On Occupation Kernels, Liouville Operators, and Dynamic Mode Decomposition
Using the newly introduced occupation kernels,'' the present manuscript develops an approach to dynamic mode decomposition (DMD) that treats continuous time dynamics, without discretization, through the Liouville operator. The technical and theoretical differences between Koopman based DMD for discrete time systems and Liouville based DMD for continuous time systems are highlighted, which includes an examination of these operators over several reproducing kernel Hilbert spaces.
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10281591
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2021 American Control Conference (ACC)
2. Abstract For $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , we define a weak $$\gamma$$ γ - Liouville quantum gravity ( LQG ) metric to be a function $$h\mapsto D_h$$ h ↦ D h which takes in an instance of the planar Gaussian free field and outputs a metric on the plane satisfying a certain list of natural axioms. We show that these axioms are satisfied for any subsequential limits of Liouville first passage percolation. Such subsequential limits were proven to exist by Ding et al. (Tightness of Liouville first passage percolation for $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , 2019. ArXiv e-prints, arXiv:1904.08021 ). It is also known that these axioms are satisfied for the $$\sqrt{8/3}$$ 8 / 3 -LQG metric constructed by Miller and Sheffield (2013–2016). For any weak $$\gamma$$ γ -LQG metric, we obtain moment bounds for diameters of sets as well as point-to-point, set-to-set, and point-to-set distances. We also show that any such metric is locally bi-Hölder continuous with respect to the Euclidean metric and compute the optimal Hölder exponents in both directions. Finally, we show that LQG geodesics cannot spend a long time near a straightmore »