Abstract: Researchers in mathematical physics, financial mathematics, and mathematical economics are finding that in many instances models that incorporate jumps are more realistic than those that do not. These models lead to some interesting mathematics.

I'll discuss several types of jump processes and some integral operators that arise. It turns out there are connections with singular integrals and pseudo-differential operators. I'll also talk about Harnack inequalities for functions that are harmonic with respect to these integral operators. Since the operators are non-local, quite different techniques needed to be developed.