Invariance of Gaussian RKHSs under Koopman operators of stochastic differential equations with constant matrix coefficients
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914807742988288 |
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| author | Philipp, Friedrich Schaller, Manuel Worthmann, Karl Peitz, Sebastian Nüske, Feliks |
| author_facet | Philipp, Friedrich Schaller, Manuel Worthmann, Karl Peitz, Sebastian Nüske, Feliks |
| contents | We consider the Koopman operator semigroup $(K^t)_{t\ge 0}$ associated with stochastic differential equations of the form $dX_t = AX_t\,dt + B\,dW_t$ with constant matrices $A$ and $B$ and Brownian motion $W_t$. We prove that the reproducing kernel Hilbert space $\bH_C$ generated by a Gaussian kernel with a positive definite covariance matrix $C$ is invariant under each Koopman operator $K^t$ if the matrices $A$, $B$, and $C$ satisfy the following Lyapunov-like matrix inequality: $AC^2 + C^2A^\top\le 2BB^\top$. In this course, we prove a characterization concerning the inclusion $\bH_{C_1}\subset\bH_{C_2}$ of Gaussian RKHSs for two positive definite matrices $C_1$ and $C_2$. The question of whether the sufficient Lyapunov-condition is also necessary is left as an open problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_14429 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Invariance of Gaussian RKHSs under Koopman operators of stochastic differential equations with constant matrix coefficients Philipp, Friedrich Schaller, Manuel Worthmann, Karl Peitz, Sebastian Nüske, Feliks Probability Dynamical Systems We consider the Koopman operator semigroup $(K^t)_{t\ge 0}$ associated with stochastic differential equations of the form $dX_t = AX_t\,dt + B\,dW_t$ with constant matrices $A$ and $B$ and Brownian motion $W_t$. We prove that the reproducing kernel Hilbert space $\bH_C$ generated by a Gaussian kernel with a positive definite covariance matrix $C$ is invariant under each Koopman operator $K^t$ if the matrices $A$, $B$, and $C$ satisfy the following Lyapunov-like matrix inequality: $AC^2 + C^2A^\top\le 2BB^\top$. In this course, we prove a characterization concerning the inclusion $\bH_{C_1}\subset\bH_{C_2}$ of Gaussian RKHSs for two positive definite matrices $C_1$ and $C_2$. The question of whether the sufficient Lyapunov-condition is also necessary is left as an open problem. |
| title | Invariance of Gaussian RKHSs under Koopman operators of stochastic differential equations with constant matrix coefficients |
| topic | Probability Dynamical Systems |
| url | https://arxiv.org/abs/2405.14429 |