diff 56bc728..f6bf760 hard_sphere_boltzmann_fluctuations_v2.txt
diff --git a/hard_sphere_boltzmann_fluctuations_v2.txt b/hard_sphere_boltzmann_fluctuations_v2.txt
index c787ac1..466532f 100644
--- a/hard_sphere_boltzmann_fluctuations_v2.txt
+++ b/hard_sphere_boltzmann_fluctuations_v2.txt
@@ -72,10 +72,10 @@ f_0(z), f_t(z) : initial one-particle density and the regular Boltzmann
solution, both in z=(x,v); regularity on [0,T] is
the kinetic assumption.
-|cdot|_{Bol,beta} : sup norm with a Gaussian velocity weight
+|·|_{Bol,beta} : sup norm with a Gaussian velocity weight
e^{beta|v|^2}, summed over unit spatial cells so mass is controlled.
-B, Deltaphi : collision kernel [(v-v_*)cdotomega]_+ and the change
+B, Deltaphi : collision kernel [(v-v_*)·omega]_+ and the change
Deltaphi of a test across one collision.
A_t, C_t : linearized Boltzmann drift and collision-noise covariance:
@@ -105,7 +105,7 @@ around the other group. Their relative motion is a straight line, and a line
meets that tiny sphere in a window whose cross section is a disk of area of
order ε^2, hit during a sliding time interval. This is the source of the
factor ε^2 per merging collision, the merger Jacobian
-ε^2[(v-v_*)cdotomega] d t d omega. The gas carries activity
+ε^2[(v-v_*)·omega] d t d omega. The gas carries activity
mu=ε^{-2} per particle, so a connected diagram with h marked
particles and m lines weighs mu^{m-h}, and each merger pays one ε^2:
the two powers of ε cancel line by line. The initial excluded-volume link
@@ -132,16 +132,16 @@ It is a supremum with a Gaussian velocity weight, summed over unit spatial
cells so that it also controls the total mass. Hard-sphere collisions use
the kernel and scattering
-B(v-v_*,omega) =[(v-v_*)cdotomega]_+, omegainmathbb S^2bbS^2
+B(v-v_*,omega) =[(v-v_*)·omega]_+, omegainS^2
-v' =v-[(v-v_*)cdotomega]omega,
-v_*' =v_*+[(v-v_*)cdotomega]omega
+v' =v-[(v-v_*)·omega]omega,
+v_*' =v_*+[(v-v_*)·omega]omega
with [a]_+=max{a,0} and d omega the unnormalized surface measure.
The Boltzmann equation reads
-(∂_t+vcdotnabla_x)f=Q(f,f),
-Q(f,f)=int_{R^3timesmathbb S^2}bbS^2B(f'f_*'-ff_*) d v_* d omega
+(∂_t+v·nabla_x)f=Q(f,f),
+Q(f,f)=int_{R^3timesS^2}'f_*'-ff_*) d v_* d omega
all densities evaluated at the same x.
@@ -149,12 +149,12 @@ Fix the sphere diameter ε>0 and set the mean density
mu=ε^{-2}. The grand-canonical initial law, including N=0, is
P_ε(N, d Z_N)
-=mathcal Z_ε^calZ_ε^{-1}(mu^N)/(N!)
-1 _{mathcal D_N^ε}calD_N^ε(Z_N)prod_{i=1}^N f_0(z_i) d z_i
+=Z_ε^calZ_ε^{-1}(mu^N)/(N!)
+1 _{D_N^ε}calD_N^ε(Z_N)prod_{i=1}^N f_0(z_i) d z_i
with the non-overlap set
-mathcal D_N^ε=calD_N^ε={(z_i):|x_i-x_j|>ε, ine j} and
-1lemathcal Z_εcalZ_εle e^mu. Given the configuration, spheres move by
+D_N^ε=calD_N^ε={(z_i):|x_i-x_j|>ε, ine j} and
+1leZ_εcalZ_εle e^mu. Given the configuration, spheres move by
free flight and deterministic elastic collisions, defined almost surely
by Deng--Hani--Ma, Proposition 1.2. There is no random collision rule and no
resampling: the only randomness is the initial configuration.
