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Exact packing measure of the range of ψ-Super Brownian motions

Abstract : 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>2γγ/(γγ−1), where γγ∈(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/rloglog1/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 Duquesne (Ann Probab 37(6):2431–2458, 2009) that treats the quadratic branching case. For a wide class of ψ, the constant 2γγ/(γγ−1) is shown to be equal to the packing dimension of the total range
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Submitted on : Monday, December 14, 2015 - 12:35:26 PM
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Xan Duhalde, Thomas Duquesne. Exact packing measure of the range of ψ-Super Brownian motions. Probability Theory and Related Fields, Springer Verlag, 2015, pp.1-52. ⟨10.1007/s00440-015-0680-2⟩. ⟨hal-01242935⟩



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