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

Zbl 0716.00011