Harnack inequalities and H\"older estimates for master equations

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