2020/01/20 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.2001.07269
openalex publication_date 2020/01/20 · openalex created_date 2020/01/30 · openalex updated_date 2026/07/28
We use a renormalization of the total mass of the exit measure from the complement of a small ball centered at x∈ ℝd for d≤ 3 to give a new construction of the total local time Lx of super-Brownian motion at x. In \citeHong20 a more singular renormalization of the total mass of the exit measure concentrating on x, where the exit measure is positive but unusually small, is used to build a boundary local time supported on the topological boundary of the range of super-Brownian motion. Our exit measure construction of Lx motivates this renormalization. We give an important step of this construction here by establishing the convergence of the associated mean measure to an explicit limit; this will be used in the construction of the boundary local time in \citeHong20. Both our results rely on the behaviour of solutions to the associated semilnear elliptic equation with singular initial data and on Le Gall's special Markov property for exit measures.