We study an infinite-dimensional Ornstein-Uhlenbeck process $(X_t)$ in a given Hilbert space $H$. This is driven by a cylindrical symmetric L\'evy process without a Gaussian component and taking values in a Hilbert space $U$ which usually contains $H$. We give if and only if conditions under which $X_t$ takes values in $H$ for some $t>0$ or for all $t>0$. Moreover, we prove irreducibility for $(X_t)$ and study existence and uniqueness of invariant measures.
On linear evolution equations for a class of cylindrical Lévy noises
PRIOLA, Enrico;
2010-01-01
Abstract
We study an infinite-dimensional Ornstein-Uhlenbeck process $(X_t)$ in a given Hilbert space $H$. This is driven by a cylindrical symmetric L\'evy process without a Gaussian component and taking values in a Hilbert space $U$ which usually contains $H$. We give if and only if conditions under which $X_t$ takes values in $H$ for some $t>0$ or for all $t>0$. Moreover, we prove irreducibility for $(X_t)$ and study existence and uniqueness of invariant measures.
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