Non-symmetric stable processes: Dirichlet heat kernel, Martin kernel and Yaglom limit

We study a $d$-dimensional non-symmetric strictly $\alpha$-stable Lévy process $\mathbf{X}$, whose spherical density is bounded and bounded away from the origin. First, we give sharp two-sided estimates on the transition density of $\mathbf{X}$ killed when leaving an arbitrary $\kappa$-fat set. We apply these results to get the existence of the Yaglom limit for arbitrary $\kappa$-fat cone. In the meantime we also obtain the spacial asymptotics of the survival probability at the vertex of the cone expressed by means of the Martin kernel for $\mathrm{\Gamma }$ and its homogeneity exponent. Our results hold for the dual process $\widehat{\mathbf{X}}$, too.