Random Hamiltonians I: Probability measures and random walks on the Hamiltonian diffeomorphism group
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909823465947136 |
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| author | Dawid, Adrian |
| author_facet | Dawid, Adrian |
| contents | We construct a family of probability measures on the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M,ω)$. We show that these measures are Borel measures with respect to the topology induced by the Hofer metric. Further, we show that these measures turn any Hofer-Lipschitz function into a random variable with finite expectation. These measures have (for suitable choices of parameters) several desirable properties, such as full support on $\text{Ham}(M,ω)$, explicit estimates of the measure of Hofer-balls, and certain controls under the action of the group. We also define a family of probability measures on the space of autonomous Hamiltonian diffeomorphisms. These measures have similar properties and give rise to a random walk on the group $\text{Ham}(M,ω)$. Finally, we show that under certain limits this construction gives rise to probability measures on the space of Hamiltonian homeomorphisms and on the metric completion of $\text{Ham}(M,ω)$ with respect to the Hofer metric and the spectral metric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_03190 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random Hamiltonians I: Probability measures and random walks on the Hamiltonian diffeomorphism group Dawid, Adrian Symplectic Geometry Dynamical Systems Probability We construct a family of probability measures on the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M,ω)$. We show that these measures are Borel measures with respect to the topology induced by the Hofer metric. Further, we show that these measures turn any Hofer-Lipschitz function into a random variable with finite expectation. These measures have (for suitable choices of parameters) several desirable properties, such as full support on $\text{Ham}(M,ω)$, explicit estimates of the measure of Hofer-balls, and certain controls under the action of the group. We also define a family of probability measures on the space of autonomous Hamiltonian diffeomorphisms. These measures have similar properties and give rise to a random walk on the group $\text{Ham}(M,ω)$. Finally, we show that under certain limits this construction gives rise to probability measures on the space of Hamiltonian homeomorphisms and on the metric completion of $\text{Ham}(M,ω)$ with respect to the Hofer metric and the spectral metric. |
| title | Random Hamiltonians I: Probability measures and random walks on the Hamiltonian diffeomorphism group |
| topic | Symplectic Geometry Dynamical Systems Probability |
| url | https://arxiv.org/abs/2510.03190 |