It is shown that the Hausdorff dimension of an invariant measure generated by a Poisson driven stochastic differential equation is greater than or equal to 1.

We consider the fractional Laplacian $-{(-\Delta )}^{\alpha /2}$ on an open subset in ${ℝ}^d$ with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such a Dirichlet fractional Laplacian in $C^{1,1}$ open sets. This heat kernel is also the transition density of a rotationally symmetric $\alpha$-stable process killed upon leaving a $C^{1,1}$ open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

We give sharp estimates for the transition density of the isotropic stable Lévy process killed when leaving a right circular cone.