StoppedMomentsCubeModel #
Ported from scottnarmstrong/Superdiffusion at 7f20476385d4af418f110303085ad0740c9eba46. Module imports, support namespace and local option directives adapted for Lean Pool.
The stopped moments of the stream process on a triadic cube, at the model #
The stopped-moment bounds of the displacement estimates are composed here at the model: the
process is the stream process of a full sample, the cube is the triadic cube centred at the
origin at a scale m, the observables are the linear and quadratic data cut off by a smooth
function equal to one on the closed cube, and the corrector bounds are the ones of the
renormalization theorem at the sample.
The objects the quenched moment composition takes as free parameters are defined:
cubeCutoff d m— a smooth cut-off between0and1, equal to one on the closed cube of scalemcentred at the origin and supported in the open cube of the next scale;cubeExitProbability M omega m t— the probability that the stream process started at the origin has left the cube of scalemby the timet;cubeStoppedMean M omega m t— the vector whosei-th coordinate is the mean, at the exit time from the cube truncated att, of the cut-off linear observable in the directioni;
and the two stopped bounds are proved for them, in the exact form that composition consumes:
- the squared norm of the stopped mean vector is at most
4 d (3^m)² EB²; - in every direction, the stopped mean of the cut-off quadratic observable is within
10 (3^m)² EB + 2 sigmaBar ttimes the exit probability of2 sigmaBar t.
Both are proved under the corrector clause of the renormalization theorem at the sample and the
scale m, with the effective diffusivity sigmaBar and the error value EB of that clause; the
clause holds almost surely at every scale for the families of
exists_renormalizationFamily_corrector, and the almost-sure form on the full-sample carrier is
recorded at the end. The side condition
d sigmaBar t ≤ (3^m)² of the composition is derived from the defining property of the
confinement scale.
Main definitions #
Main results #
vecNormSq_cubeStoppedMean_le— the squared stopped mean.abs_cubeStoppedQuadratic_sub_le— the stopped second moment in every direction.dim_mul_sigmaBar_mul_le_three_zpow_sq_of_isConfinementScale— the side condition.ae_renormalizationCorrector_fullSample— the corrector clause holds almost surely on the full-sample carrier, at every scale.
References #
- ABK26, Steps 1 and 3 of the superdiffusive displacement bounds of Section 5.4.
1. The cut-off of the cube #
A smooth cut-off between 0 and 1, equal to one on the closed cube of scale m centred at
the origin and supported in the open cube of the next scale.
The cut-off of the cube of scale m centred at the origin: smooth, between 0 and 1,
equal to one on the closed cube, supported in the open cube of the next scale.
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2. The bounds for the cut-off observables #
A point of the cube of scale m + 1 has supremum norm below half its side.
The cut-off linear observables are uniformly bounded, in every direction.
The cut-off quadratic observables are uniformly bounded, in every direction.
3. The basis directions #
4. The process quantities #
The exit probability: the probability that the stream process started at the origin has
left the cube of scale m centred at the origin by the time t.
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- One or more equations did not get rendered due to their size.
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The stopped mean vector: its i-th coordinate is the mean, under the stream process
started at the origin, of the cut-off linear observable in the direction i at the position of
the path at its exit time from the cube of scale m centred at the origin, truncated at t.
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- One or more equations did not get rendered due to their size.
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5. The squared stopped mean #
The squared norm of the stopped mean vector. Each coordinate is the stopped mean of the
cut-off linear observable in that direction, which is within 2 · 3^m EB of zero by the stopped
mean bound and the corrector estimate at the linear datum; summing the squares over the d
directions gives 4 d (3^m)² EB².
6. The stopped second moment #
The stopped second moment in every direction. The stopped mean of the cut-off quadratic
observable in the direction i is within 10 (3^m)² EB plus 2 sigmaBar t times the exit
probability of 2 sigmaBar t, by the stopped second-moment bound with the corrected observable
and the corrector estimate at the quadratic datum.
7. The side condition of the composition #
The confinement scale dominates the diffusive scale. At a confinement scale m for the
length K |log γ|^{1/2} R(t), with K² ≥ 2 d, the running diffusivity satisfies
d sigmaBar t ≤ (3^m)², once the disorder parameter is small enough that
d (1 + 2 C) γ^{1/2} |log γ| ≤ 1/2.
8. The corrector clause on the full-sample carrier #
The corrector clause holds almost surely on the full-sample carrier, at every scale. The clause of the renormalization families is almost sure on the cutoff-sample carrier at each scale; countability of the scales and the identification of the full-sample law with the cutoff-sample law transport it.