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April, 1992 On the Parabolic Martin Boundary of the Ornstein-Uhlenbeck Operator on Wiener Space
Michael Rockner
Ann. Probab. 20(2): 1063-1085 (April, 1992). DOI: 10.1214/aop/1176989818


We study the positive parabolic functions of the Ornstein-Uhlenbeck operator on an abstract Wiener space $E$ using the approach developed by Dynkin. This involves first proving a characterization of the entrance space of the corresponding Ornstein-Uhlenbeck semigroup and deriving an integral representation for an arbitrary entrance law in terms of extreme ones. It is shown that the Cameron-Martin densities are extreme parabolic functions, but that if $\dim E = \infty$, not every positive parabolic function has an integral representation in terms of those (which is in contrast to the finite-dimensional case). Furthermore, conditions for a parabolic function to be representable in terms of Cameron-Martin densities are proved.


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Michael Rockner. "On the Parabolic Martin Boundary of the Ornstein-Uhlenbeck Operator on Wiener Space." Ann. Probab. 20 (2) 1063 - 1085, April, 1992.


Published: April, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0761.60067
MathSciNet: MR1159586
Digital Object Identifier: 10.1214/aop/1176989818

Primary: 60J45
Secondary: 31C25 , 60J50

Keywords: Abstract Wiener space , Dirichlet spaces , entrance space , infinite-dimensional Ornstein-Uhlenbeck process and operator , integral representation of convex sets , Martin boundary , parabolic function

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 2 • April, 1992
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