Diffusion via evaluation map: approximation of the uniform measure

Establish that for any epsilon > 0 and any point p in a closed symplectic manifold (M, ω), there exists a centered, time-symmetric law-defining datum D (as defined in the paper) such that the push-forward of the probability measure μ^D on Ham(M, ω) under the evaluation map ev_p: Ham(M, ω) → M is within epsilon of the uniform probability measure on M in a specified metric (for example, total variation or Wasserstein distance).

Background

In the simulations section, the author observes a diffusion-like phenomenon: points concentrated in a small region of the torus spread out rapidly under time-evolution by random Hamiltonian flows sampled from the constructed measures. Motivated by these observations, the author formulates a conjectural description of this diffusion effect in terms of the evaluation map from the Hamiltonian diffeomorphism group to the manifold.

The conjecture seeks a rigorous probabilistic statement asserting that, for an appropriate choice of law-defining datum D, the push-forward of the measure μ^D under ev_p approximates the uniform distribution on M arbitrarily well, thereby formalizing the numerically suggested diffusion behavior.