stable processes - Iyer, Sarvesh Ravichandran (Bengaluru: Indian Statistical Institute, 2024)
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.
Elliptic Harnack inequality and geometric stable processes -- The conformal walk dimension of geometric stable processes -- The Martingale Problem -- Future work
Thesis (Ph.D.) - Indian Statistical Institute, 2024
Includes bibliography
Guided by Prof. Siva Athreya
Keywords: Jump Processes; Harnack Inequalities; Martingale Problem; Geometric Stable Process