Abstract
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.
Highlights
Introduction and main resultsThe local time of super-Brownian motion (SBM) has been well studied by many authors, e.g., Adler and Lewin [1], Barlow, Evans and Perkins [2], Krone [9], Sugitani [14], etc
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
A super-Brownian motion X = (Xt, t ≥ 0) starting at μ ∈ MF is a continuous MF -valued strong Markov process defined on some filtered probability space (Ω, F, Ft, P ) with X0 = μ a.s
Summary
The local time of super-Brownian motion (SBM) has been well studied by many authors, e.g., Adler and Lewin [1], Barlow, Evans and Perkins [2], Krone [9], Sugitani [14], etc. A super-Brownian motion X = (Xt, t ≥ 0) starting at μ ∈ MF is a continuous MF -valued strong Markov process defined on some filtered probability space (Ω, F , Ft, P ) with X0 = μ a.s. Write μ(φ) = φ(x)μ(dx) for any measure μ. Has the law, PX0 , of a super-Brownian motion X starting from X0 It follows from (1.5) that the total local time Lx under PX0 may be decomposed as. The following result gives a new construction of the total local time Lx in terms of the local asymptotic behavior of the exit measures at x This result is useful in the construction of a boundary local time measure whose support is the topological boundary of the range of super-Brownian motion in d = 2 and d = 3 (see [7]).
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