My research currently lies in the field of probabilistic potential theory. I am primarily interested in processes associated to integro-differential operators with kernels possessing critically low singularities, and in questions such as the presence of Harnack inequalities, continuity of harmonic functions and well-posedness of martingale problems.
The processes considered in my research include a large class of subordinate Brownian motions, in particular the geometric stable processes. Probabilistic potential theory lies in the intersection of many broad fields such as partial differential equations and stochastic processes. For example, any Feller process possesses an infinitesimal generator which gives rise to a class of Dirichlet problems. Solutions to these Dirichlet problems can be expressed and studied using functionals of the process, and probabilistic estimates may be used to study their regularity. Similarly, there are many connections between potential theory and stochastic processes. Thus, the field is filled with numerous problems and utilizes techniques from various areas.
The following papers are the result of the projects pursued throughout the duration of my PhD.
1. Elliptic Harnack Inequality and Conformal Walk Dimension for the Geometric Stable ProcessÂ
Joint work with Prof. Siva Athreya and Prof. Mathav Murugan. In preparation.
2. Martingale Problem for a Class of Geometric Stable-like Processes. In preparation.