Abstract: 

We consider divergence form operators L=divAgrad which are symmetric and uniformly elliptic. My masters research (2020-2022) with Pierre Portal has led us to conjecture that the L-mean value sets not only provide a geometric description of L-harmonic functions but also the process X generated by L. In my final presentation, I will illustrate how L-mean value sets are obtained constructively via Luis Caffarelli’s obstacle method. I will then show how the mean value sets are used to prove the Feynman-Kac formula for the classical Dirichlet problem with continuous boundary data on a smooth domain. Finally, I will end with simulations of L-mean value sets and suggest their use in developing random walk approximations for the process X.