Local behavior of local times of super Brownian motion

2017/06/08 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.1706.02759

openalex publication_date 2017/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For x∈ R^d- \0\, in dimension d=3, we study the asymptotic behavior of the local time L_t^x of super-Brownian motion X starting from δ₀ as x → 0. Let ψ(x)=((1/2π²) log (1/|x|))^1/2 be a normalization, Theorem 1 implies that (L_t^x-(1/2π|x|))/ψ(x) converges in distribution to a standard normal distributed random variable as x→ 0. For dimension d=2, Theorem 2 implies that L^x_t-(1/π)log(1/|x|) is L¹ bounded as x→ 0. To do this, we prove a Tanaka formula for the local time which refines a result in Barlow, Evans and Perkins.

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