A class of random measures and their fractal geometry

Typical random measures Ξ belonging to our class are • occupation measures of stable subordinators with stability index 0 < α < 1, • states of a Dawson-Watanabe superprocesses in R, d ≥ 2, • intersection local times of two Brownian paths in R, d = 2, 3. Very roughly, with some modification in the critical cases d = 2, the following basic common properties of these examples can be identified: • The local hitting properties are related to the local intensity, i.e. for some scaling index α > 0 we have, α P{ΞB (x) > 0} ∼ EΞ(B (x)). • Given that Ξ charges a small ball B, its neighbourhood looks like a translation of the Palm distribution P associated with a stationary version of the process. • Given that Ξ charges two balls with distance of larger order than their size, the behaviour of Ξ inside these balls is (up to constant factors) conditionally independent. • Local self-similarity holds with scaling index α, Ξ(r ·) ≈ rΞ( · ) under P. • There is a finite annular lacunarity index ξ such that P{Ξ(B1 \Br) = 0} ≈ r as r ↓ 0. The indices associated with our examples are the stability index α and ξ = 2α in the case of stable subordinators; α = 2, ξ = 4 for the superprocess example; and in the intersection example α = 2, ξ = 35 12 if d = 2, α = 1, 1 < ξ < 2 unknown if d = 3. The lacunarity index in the planar case of the intersection example goes back to the seminal work of Lawler, Schramm and Werner.