@@ -183,18 +183,18 @@ Define the pair difference and the two limiting operators
Deltaphi =phi(x,v')+phi(x,v_*')-phi(x,v)-phi(x,v_*)
(A_tphi)(x,v)
-=vcdotnabla_xphi
-+int_{R^3timesmathbb S^2}bbS^2B(v-v_*,omega)f_t(x,v_*)Deltaphi
+=v·nabla_xphi
++int_{R^3timesS^2}-v_*,omega)f_t(x,v_*)Deltaphi
d v_* d omega
C_t(phi,psi)
-=tfrac12int_{R^3times R^3times R^3timesmathbb S^2}bbS^2
+=tfrac12int_{R^3times R^3times R^3timesS^2}
B(v-v_*,omega)f_t(x,v)f_t(x,v_*)Deltaphi Deltapsi
d x d v d v_* d omega
A_t is the linearized Boltzmann generator; the collision part carries
no extra factor 1/2 because of the pre/post substitution together with
-omegamapsto-omega (the collision normalization, below). The quadratic form
+omega|->-omega (the collision normalization, below). The quadratic form
C_t is the collision noise, nonnegative because f_tge0.
Theorem (Finite-dimensional fluctuation limit). Under the kinetic assumption, the centered Gaussian field zeta solving the
@@ -252,7 +252,7 @@ the sharp connected bounds). The h-th cumulant is O(mu^{1-h}),
with the leading h=1,2 terms identified as signed sums of trees
weighted by f_{itau}, contact flux, and pm1 signs. This is proved
by chronological merger coordinates: each collision that merges two
-clusters contributes ε^2[(v-v_*)cdotomega]_+, and summing all
+clusters contributes ε^2[(v-v_*)·omega]_+, and summing all
clusters costs CT/sqrt L<1.
4. Gaussian limit for the model dynamics
@@ -285,13 +285,13 @@ divided by mu^h:
int F_{h,0}^{ε}(Z_h)Phi(Z_h) d Z_h
=mu^{-h}Esum_{i_1,...,i_h distinct}Phi(z_{i_1},...,z_{i_h})
-Write [h]={1,...,h}, Z_B=(z_j)_{jin B}, and let mathcal P(V)calP(V)
+Write [h]={1,...,h}, Z_B=(z_j)_{jin B}, and let P(V)
be the set of set-partitions of V. The factorial cumulant
(connected) densities c_{h,0}^{ε} are defined by the cumulant
inversion of the factorial hierarchy,
F_{h,0}^{ε}(Z_h)
-=sum_{piinmathcal P([h])}calP([h])prod_{Binpi}c_{|B|,0}^{ε}(Z_B)
+=sum_{piinP([h])}calP([h])prod_{Binpi}c_{|B|,0}^{ε}(Z_B)
The one-point function is g_0=c_{1,0}^{ε}=F_{1,0}^{ε}. Set
@@ -318,14 +318,14 @@ E(N)_plemu^p, sup_{0<εleε_0}E(N/mu)^p<∞
of h,
|c_{h,0}^{ε}(Z_h)|le(Ch)^{Ch}prod_{j=1}^hf_0(z_j)
-sum_{mathcal T}calTprod_{{i,j}inmathcal T}calT
+sum_{T}{{i,j}inT}
( 1 _{|x_i-x_j|leε}+ε^υ)
-the sum over trees mathcal TcalT on [h]. For fixed hge2 and any
+the sum over trees T on [h]. For fixed hge2 and any
D>0 there is C_{D,h} with
|c_{h,0}^{ε}(Z_h)|le C_{D,h}prod_{j=1}^hf_0(z_j)
-sum_{mathcal T}calTprod_{{i,j}inmathcal T}calT
+sum_{T}{{i,j}inT}
( 1 _{|x_i-x_j|le C_{D,h}ε}+ε^D)
and
@@ -342,12 +342,12 @@ lim_{εto0}sqrtmu(Epi_0^ε(phi)-int f_0phi)
Proof.
