Abstract
For a function $\ell$ satisfying suitable integrability (but not continuity) requirements, we construct a process $(B^\ell_u, 0 \leq u \leq 1)$ interpretable as Brownian excursion conditioned to have local time $\ell(\cdot)$ at time $1$. The construction is achieved by first defining a non-homogeneous version of Kingman's coalescent and then applying the general theory in Aldous (1993) relating excursion-type processes to continuum random trees. This complements work of Warren and Yor (1997) on the Brownian burglar.
Highlights
Let (Bu, 0 ≤ u ≤ 1) be standard Brownian excursion and (Ls, 0 ≤ s < ∞) its local time, more precisely its local time at time 1: hLs ds = 1(Bu≤h) du, h ≥ 0.Biane - Yor [4] give an extensive treatment, including an elegant description of the law of L as a random time-change of the Brownian excursion: t ( Ls/2, s ≥ 0)=d (Bτ−1 (s), s ≥ 0) for τ (t) =1/Bs ds where =d indicates equality in law
References to further papers on standard Brownian excursion can be found in those references
The construction does not directly involve any “Brownian” ingredients, but the theorem shows that B can be interpreted as Brownian excursion conditioned to have local time
Summary
Let (Bu, 0 ≤ u ≤ 1) be standard Brownian excursion and (Ls, 0 ≤ s < ∞) its local time, more precisely its local time at time 1: h. Applying the general correspondence [2] between consistent families of trees and excursion functions, we obtain (section 2.2) a Cexc[0, 1]-valued process B. The construction does not directly involve any “Brownian” ingredients, but the theorem (proved in section 3.2) shows that B can be interpreted as Brownian excursion conditioned to have local time. If B is standard Brownian excursion and L its local time, ψ( ) is a version of the conditional law of B given L =. By the continuity assertion of the construction, ψ( ) is specified uniquely “by continuity” for all ∈ L We emphasize this uniqueness because our definition of ψ( ) will be somewhat indirect, and without knowing continuity one might suspect there could be different extensions of ψ from the set of “typical paths of L” to larger spaces such as L. Perhaps the most general setting is where ds/ (s) is a sigma-finite measure on (0, s∗)
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