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.
Publié le : 1992-04-14
Classification:
Martin boundary,
entrance space,
parabolic function,
infinite-dimensional Ornstein-Uhlenbeck process and operator,
abstract Wiener space,
integral representation of convex sets,
Dirichlet spaces,
60J45,
31C25,
60J50
@article{1176989818,
author = {Rockner, Michael},
title = {On the Parabolic Martin Boundary of the Ornstein-Uhlenbeck Operator on Wiener Space},
journal = {Ann. Probab.},
volume = {20},
number = {4},
year = {1992},
pages = { 1063-1085},
language = {en},
url = {http://dml.mathdoc.fr/item/1176989818}
}
Rockner, Michael. On the Parabolic Martin Boundary of the Ornstein-Uhlenbeck Operator on Wiener Space. Ann. Probab., Tome 20 (1992) no. 4, pp. 1063-1085. http://gdmltest.u-ga.fr/item/1176989818/