InjectionAssembly #
Ported from scottnarmstrong/Superdiffusion at 7f20476385d4af418f110303085ad0740c9eba46. Module imports, support namespace and local option directives adapted for Lean Pool.
The injection estimate on a good cube #
On the intersection of the good cube event and the large-scale event, the
rough-field solution at any scale m ≥ n is close, on the cube y + □_n, to the
comparator solution at scale n, and inherits its 1/2-Hölder regularity.
The proof is the perturbation iteration: the rough-field solution at scale m
is the sum of a series of increments, each obtained from the previous one by
solving a Laplace problem forced by w_j ∇ · (k_m - k_n) and then the
rough-field problem at scale n with the resulting gradient as datum. The
large-scale event caps the size of ∇ · (k_m - k_n) on the cube uniformly in
m; the good cube event caps the regularity of the solution map at scale n;
and the annealed normalization turns the product of the two caps into a factor
≤ ε once the threshold of the large-scale event is calibrated as
C⁻¹ ε γ^{-1/2}. The series therefore converges geometrically in the
1/2-Hölder gauge, and converges in L² to the rough-field solution, so its
sum is that solution. The supremum estimate follows from the Hölder estimate
because each increment has zero trace on the cube, and the triangle inequality
against the good event closes both conclusions.
The constant and the disorder regime are explicit:
C(d, c⋆, C_reg) = 1 + 3 C_reg + 96 d² C_step C_reg c⋆^{-1/2} ,
γ₀(d, c⋆) = C_land^{-10} c⋆^{10} ,
where C_step is the constant of one iteration step and C_land is the
constant of the annealed-gauge comparison 3^{γ j} σ_{j-1}^{-1} ≤ 16 c⋆^{-1/2} γ^{1/2}, whose regime is exactly γ ≤ γ₀.
Main results #
injection_in_L_infty_v2_provider, at every scalen : ℤ.
References #
- ABK26, Proposition 5.1 (the injection estimate) and its proof.
1. An arithmetic helper #
From the normalized gauge to a Hölder bound. The normalization factor
σ_n 3^{-n/2} is positive, so an ℝ≥0∞ bound on the normalized gauge of a
function is a two-point bound with the reciprocal constant.
2. The injection estimate #
Proposition 5.1. On the good cube event and the large-scale event with
threshold C⁻¹ ε γ^{-1/2}, the rough-field solution at any scale m ≥ n is
within C ε 3^{n/2} [g] of the comparator in the normalized L^∞ gauge of the
cube, and its continuous representative obeys the normalized 1/2-Hölder bound
C 3^{n/2} [g].
The constant is C(d, c⋆, C_reg) and the disorder regime γ ≤ γ₀(d, c⋆) is the
one carried by the annealed-gauge comparison.
The scale n ranges over all of ℤ. No step below uses a lower bound on it:
every power of 3 that occurs is a real power, the annealed gauge σ_n is
positive at every integer scale, and the cubes and the two events are defined
at every n : ℤ.