تقرير
Invariance of Gaussian RKHSs under Koopman operators of stochastic differential equations with constant matrix coefficients
| العنوان: | Invariance of Gaussian RKHSs under Koopman operators of stochastic differential equations with constant matrix coefficients |
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| المؤلفون: | Philipp, Friedrich, Schaller, Manuel, Worthmann, Karl, Peitz, Sebastian, Nüske, Feliks |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Probability, Mathematics - Dynamical Systems |
| الوصف: | 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. Comment: 11 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2405.14429 |
| رقم الأكسشن: | edsarx.2405.14429 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |