2014/07/18 by Thomas Duquesne, Duquesne, Thomas, Xan Duhalde +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1407.4913
43 pages
arxiv created 2014/07/18 · arxiv updated 2014/07/21
We consider super processes whose spatial motion is the d-dimensional Brownian motion and whose branching mechanism ψ is critical or subcritical; such processes are called ψ-super Brownian motions. If d > 2\bgamma/(\bgamma - 1), where \bgamma ∈ (1,2] is the lower index of ψ at ∞, then the total range of the ψ-super Brownian motion has an exact packing measure whose gauge function is g(r) = (loglog1/r) / φ-1 ( (1/rloglog 1/r)2), where φ = ψ^′ ∘ ψ -1. More precisely, we show that the occupation measure of the ψ-super Brownian motion is the g-packing measure restricted to its total range, up to a deterministic multiplicative constant only depending on d and ψ. This generalizes the main result of \citeDuq09 that treats the quadratic branching case. For a wide class of ψ, the constant 2\bgamma/(\bgamma - 1) is shown to be equal to the packing dimension of the total range.