(i) Factorial bound and moments. For fixed Z_h, let
-mathcal Z_ε[Z_h]calZ_ε[Z_h] be the partition function of the remaining
+Z_ε[Z_h]calZ_ε[Z_h] be the partition function of the remaining
particles, still mutually excluding and additionally avoiding the balls
B(x_j,ε). Directly from the grand-canonical law,
-F_{h,0}^{ε}(Z_h)= 1 _{mathcal D_h^ε}calD_h^ε(Z_h)prod_{j=1}^hf_0(z_j)
-(mathcal Z_ε[Z_h])/(calZ_ε[Z_h])/(mathcal Z_ε)calZ_ε)
+F_{h,0}^{ε}(Z_h)= 1 _{D_h^ε}calD_h^ε(Z_h)prod_{j=1}^hf_0(z_j)
+(Z_ε[Z_h])/(calZ_ε[Z_h])/(Z_ε)calZ_ε)
The ratio is le1, giving the factorial upper bound. For h=1 it is
the probability that B(x_1,ε) is empty of particle centers; by the
@@ -359,23 +359,23 @@ as a positive combination of (N)_j gives the moment bounds.
(ii) The Penrose tree bound. Write chi_{ij}= 1 _{|x_i-x_j|le
ε} and set
-mathcal U_1=1,calU_1=1,
-mathcal U_n(Z_n)=calU_n(Z_n)=sum_{H conn. on [n]}prod_{{i,j}in H}(-chi_{ij}) (nge2).
+U_1=1,
+U_n(Z_n)={H conn. on [n]}prod_{{i,j}in H}(-chi_{ij}) (nge2).
The tree bound is
-|mathcal U_n(Z_n)|calU_n(Z_n)|lesum_{mathcal TcalT tree}prod_{{i,j}inmathcal T}calTchi_{ij}
+|U_n(Z_n)|calU_n(Z_n)|lesum_{T tree}prod_{{i,j}inT}{ij}
This is the Penrose forest scheme. Root a connected labelled graph at
vertex 1, assign graph-distance levels, and let the parent of each
vertex be its least-labelled neighbour in the preceding level. For a
-fixed resulting tree mathcal T,calT, the graphs giving it are exactly
-mathcal T⊆calT⊆ H⊆ R(mathcal T),calT), where R(mathcal T)calT)
+fixed resulting tree T, the graphs giving it are exactly
+T⊆calT⊆ H⊆ R(T), where R(T)
adds all same-level edges and every edge to a preceding-level vertex with
label ge that of the chosen parent. These intervals partition all
connected graphs, and summing one interval gives
-prod_{einmathcal T}calT(-chi_e)prod_{ein R(mathcal T)∖calT)∖mathcal T}calT(1-chi_e),
+prod_{einT}-chi_e)prod_{ein R(T)∖calT)∖T}-chi_e),
whose second factor lies in [0,1], proving the tree bound.
@@ -387,12 +387,12 @@ classical systems, in Statistical Mechanics (Benjamin, 1967).
Integrating a tree from its leaves, keeping one free vertex, gives
-int f_0^{ ⊗ n}|mathcal U_n|calU_n|le n^{n-2}b_ε^{n-1} (nge2)
+int f_0^{ ⊗ n}|U_n|calU_n|le n^{n-2}b_ε^{n-1} (nge2)
Hence the connected-graph (Mayer) series is absolutely convergent when
emu b_ε|a|_∞<1, and by exponentiation/analytic continuation
-logmathcal Z_ε[a]=calZ_ε[a]=sum_{nge1}(mu^n)/(n!)intmathcal U_n(Z_n)calU_n(Z_n)
+logZ_ε[a]=calZ_ε[a]=sum_{nge1}(mu^n)/(n!)intU_n(Z_n)
prod_{j=1}^nf_0(z_j)a(z_j) d Z_n
From the partition function to the cumulants. Apply
@@ -402,7 +402,7 @@ differentiation gives the rooted Mayer series
c_{h,0}^{ε}(Z_h)=prod_{j=1}^hf_0(z_j)
sum_{nge0}(mu^n)/(n!)int f_0^{ ⊗ n}(Z_{h+1:h+n})
-mathcal U_calU_{h+n}(Z_{h+n}) d Z_{h+1:h+n}
+U_{h+n}(Z_{h+n}) d Z_{h+1:h+n}
For bounds uniform in h: delete h-1 edges of a tree on h+n vertices
so each component contains one fixed root (possible by repeatedly cutting
@@ -452,7 +452,7 @@ S_phi=sum_iphi(z_i) and let kappa_k denote ordinary cumulants.
