RenormalizationAtRandomScale #
Ported from scottnarmstrong/Superdiffusion at 7f20476385d4af418f110303085ad0740c9eba46. Module imports, support namespace and local option directives adapted for Lean Pool.
The renormalization of the generator, read at the random confinement scale #
The superdiffusive displacement bounds evaluate the effective diffusivity and the error random variable of the renormalization theorem at the random confinement scale. This file makes that reading available.
The renormalization theorem produces, for each scale separately, an effective diffusivity and an error random variable. Assembling them into families indexed by the scales is one application of choice over the integers, and is the first result below; the families are stated with the effective diffusivity profile of the intrinsic-scale file, so that the scale arithmetic applies to them verbatim.
The remaining results evaluate at the random scale. The confinement scale is
above the intrinsic scale, so the scale arithmetic applies pointwise at
3^{m_t}; and the moments of the error family at the random scale follow from
the uniform-in-scale moment bound together with the level-set decomposition.
The moment bound of the renormalization theorem carries the factor
γ^{1/2}|log γ|³ and the p-dependence p^{1/2} + |log γ|^{1/2}, as in the
source; the additive p-dependence is the one the proof of that theorem yields
over the whole range of exponents, and it agrees with p^{1/2} up to a
constant once p is at least of order |log γ|. It is inherited verbatim
here: the bound below reads
(C(p^{1/2} + |log γ|^{1/2}) γ^{1/2} |log γ|³)^p.
References #
- ABK26, the renormalization of the generator and the superdiffusive displacement bounds of Section 5.4.
2. The scale arithmetic at the confinement scale #
The scale arithmetic at the confinement scale. The confinement scale is
above the intrinsic scale, because the deterministic factor
K |log γ|^{1/2} R(t) in its definition is at least R(t); so the comparison
between R(t)² and the displacement holds at 3^{m_t}.
3. The moments of the confinement scale in terms of the intrinsic scale #
The moments of the confinement scale, in the form in which they are used:
3^{m_t} has moments of order p bounded by a constant multiple of
|log γ|^{1/2} R(t).
4. The moments of the error family at the confinement scale #
The moments of the renormalization error at the confinement scale. The
moment bound of the renormalization theorem is uniform in the scale, so the
level-set decomposition of the random scale spends it at the exponent 2p and
returns the same profile at the exponent p, with an explicit constant.
The profile is the one the renormalization theorem carries: the factor
p^{1/2} + |log γ|^{1/2} and the power |log γ|³.
5. The same moments at the chosen confinement scale #
The moments of the chosen confinement scale. The hypotheses of the moment bound already make the chosen scale measurable and almost surely minimal, so the scale is no longer a binder.
The moments of the renormalization error at the chosen confinement scale. As above, with the error family evaluated at the chosen scale.