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Martin boundaries on Wiener space. (English) Zbl 0726.60078

Diffusion processes and related problems in analysis, Vol. I: Diffusions in analysis and geometry, Proc. Int. Conf., Evanston/IL (USA) 1989, Prog. Probab. 22, 3-16 (1990).
[For the entire collection see Zbl 0716.00011.]
Let (X,\({\mathbb{P}})\) be the infinite-dimensional Brownian motion with the state space \(B:=\{x\in {\mathcal C}_{[0,1]}| x(0)=0\}.\) Denote with \({\mathcal P}\) the class of probability measures, \({\mathbb{Q}}\), on \(\Omega:={\mathcal C}([0,\infty),B)\) such that for all \(t\geq 0\), \({\mathbb{Q}}(\cdot | \hat F_ t)={\mathbb{P}}(\cdot | \hat F_ t),\) where \(\hat F_ t:=\sigma (X_ s;\;s\geq t).\) It is proved that for any \({\mathbb{Q}}\in {\mathcal P}\) there is exactly one probability measure \(\nu\) on B such that \[ (*)\;{\mathbb{Q}}=\int_{B}{\mathbb{P}}^ y\nu (dy), \] where \({\mathbb{P}}^ y\) is the measure associated with \(X_ t+yt\). Conversely, any probability measure \(\nu\) on B induces via (*) a measure \({\mathbb{Q}}\in {\mathcal P}\). Therefore, B is called the space-time Martin boundary of X. The extremal space-time harmonic functions can now be characterized by absolute continuity, which shows that only points in the Cameron-Martin space carries such a function. On the contrary to the finite-dimensional case it is seen that there are space-time harmonic functions, which cannot be represented in terms of extremals. The structure of h-processes is given by characterising the drift term. The infinite-dimensional Ornstein-Uhlenbeck processes are also discussed.
Reviewer: P.Salminen (Åbo)

MSC:

60J65 Brownian motion
60J50 Boundary theory for Markov processes

Citations:

Zbl 0716.00011