More specifically, I am dealing with probabilistic/analytic inequalities that deal with specific quantities of an n-dimensional region with finite volume, namely the first exit time distribution for an α-stable symmetric process, 0<α≤2. In this case, the class of regions I consider here are bounded Lipschitz domains. I then see how I can apply symmetrization techniques (in particular, Steiner symmetrization) to increase the first exit time distribution and see what type of convergence results I obtain in extending to a countable sequence of such symmetrizations. The idea is to use this as a means of proving the Polya-Szegö conjecture in greater generality beyond the case of the first exit time of Brownian motion (when α=2).

Though this description does seem relatively abstract, there is a way to visualize it as a real-world problem, which can be seen here ("Can you hear the shape of a drum?"). You can also see some models of circular membranes with fixed boundaries here to visualize the various types of modes/frequencies of a drum in the case of a 2-D circle.

To see what happens more precisely regarding the mathematics behind these details, you can view a survey paper I've been working on here (there may be typos; just FYI). I also gave a talk at the Purdue Probability Seminar in the Spring 2023 semester with slide presentation, so you may also use this to see what I've been working on.