Renormalization of local times of super-Brownian motion

2017/11/17 by Jieliang Hong, Hong, Jieliang
Economics, Econometrics and Finance · Mathematics · #60J68 #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1711.06447

openalex publication_date 2017/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For the local time L_t^x of super-Brownian motion X starting from δ₀, we study its asymptotic behavior as x→ 0. In d=3, we find a normalization ψ(x)=(1/(2π²) log (1/|x|))^1/2 such that (L_t^x-1/(2π|x|))/ψ(x) converges in distribution to standard normal as x→ 0. In d=2, we show that L_t^x-(1/π)log (1/|x|) converges a.s. as x→ 0. We also consider general initial conditions and get similar renormalization results. The behavior of the local time allows us to derive a second order term in the asymptotic behavior of a related semilinear elliptic equation.

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