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Symmetric Ornstein-Uhlenbeck Semigroups and their Generators
Probability · Submitted 30 May 2002
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We provide necessary and sufficient conditions for a Hilbert space-valued Ornstein-Uhlenbeck process to be reversible with respect to its invariant measure $\mu$.
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The abstract, sentence by sentence
We provide necessary and sufficient conditions for a Hilbert space-valued Ornstein-Uhlenbeck process to be reversible with respect to its invariant measure $\mu$. For a reversible process the domain of its generator in $L^p(\mu )$ is characterized in terms of appropriate Sobolev spaces thus extending the Meyer equivalence of norms to any symmetric Ornstein-Uhlenbeck operator. We provide also a formula for the size of the spectral gap of the generator. Those results are applied to study the Ornstein-Uhlenbeck process in a chaotic environment. Necessary and sufficient conditions for a transition semigroup $(R_t)$ to be compact, Hilbert-Schmidt and strong Feller are given in terms of the coefficients of the Ornstein-Uhlenbeck operator. We show also that the existence of spectral gap implies a smoothing property of $R_t$ and provide an estimate for the (appropriately defined) gradient of $R_t\phi$. Finally, in the Hilbert-Schmidt case, we show that for any $\phi\in L^p(\mu)$ the function $R_t\phi$ is an (almost) classical solution of a version of the Kolmogorov equation.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We provide necessary and sufficient conditions for a Hilbert space-valued Ornstein-Uhlenbeck process to be reversible with respect to its invariant measure $\mu$. | result 0.50 | 0.95 | 0.50 | asserted 1.00 |
| s1 | For a reversible process the domain of its generator in $L^p(\mu )$ is characterized in terms of appropriate Sobolev spaces thus extending the Meyer equivalence of norms to any symmetric Ornstein-Uhlenbeck operator. | result 0.78 | 0.89 | 0.24 | asserted 1.00 |
| s2 | We provide also a formula for the size of the spectral gap of the generator. | result 0.75 | 0.95 | 0.13 | asserted 1.00 |
| s3 | Those results are applied to study the Ornstein-Uhlenbeck process in a chaotic environment. | contribution 0.91 | 0.85 | 0.05 | asserted 0.96 |
| s4 | Necessary and sufficient conditions for a transition semigroup $(R_t)$ to be compact, Hilbert-Schmidt and strong Feller are given in terms of the coefficients of the Ornstein-Uhlenbeck operator. | result 0.87 | 0.92 | 0.12 | asserted 1.00 |
| s5 | We show also that the existence of spectral gap implies a smoothing property of $R_t$ and provide an estimate for the (appropriately defined) gradient of $R_t\phi$. | result 0.96 | 0.95 | 0.11 | asserted 1.00 |
| s6 | Finally, in the Hilbert-Schmidt case, we show that for any $\phi\in L^p(\mu)$ the function $R_t\phi$ is an (almost) classical solution of a version of the Kolmogorov equation. | result 0.97 | 0.95 | 0.10 | asserted 1.00 |
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Original abstract
We provide necessary and sufficient conditions for a Hilbert space-valued Ornstein-Uhlenbeck process to be reversible with respect to its invariant measure $\mu$. For a reversible process the domain of its generator in $L^p(\mu )$ is characterized in terms of appropriate Sobolev spaces thus extending the Meyer equivalence of norms to any symmetric Ornstein-Uhlenbeck operator. We provide also a formula for the size of the spectral gap of the generator. Those results are applied to study the Ornstein-Uhlenbeck process in a chaotic environment. Necessary and sufficient conditions for a transition semigroup $(R_t)$ to be compact, Hilbert-Schmidt and strong Feller are given in terms of the coefficients of the Ornstein-Uhlenbeck operator. We show also that the existence of spectral gap implies a smoothing property of $R_t$ and provide an estimate for the (appropriately defined) gradient of $R_t\phi$. Finally, in the Hilbert-Schmidt case, we show that for any $\phi\in L^p(\mu)$ the function $R_t\phi$ is an (almost) classical solution of a version of the Kolmogorov equation.
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