Abstract
For the local time $L_t^x$ of super-Brownian motion $X$ starting from $\delta_0$, we study its asymptotic behavior as $x\to 0$. In $d=3$, we find a normalization $\psi(x)=(1/(2\pi^2) \log (1/|x|))^{1/2}$ such that $(L_t^x-1/(2\pi|x|))/\psi(x)$ converges in distribution to standard normal as $x\to 0$. In $d=2$, we show that $L_t^x-(1/\pi)\log (1/|x|)$ converges a.s. as $x\to 0$. 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.
Highlights
Introduction and main results1.1 IntroductionSuper-Brownian motion arises as a scaling limit of critical branching random walks
For the local time Lxt of super-Brownian motion X starting from δ0, we study its asymptotic behavior as x → 0
A Super -Brownian Motion X starting at μ ∈ MF (Rd) is a continuous MF (Rd)-valued strong
Summary
Super-Brownian motion arises as a scaling limit of critical branching random walks. Let MF = MF (Rd) be the space of finite measures on (Rd, B(Rd)) equipped with the topology of weak convergence of measures, and (Ω, F, Ft, P ) be a filtered probability space. Mytnik and Perkins (2017) obtain the exact Hausdorff dimension of the boundary of super-Brownian motion, defined as the boundary of the set of points where the local time is positive. Lxt is called the local time of X at point x ∈ Rd and time t ≥ 0, which is jointly lower semicontinuous and is monotone increasing in t ≥ 0 Theorem II.7.3(a) in Perkins (2002) gives the existence of a σ-finite measure Nx0 on C([0, ∞), MF (Rd)), and it is defined to be the weak limit of N PδNx0 (X·N ∈ ·) as N → ∞, where X·N under PδNx0 is the approximating branching particle system starting from a single particle at x0 (see Ch.p. II. of Perkins (2002)). Perhaps surprisingly Lxt will be jointly continuous on {(t, x) : t ≥ 0, x ∈ Rd}, Nx0-a.e
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