Poisson representation of Brownian bridge.

Resumen:  We consider Brownian motion $(B(t))$, for $t\in[0,1]$, and Brownian bridge $BB(t)$, the Brownian motion conditioned to return to $0$ at time~$1$. The following identity is well known,(1)\,\hfill   law of $(BB(t))_{t\in[0,1]}= $ law of $(B(t)- tB(1))_{t\in[0,1]}$. \hfill\
A centered and rescaled Poisson point process $B^\varepsilon(t)$ converges to Brownian motion, where $\varepsilon$ is the scaling parameter going to $0$. For  each $\varepsilon>0$, we construct a coupling $(B^\varepsilon(t),BB^\varepsilon (t))$ satisfying an almost sure version of (1).  Taking  $\varepsilon\to0$ one shows (1).

Posted on Oct 15, 2025 in Seminario de Probabilidades de Chile, Seminars