Small-space and large-time estimates and asymptotic expansion of the distribution function and (the derivatives of) the density function of hitting times of points for symmetric Lévy processes are studied. The Lévy measure is assumed to have completely monotone density function, and a scaling-type condition inf ξΨ ′′ (ξ)/Ψ ′ (ξ) > 0 is imposed on the Lévy-Khintchine exponent Ψ. Proofs are based on generalised eigenfunction expansion for processes killed upon hitting the origin. R

We prove the existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular (possibly disconnected) domains of harmonicity, in the context of general metric measure spaces. As a corollary, we prove the uniqueness of the Martin kernel at each boundary point, that is, we identify the Martin boundary with the topological boundary. We also prove a Martin representation theorem for harmonic functions. Examples covered by our results include: strictly stable Lévy processes in R d with positive continuous density of the Lévy measure; stable-like processes in R d and in domains; and stable-like subordinate diffusions in metric measure spaces.

We prove existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular domains, in the context of general metric measure spaces. As a corollary, we prove uniqueness of the Martin kernel at each boundary point, that is, we identify the Martin boundary with the topological boundary. We also prove a Martin representation theorem for harmonic functions. Examples covered by our results include: strictly stable Lévy processes in R d with positive continuous density of the Lévy measure; stable-like processes in R d and in domains; and stable-like subordinate diffusions in metric measure spaces.

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