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

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

Zbl 0754.00008
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