Substituting a=e^{uphi} in the logarithmic partition formula and
differentiating k times at u=0,
-kappa_k(S_phi)=sum_{nge1}(mu^n)/(n!)int f_0^{ ⊗ n}mathcal U_ncalU_n
+kappa_k(S_phi)=sum_{nge1}(mu^n)/(n!)int f_0^{ ⊗ n}U_n
(sum_{j=1}^nphi(z_j))^k,
and the rooted tree integral bound gives kappa_k(S_phi)=muint f_0phi^k+
@@ -489,10 +489,10 @@ sup_{0le tle T}int_{R^6}f_t(x,v)e^{q|v|^2} d x d v<∞
Proof.
Let Q^+ be the gain part of Q and P_tg=Q^+(g,f_t) with the second
-factor fixed. With mathsf T_tg(x,v)=g(x-tv,v)sfT_tg(x,v)=g(x-tv,v) and
-(mathcal Vg)_t=calVg)_t=int_0^tmathsf T_sfT_{t-s}P_sg_s d s, positivity and the
+factor fixed. With T_tg(x,v)=g(x-tv,v)sfT_tg(x,v)=g(x-tv,v) and
+(Vg)_t= T_sfT_{t-s}P_sg_s d s, positivity and the
transport Duhamel formula give
-f_tlesum_{j<n}(mathcal V^jcalV^jmathsf T_sfT_cdot f_0)_t+(mathcal V^nf)_t.calV^nf)_t.
+f_tlesum_{j<n}(V^j T_sfT_· f_0)_t+(V^nf)_t.
On the Gaussian decay spaces Y_a with |g|_{Y_a}=sup e^{a|v|^2}|g|,
energy conservation gives for 0<b<ale2beta
@@ -501,7 +501,7 @@ energy conservation gives for 0<b<ale2beta
Distributing the exponent from beta to beta/2 over n factors and
using that the time simplex has volume T^n/n!,
-sup_{tle T}|(mathcal V^nf)_tcalV^nf)_t|_{Y_{beta/2}}le M(CT)^n/n!(1+
+sup_{tle T}|(V^nf)_t|_{Y_{beta/2}}le M(CT)^n/n!(1+
√(2n/beta))^nto0, so f is dominated pointwise by the positive
gain series.
@@ -519,13 +519,13 @@ E_a^0={hin C(R^6):e^{-a|v|^2}hin C_0(R^6)},
|h|_a=sup e^{-a|v|^2}|h|
on which smooth compactly supported tests are dense. Let
-mathsf S_uh(x,v)=h(x+uv,v)sfS_uh(x,v)=h(x+uv,v) be free test transport and
-K_t=A_t-vcdotnabla_x.
+S_uh(x,v)=h(x+uv,v)sfS_uh(x,v)=h(x+uv,v) be free test transport and
+K_t=A_t-v·nabla_x.
Lemma (Backward evolution). For 0le a<b<2beta and 0le rle tle T there is a bounded backward
propagator U(r,t):E_a^0to E_b^0 solving
-U(r,t)h=mathsf S_sfS_{t-r}h+int_r^tmathsf S_sfS_{s-r}K_sU(s,t)h d s
+U(r,t)h=S_{t-r}h+int_r^tS_{s-r}K_sU(s,t)h d s
with the composition law at intermediate exponents and uniqueness as a
mild propagator in these scales. For phiin C_c^∞,
@@ -551,7 +551,7 @@ so the series converges for every finite T. Splitting a time simplex at
an intermediate time gives composition, and the same estimate on an
iterated remainder gives uniqueness with a positive exponent allowance.
