Fractional diffusion in the full space: decay and regularity

We consider fractional partial differential equations posed on the full space $$\mathbb {R}^d$$ R d . Using the well-known Caffarelli-Silvestre extension to $$\mathbb {R}^d \times \mathbb {R}^+$$ R d × R + as equivalent definition, we derive existence and uniqueness of weak solutions. We show that solutions to a truncated extension problem on $$\mathbb {R}^d \times (0,\mathcal {Y})$$ R d × ( 0 , Y ) converge to the solution of the original problem as $$\mathcal {Y}\rightarrow \infty $$ Y → ∞ . 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.