A noncanonical decomposition of the Brownian sheet. (Une décomposition non-canonique de drap brownien.) (French) Zbl 0767.60082

Jeulin, Th.; Yor, M.

Séminaire de probabilités XXVI, Lect. Notes Math. 1526, 322-347 (1992).

[For the entire collection see Zbl 0754.00008.]
Let \((B_ t,t\geq 0)\) be a real-valued Brownian motion started at 0. The authors have proven in a previous work that the process \(\tilde B_ \bullet=B_ \bullet-\int^ \bullet_ 0(B_ s/s)ds\) is again a Brownian motion, and that for each \(t\geq 0\), \(B_ t\) is independent of \((\tilde B_ s, s\leq t)\). The first result of the paper under review concerns the inverse problem. Specifically, the set \({\mathcal I}\) of probability measures on the canonical space \({\mathcal C}(\mathbb{R}_ +,\mathbb{R})\) for which the coordinate process \(X\) satisfies
(i) \(\tilde X_ \bullet=X_ \bullet-\int^ \bullet_ 0(X_ s/s)ds\) is a Brownian motion,
(ii) for each \(t\geq 0\), \(X_ t\) is independent of \((\tilde X_ s, s\leq t)\)
is identified with the set of space-time harmonic transforms of the Wiener measure. Equivalently, a probability measure \(P\) is in \({\mathcal I}\) iff it is the law of \((B_ t+Yt, t\geq 0)\), where \(Y\) is a random variable independent of \((B_ t,t\geq 0)\).
Then an analog for the Brownian sheet \((B_{s,t}; s\geq 0, t\geq 0)\) is discussed. The relevant transformation is now \(T_ 2\), which is specified by \[ T_ 2(B)_{s,t}=B_{s,t}-\int^ s_ 0{du\over u}B_{u,t}-\int^ t_ 0{dv\over v}B_{s,v}+\int^ s_ 0{du\over u}\int^ t_ 0{dv\over v}B_{u,v}. \] Typically, \(\tilde B\) is again a Brownian sheet, and ergodic properties of the transformation \(T_ 2\) are studied. Finally, the set \({\mathcal I}^{(2)}\) of probability measures on the canonical space \({\mathcal C}(\mathbb{R}^ 2_ +,\mathbb{R})\) for which
(i) \(T_ 2(X)\) is a Brownian sheet,
(ii) \(\forall s,t\geq 0\), \((T_ 2(X)_{u,v})_{u\leq s,v\leq t}\) is independent of \(((X_{u,t},X_{s,v}))_{u\leq s,v\leq t}\)
is characterized. The main difference in comparison with the first result is that there are now probability measures in \({\mathcal I}^{(2)}\) which are not locally equivalent to the law of the Brownian sheet.

Reviewer: J.Bertoin (Paris)

MSC:

60J65

Brownian motion

Keywords:

space-time harmonic transform; semimartigale decomposition; Brownian motion; Brownian sheet

Citations:

Zbl 0754.00008