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Random Hamiltonians I: Probability measures and random walks on the Hamiltonian diffeomorphism group

Adrian Dawid

TL;DR

This work introduces a probabilistic framework for random Hamiltonian diffeomorphisms on a closed symplectic manifold by pushing forward Gaussian measures from the Hamiltonian function space to the Hamiltonian diffeomorphism group via the Hofer topology. It develops law-defining data $\mathcal{D}$ to build Gaussian Hamiltonians $H$ and derives a family of Hofer-Borel measures $\mu_{\mathrm{Ham}}^{\mathcal{D}}$, proving finite expectations for Hofer-Lipschitz observables and sub-Gaussian tail bounds; under exhaustive data these measures have full support and admit an autonomous counterpart yielding random walks. The paper further analyzes limiting behavior as the regularity parameter $\mathfrak{r}$ varies, obtaining weak limits on Hofer and spectral metric completions, $C^0$-Hamiltonian spaces, and Hamiltonian homeomorphism groups, thereby connecting probabilistic geometry with $C^0$-symplectic topics. Simulations on $\mathbb{T}^2$ suggest diffusion-like spreading and Crofton-type patterns for Lagrangian intersections, guiding future rigorous analysis and providing concrete numerical insight into the random-Hamiltonian framework.

Abstract

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.

Random Hamiltonians I: Probability measures and random walks on the Hamiltonian diffeomorphism group

TL;DR

This work introduces a probabilistic framework for random Hamiltonian diffeomorphisms on a closed symplectic manifold by pushing forward Gaussian measures from the Hamiltonian function space to the Hamiltonian diffeomorphism group via the Hofer topology. It develops law-defining data to build Gaussian Hamiltonians and derives a family of Hofer-Borel measures , proving finite expectations for Hofer-Lipschitz observables and sub-Gaussian tail bounds; under exhaustive data these measures have full support and admit an autonomous counterpart yielding random walks. The paper further analyzes limiting behavior as the regularity parameter varies, obtaining weak limits on Hofer and spectral metric completions, -Hamiltonian spaces, and Hamiltonian homeomorphism groups, thereby connecting probabilistic geometry with -symplectic topics. Simulations on suggest diffusion-like spreading and Crofton-type patterns for Lagrangian intersections, guiding future rigorous analysis and providing concrete numerical insight into the random-Hamiltonian framework.

Abstract

We construct a family of probability measures on the group of Hamiltonian diffeomorphisms of a closed symplectic manifold . 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 , 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 . 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 with respect to the Hofer metric and the spectral metric.

Paper Structure

This paper contains 25 sections, 48 theorems, 180 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main results and strategy
  3. Limiting behavior and $C^0$-symplectic geometry
  4. Previous approaches and related work
  5. Future work
  6. Acknowledgements
  7. Background
  8. Symplectic geometry
  9. Sobolev spaces
  10. Gaussian processes
  11. Gaussian measures
  12. Elementary properties of Gaussian processes
  13. Construction of the Measure
  14. Law-defining data
  15. Gaussian measure on $C^\infty_0$
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\mathcal{D}$ be any law-defining datum . Then the measure $\mu_{\mathop{\mathrm{Ham}}\nolimits}^{\mathcal{D}}$ is a Hofer-Borel probability measure on $\mathop{\mathrm{Ham}}\nolimits(M,\omega)$, i.e. $\mu_{\mathop{\mathrm{Ham}}\nolimits}^{\mathcal{D}}(\mathop{\mathrm{Ham}}\nolimits(M,\omega)) =

Figures (5)

  • Figure 1: Various enlargements of the space of Hamiltonian diffeomorphisms and their relations.
  • Figure 2: Sample paths of a Gaussian process described in Example \ref{['ex:gaussian-process-ex1']}. The values of $\alpha$ are (from left to right): $1/2, 10$ and $50$.
  • Figure 3: Visualization of $X_H(t,\cdot)$ for $\mathcal{D}_2$ on $\mathbb{T}^2$. The rows represent independent draws. The different columns represent evaluations of the Hamiltonian vector fields at $t=0,0.25,0.5,0.75$ from left to right. An animated version of this figure is available online at https://www.dpmms.cam.ac.uk/ apd55/random-hamiltonians/.
  • Figure 4: First row: Here we sample $100$ Hamiltonian functions from $\mu_{\mathcal{H}}^{\mathcal{D}}$. Then their time-$t$ flows are applied to $100$ points in a radius $0.1$ ball around $(0.5,0.5)$. Their positions are shown at $t=0,0.05, 0.1, 0.25$ from left to right. The different colors represent different draws. Second row: The torus is divided into a $10 \times 10$ grid and the number of points (shown in the first row) in each grid cell is counted. Darker colors represent more points.
  • Figure 5: These curves represent $\varphi(K)$ for twelve samples $\varphi$ from $\mu_{\mathop{\mathrm{Ham}}\nolimits}^{\mathcal{D}}$ with $\mathfrak{r} = 0.14$.

Theorems & Definitions (119)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 109 more