Positivity of stable densities

Abstract:

We settle two conjectures of Taylor about the positivity of the densities $p(t,x)$ of a drift-free, nondegenerate, stable process on $d$-dimensional Euclidean space ${R^d}$ starting at the origin. If $0 < \alpha < 1$ and $p(1,0) = 0$, we show that $x:\;p(t,x) > 0$ for some $t > 0$ is an open convex cone with vertex 0 and that $p(t,x) > 0$ for all $t > 0$ for each $x$ in this cone. If $\alpha = 1$ we show that $p(t,x) > 0$ for all $t > 0$ and all $x \in {R^d}$.