Harnack inequalities and Hölder estimates for master equations

Abstract

We study master equations of the form $$({\partial }_t+L{)}^{s}u=fin\mathbb{R}\times \Omega$$ where $L$ is a divergence form elliptic operator and $\Omega \subseteq {\mathbb{R}}^n$. These are nonlocal equations of order $2s$ in space and $s$ in time that take into account the values of $u$ everywhere in $\Omega$ and for past times. We show parabolic interior and boundary Harnack inequalities and local parabolic Hölder continuity of solutions. To this end, we prove a characterization of fractional powers of parabolic operators ${\partial }_t+L$ with a degenerate parabolic extension problem.

28 pages. To appear in SIAM Journal of Mathematical Analysis