2018/04/23 by Hughes, Thomas
#28A78 (Secondary) #60H15 #60J55 #60J68 (Primary) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1804.08687
For a one-dimensional super-Brownian motion with density X(t,x), we construct a random measure Lt called the boundary local time which is supported on ∂ \x:X(t,x) = 0\ =: BZt, thus confirming a conjecture of Mueller, Mytnik and Perkins (2017). Lt is analogous to the local time at 0 of solutions to an SDE. We establish first and second moment formulas for Lt, 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 dim(BZt) = 2-2λ0> 0 with positive probability, a recent result of Mueller, Mytnik and Perkins (2017), where -λ0 is the lead eigenvalue of a killed Ornstein-Uhlenbeck operator that characterizes the left tail of X(t,x). In a companion work, the author and Perkins use the boundary local time and some of its properties proved here to show that dim(BZt) = 2-2λ0 a.s. on \Xt(ℝ) > 0 \.