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A boundary local time for one-dimensional super-Brownian motion and applications. (English) Zbl 1467.60063
Summary: For a one-dimensional super-Brownian motion with density \(X(t,x)\), we construct a random measure \(L_{t}\) called the boundary local time which is supported on \(BZ_{t} := \partial \{x:X(t,x) = 0\}\), thus confirming a conjecture of C. Mueller et al. [Ann. Probab. 45, No. 6A, 3481–3534 (2017; Zbl 1412.60123)]. \(L_{t}\) is analogous to the local time at 0 of solutions to an SDE. We establish first and second moment formulas for \(L_{t}\), some basic properties, and a representation in terms of a cluster decomposition. Via the moment measures and the energy method we give a more direct proof that \(\text{dim} (BZ_{t}) = 2-2\lambda _{0}> 0\) with positive probability, a recent result of Mueller et al. [loc. cit.], where \(-\lambda_{0}\) is the lead eigenvalue of a killed Ornstein-Uhlenbeck operator that characterizes the left tail of \(X(t,x)\). In a companion work [T. Hughes and E. Perkins, Ann. Inst. Henri Poincaré, Probab. Stat. 55, No. 4, 2395–2422 (2019; Zbl 1434.60234)], the author and Perkins use the boundary local time and some of its properties proved here to show that \(\text{dim} (BZ_{t}) = 2-2\lambda_{0}\) a.s. on \(\{X_{t}(\mathbb{R} ) > 0 \}\).

60J68 Superprocesses
60J55 Local time and additive functionals
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
28A78 Hausdorff and packing measures
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