Crofton-type formula for expected Lagrangian intersections under random Hamiltonians

Establish a Crofton-type formula for random Hamiltonian diffeomorphisms: for a closed symplectic manifold (M, ω), a centered, time-symmetric, exhaustive or periodically exhaustive law-defining datum D, and closed Lagrangian submanifolds L, K ⊂ M, prove the existence of a constant C > 0 and a smooth function ρ: M → (0, ∞) with ∫_M ρ ω^n = vol(M), depending only on D and L, such that the expectation E_{φ∼μ^D}[ #(L ∩ φ(K)) ] equals C · vol_{ρ·g}(K), where g is the Riemannian metric associated to ω and the almost complex structure J specified in D; in particular, derive constants C′, C″ > 0 so that C′ · vol_g(K) ≤ E_{φ∼μ^D}[ #(L ∩ φ(K)) ] ≤ C″ · vol_g(K) for all closed Lagrangian submanifolds K ⊂ M.

Background

The author draws an analogy with classical Crofton formulas in integral geometry and with an established result on complex projective space, where the expected number of intersections between a fixed Lagrangian and the image of another under a compact Lie group action is proportional to the volume of the latter.

Using the probability measures μ^D on Ham(M, ω) built from Gaussian processes, the paper conjectures a symplectic Crofton-type formula relating the expected intersection count to a weighted volume of the Lagrangian K under a modified metric. The conjecture is supported by numerical experiments reported in the paper.