Elliptic Harnack inequality, conformal walk dimension and Martingale problem for geometric stable processes/ Sarvesh Ravichandran Iyer

Thesis (Ph.D.) - Indian Statistical Institute, 2024

Includes bibliography

Elliptic Harnack inequality and geometric stable processes -- The conformal walk dimension of geometric stable processes -- The Martingale Problem -- Future work

Guided by Prof. Siva Athreya

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.