The Packing Measure of the Range of Super-Brownian Motion

TL;DR

This paper establishes an exact packing measure for the total range of Super-Brownian motion in dimensions five and higher, linking the occupation measure to a specific gauge function with a precise constant.

Contribution

It provides a rigorous proof that the total occupation measure of Super-Brownian motion equals its $g$-packing measure with a dimension-dependent constant in super-critical dimensions.

Findings

The total range has an exact packing measure with respect to a specific gauge function.

The occupation measure equals the $g$-packing measure up to a constant.

The result applies in super-critical dimensions $d \,\geq\, 5$.

Abstract

We prove that the total range of Super-Brownian motion with quadratic branching mechanism has an exact packing measure with respect to the gauge function $g(r)=r^4 (\log \log1/r)^{-3}$ in super-critical dimensions $d\geq 5$. More precisely, we prove that the total occupation measure of Super-Brownian motion is equal to the $g$-packing measure restricted to its range, up to a deterministic multiplicative constant that only depends on space dimension $d$.