Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation

Abstract

We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator $$(\partial_t-\Delta)^su(t,x)=f(t,x),\quad\hbox{for}~0<s<1.$$ This nonlocal equation of order $s$ in time and $2s$ in space arises in Nonlinear Elasticity, Semipermeable Membranes, Continuous Time Random Walks and Mathematical Biology. It plays for space-time nonlocal equations like the generalized master equation the same role as the fractional Laplacian for nonlocal in space equations. We obtain a pointwise integro-differential formula for $(\partial_t-\Delta)^su(t,x)$ and parabolic maximum principles. A novel extension problem to characterize this nonlocal equation with a local degenerate parabolic equation is proved. We show parabolic interior and boundary Harnack inequalities, and an Almgrem-type monotonicity formula. H\"older and Schauder estimates for the space-time Poisson problem are deduced using a new characterization of parabolic H\"older spaces. Our methods involve the \textit{parabolic language of semigroups} and the Cauchy Integral Theorem, which are original to define the fractional powers of $\partial_t-\Delta$. Though we mainly focus in the equation $(\partial_t-\Delta)^su=f$, applications of our ideas to variable coefficients, discrete Laplacians and Riemannian manifolds are stressed out.