Sobolev type estimates for nonlocal equations in nondivergence form

Non­local equations such as integro­-differential equations with jump Lévy processes and fractional time derivatives have received increasing attention in the last two decades. They arise from
models in physics, engineering, and finance that involve long range interactions or "memory" effects. Such equations have been studied by using both probabilistic and analytic methods. 
In this talk, I will present some recent results on Sobolev estimates for nonlocal (fractional) elliptic and parabolic equations in nondivergence form. We considered equations with time fractional derivatives of the Caputo type, or with nonlocal derivatives in the space variables (eg. the fractional Laplacian), or both. This is based on joint work with Doyoon Kim (Korea University), Yanze Liu (Brown), and Pilgyu Jung (Korea University).