For compactly supported smooth phi, differentiability of the series at
-its upper end (its final transport gives U(r,t)(vcdotnabla_xphi);
+its upper end (its final transport gives U(r,t)(v·nabla_xphi);
the simplex faces give U(r,t)K_tphi) converges with the Dyson bound,
giving ∂_tU(r,t)phi=U(r,t)A_tphi and hence the upper-end identity.
@@ -603,8 +603,8 @@ d s+W_t(phi).
Uniqueness. On any candidate's probability space, construct the
backward mild field from its own initial field and noise. The difference
D_t has zero initial field and noise, mean-square continuity, and the
-transport mild identity D_t(phi)=int_0^tD_s(K_smathsf S_sfS_{t-s}phi) d s
-(finite-support Riemann sums with mathsf S_sfS_{t-s}phi). Iterating n
+transport mild identity D_t(phi)=int_0^tD_s(K_sS_{t-s}phi) d s
+(finite-support Riemann sums with S_{t-s}phi). Iterating n
times with 0le a<b<beta bounds its L^2 norm by
C_b|phi|_a(CT)^n/n!(1+√(n/(b-a)))^nto0, proving uniqueness in
the class and uniqueness in law.
@@ -710,9 +710,9 @@ Connection in a slab means sharing a component of its atom graph; the last
alternative of (iv) covers a straight line with no atom there. These are
Deng--Hani--Ma, Definition 3.20(1). The complexity is
-rho=|H_0|+sum_i(|H_i|+r(mathcal M_i))calM_i))
+rho=|H_0|+sum_i(|H_i|+r(M_i))
-with r(mathcal M_i)calM_i) the cycle excess in slab i; since the O-graph is
+with r(M_i) the cycle excess in slab i; since the O-graph is
a forest this is the sum of C-cluster excesses. b denotes the number of
atoms.
@@ -720,12 +720,12 @@ Integral convention. Give each bond e a vector z_e=(x_e,v_e)
with position X_e(t)=x_e+tv_e; give atom a a time t_a. Its incoming
edges are 1,2 and serial outgoing edges 1',2'. On the contact surface
put u=v_1-v_2, omega=(X_1(t_a)-X_2(t_a))/ε, and
-mathfrak b=[-ufrakb=[-ucdotomega]_+. Define
+mathfrak b=[-ufrakb=[-u·omega]_+. Define
Delta_a^{mathrm C}rmC
=delta(|X_1-X_2|-ε)mathfrak bfrakb
prod_{j=1}^2delta(X_{j'}(t_a)-X_j(t_a))
-delta(v_{1'}-v_1+(ucdotomega)omega)delta(v_{2'}-v_2-(ucdotomega)omega)
+delta(v_{1'}-v_1+(u·omega)omega)delta(v_{2'}-v_2-(u·omega)omega)
Delta_a^{mathrm O}rmO
=delta(|X_1-X_2|-ε)mathfrak bfrakb
@@ -736,25 +736,25 @@ unnormalized. With a set E_* of integrated bonds and free ends, the
positive measure of a cut history M with kernel Q is
I_M(Q)(z_{fixed})
-:=mu^{|E_*|-2b}int_{mathcal D_M}calD_Mint Q(z_E,t_M)
+:=mu^{|E_*|-2b}int_{D_M} Q(z_E,t_M)
prod_{ain M}Delta_aprod_{ein E_*} d z_eprod_{ain M} d t_a
-where mathcal D_McalD_M enforces the slab and strictly increasing times along
+where D_M enforces the slab and strictly increasing times along
each line. For a full history |E|-2b=n= number of lines, and with h
roots the normalized integral is
-mathcal J_M(Q)=calJ_M(Q)=mu^{-h}I_M(Q):
+J_M(Q)={-h}I_M(Q):
each birth carries activity mu, each root a normalization mu^{-1},
each contact surface a factor ε^2 Deng--Hani--Ma, Proposition 7.5, Eq. (7.7).
To keep a root endpoint (x,v) at time t free, make its top end a fixed
end evaluated at (x-tv,v), removing the root activity.
