2020/01/24 by Jieliang Hong, Hong, Jieliang
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2001.09137
openalex publication_date 2020/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If Lx is the total occupation local time of d-dimensional super-Brownian motion, X, for d=2 and d=3, we construct a random measure L, called the boundary local time measure, as a rescaling of Lx e-λLx dx as λ→ ∞, thus confirming a conjecture of \citeMP17 and further show that the support of L equals the topological boundary of the range of X, \partialR. This latter result uses a second construction of a boundary local time \widetildeL given in terms of exit measures and we prove that \widetildeL=cL a.s. for some constant c>0. We derive reasonably explicit first and second moment measures for L in terms of negative dimensional Bessel processes and use it with the energy method to give a more direct proof of the lower bound of the Hausdorff dimension of \partialR in \citeHMP18. The construction requires a refinement of the L2 upper bounds in \citeMP17 and \citeHMP18 to exact L2 asymptotics. The methods also refine the left tail bounds for Lx in \citeMP17 to exact asymptotics. We conjecture that the Minkowski content of \partialR is equal to the total mass of the boundary local time L up to some constant.