Fractional diffusion in the full space: decay and regularity

We consider fractional partial differential equations posed on the full space \(\mathbb {R}^d\). Using the well-known Caffarelli-Silvestre extension to \(\mathbb {R}^d \times \mathbb {R}^+\) as equivalent definition, we derive existence and uniqueness of weak solutions. We show that solutions to a truncated extension problem on \(\mathbb {R}^d \times (0,\mathcal {Y})\) converge to the solution of the original problem as \(\mathcal {Y}\rightarrow \infty \). Moreover, we also provide an algebraic rate of decay and derive weighted analytic-type regularity estimates for solutions to the truncated problem. These results pave the way for a rigorous analysis of numerical methods for the full space problem.