-The one-slab class mathcal T_icalT_i consists of connected histories with one
+The one-slab class T_i consists of connected histories with one
top root, conditions (i)--(ii) only, no continuation or initial-link
requirement; it may have up to simΛ_i^2 lines.
-mathcal T_i^calT_i^{err} inserts one extra O-atom into a member of
-mathcal T_i.calT_i. The one-root signed tree sum contains every one-root
+T_i^{err} inserts one extra O-atom into a member of
+T_i. The one-root signed tree sum contains every one-root
particle-line tree of size <Λ_i/3, with C/O choices and sign
(-1)^{#mathrm O}rmO.
@@ -773,13 +773,13 @@ only satisfy the common Gaussian Bol envelope.
(b) If the independent input g in slab i has
|g-f_{(i-1)tau}|_{Bol,beta_{i-1}}leε^{theta_{i-1}}, then the
-signed tree sum mathsf T_i^ε[g]sfT_i^ε[g] satisfies
+signed tree sum T_i^ε[g]sfT_i^ε[g] satisfies
-|mathsf T_i^ε[g]-f_sfT_i^ε[g]-f_{itau}|_{Bol,beta_i}letfrac12ε^{theta_i}
+|T_i^ε[g]-f_sfT_i^ε[g]-f_{itau}|_{Bol,beta_i}letfrac12ε^{theta_i}
In the fixed-top-endpoint norm Bol,widetildebeta_i, the summed
-absolute contributions of cyclic mathcal T_i,calT_i, of
-mathcal T_i^calT_i^{err}, and of mathcal T_icalT_i with geΛ_i/3
+absolute contributions of cyclic T_i, of
+T_i^{err}, and of T_i with geΛ_i/3
lines are o(ε^{theta_i}). Fixed cutoff powers and a fixed
exponential-atom factor can be included, the latter absorbed by large
L.
@@ -887,7 +887,7 @@ process, not a factorization assumption.
Put
-M_i=lfloorΛ_i/2rfloor
+M_i=floor(Λ_i/2)
the expansion cluster cap. It differs from the dynamics caps
Λ_i,Gamma and belongs only to the expansion; take ε small so
@@ -914,14 +914,14 @@ where all finite subsystems are defined.
Lemma (Autonomous partition). For finite V,
-sum_{mathcal CcalCinmathcal P(V)}calP(V)prod_{Cinmathcal C}calCkappa_C
-prod_{{C,D}⊆mathcal C}calCchi(C,D)=1
+sum_{C P(V)}calP(V)prod_{CinC}
+prod_{{C,D}⊆C},D)=1
For S⊆ V and a root test H,
-H(Psi_V|_S)=sum_{mathcal CcalCinmathcal P(V)}calP(V)
-prod_{Cinmathcal C}calCkappa_Cprod_{{C,D}⊆mathcal C}calCchi(C,D)
-H(Psi_{mathcal C}calC|_S)
+H(Psi_V|_S)=sum_{C P(V)}calP(V)
+prod_{CinC}{{C,D}⊆C},D)
+H(Psi_{C}|_S)
Proof.
The collision components of the full truncated history give one summand
@@ -931,7 +931,7 @@ summand equal one, juxtapose the autonomous histories and scan incoming
contacts in time order: internal accepted contacts obey the rule, every
cross-contact fails it by chi=1, so the juxtaposition is the full
truncated history, and uniqueness identifies its components with
-mathcal C.calC.
+C.
The next lemma is finite algebra; its vertices will be autonomous clusters.
