Sample path properties of super-Brownian motion including a one-sided modulus of continuity and exact Hausdorff measure function of the range and closed support are obtained. Analytic estimates for the probability of hitting balls lead to upper bounds on the Hausdorff measure of the set of k-multiple points and a sufficient condition for a set to be "polar".

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Journal Probability Theory and Related Fields
Dawson, D.A, Iscoe, I., & Perkins, E.A. (1989). Super-Brownian motion: Path properties and hitting probabilities. Probability Theory and Related Fields, 83(1-2), 135–205. doi:10.1007/BF00333147