On the filtration of historical Brownian motion
The historical Brownian motion \(H\) is a process which takes its values in \(M_ F(C)\), the set of finite measures on \(C\), where \(C= C([0,\infty[, R^ d)\). If \(\nu\in M_ F(C)\), let \(\nu_ t\) be the image of \(\nu\) by the map \(y\to y(t)\). The super-Brownian motion \(X\) is defined by: \(X_ t\) is the image of \(H_ t\) by \(\nu\to \nu_ t\). The main result is the following: if \(d\geq 5\), then \({\mathcal F}^ X_ t= {\mathcal F}^ H_ t\) \(\forall t\geq 0\); moreover if \(d= 1\), then \({\mathcal F}^ X_ t\neq {\mathcal F}^ H_ t\). The proof is based on the fact that if \(d\geq 4\), a super- Brownian motion in \(M_ F(R^ d)\) started at \(m\neq 0\) does not intersect an independent \(R^ d\)-valued Brownian motion. If \(d= 2, 3\), the authors claim that \({\mathcal F}^ X_ t\neq {\mathcal F}^ H_ t\). The question is open when \(d= 4\).
scientific article; zbMATH DE number 731983