Exact packing measure of the range of $$\psi $$ ψ -Super Brownian motions

Abstract

We consider super processes whose spatial motion is the d-dimensional Brownian motion and whose branching mechanism $$\psi $$ is critical or subcritical; such processes are called $$\psi $$ -super Brownian motions. If $$d>2\varvec{\gamma }/(\varvec{\gamma }-1)$$ , where $$\varvec{\gamma }\in (1,2]$$ is the lower index of $$\psi $$ at $$\infty $$ , then the total range of the $$\psi $$ -super Brownian motion has an exact packing measure whose gauge function is $$g(r) = (\log \log 1/r) / \varphi ^{-1} ( (1/r\log \log 1/r)^{2})$$ , where $$\varphi = \psi ^\prime \circ \psi ^{-1}$$ . More precisely, we show that the occupation measure of the $$\psi $$ -super Brownian motion is the g-packing measure restricted to its total range, up to a deterministic multiplicative constant only depending on d and $$\psi $$ . 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 $$\psi $$ , the constant $$2\varvec{\gamma }/(\varvec{\gamma }-1)$$ is shown to be equal to the packing dimension of the total range.