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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 $$\{X_{t}(\mathbb{R} ) > 0 \}$$.

##### MSC:
 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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