@@ -941,14 +941,14 @@ Mge2, and
u(B)=sum_{substack{G connectedV(G)=B}}prod_{ein E(G)}(-a_e),
u({w})=1
-There is a finite signed remainder mathcal RcalR with
+There is a finite signed remainder R with
prod_{e⊆ W}(1-a_e)
=sum_{R⊆ I⊆ W}
-sum_{substack{mathcal BcalBinmathcal P(I)calP(I) Bcap Rne∅, |B|<M (Binmathcal B)}calB)}
-prod_{Binmathcal B}calBu(B)prod_{e⊆ W∖ I}(1-a_e)+mathcal RcalR
+sum_{substack{B P(I)calP(I) Bcap Rne∅, |B|<M (BinB)}}
+prod_{BinB}{e⊆ W∖ I}(1-a_e)+R
-|mathcal R|calR| is bounded by a sum of rooted forests, each with exactly one
+|R|calR| is bounded by a sum of rooted forests, each with exactly one
component of size in [M,2M-2] and all other components smaller than M,
all meeting R, vertices off the forest keeping their mutual compatibility
product, each selected forest with coefficient le1. Moreover
@@ -989,26 +989,26 @@ F_s=mu^{-s}sum_{nge s}(1)/((n-s)!)(pr_{[s]})_#p_n
For S=[s], V=[s+j], define the signed root-output kernel
-mathcal K_calK_{S,V}(X, d X'_S)
-=sum_{mathcal CcalCinmathcal P(V)}calP(V)prod_{Cinmathcal C}calCkappa_C(X)
-sum_{substack{mathcal BcalBinmathcal P(calP(mathcal C)calC)
+K_{S,V}(X, d X'_S)
+=sum_{C P(V)}calP(V)prod_{CinC}
+sum_{substack{B P(calP(C)
(∪_{Cin B}C)cap Sne∅, |B|<M}}
-prod_{Binmathcal B}calBu(B;X) delta_{Psi_{mathcal C}calC(X)|_S}( d X'_S)
+prod_{BinB};X) delta_{Psi_{C}|_S}( d X'_S)
and the retained operator
-(mathcal L_sF)(calL_sF)( d X'_S)
+(L_sF)( d X'_S)
=sum_{j=0}^{s(M-1)Λ-s}(mu^j)/(j!)
-int F_{s+j}( d X_V) mathcal K_calK_{S,V}(X, d X'_S)
+int F_{s+j}( d X_V) K_{S,V}(X, d X'_S)
(An empty restriction is zero; at most s retained blocks, so the bound on
j follows from the caps.) The stopped kernel replaces the inner
block sum by
-sum_{Finmathfrak F^frakF^{stop}_S(mathcal C)}calC)prod_{ein E(F)}a_e(X),
-where mathfrak F^frakF^{stop}_S(mathcal C)calC) are forests spanning
-mathcal CcalC with every component meeting S, exactly one of size
+sum_{Finmathfrak F^frakF^{stop}_S(C)}{ein E(F)}a_e(X),
+where mathfrak F^frakF^{stop}_S(C) are forests spanning
+C with every component meeting S, exactly one of size
[M,2M-2], the rest smaller, no unrooted isolated cluster; keep
-prod_Ckappa_C and the root-output map. Define mathcal E_sFcalE_sF by
+prod_Ckappa_C and the root-output map. Define E_sF by
the retained operator with this kernel and j up to
jle(s+1)(M-1)Λ-s
@@ -1017,9 +1017,9 @@ Lemma (One-slab identity and variation remainder). For a positive input law with
hierarchy after the truncated slab. If factorial measures up to
the fresh-label bound are finite, then
-F'_s=mathcal L_sF+R_s,calL_sF+R_s, |R_s|lemathcal E_sFcalE_sF
+F'_s=L_sF+R_s,calL_sF+R_s, |R_s|leE_sF
-The record version holds, and mathcal E_sFcalE_sF is a positive domination of
+The record version holds, and E_sF is a positive domination of
R_s, not a claim that R_s has this local kernel.
Proof.
@@ -1032,14 +1032,14 @@ frac1{(n-s)!}binom{n-s}{j}=frac1{j!(n-s-j)!} turns the remaining sum
into mu^{s+j}F_{s+j}; the output normalization mu^{-s} leaves
mu^j/j!. Stopped terms keep only the unreached compatibility product and
drop the rest, so positivity of the input and the multiplicity bound give
-|R_s(H)|le(mathcal E_sF)(|H|);calE_sF)(|H|); there are le s rooted components and
+|R_s(H)|le(E_sF)(|H|);calE_sF)(|H|); there are le s rooted components and
only one of size up to 2M-2 clusters, giving
the fresh-label bound. Records are untouched by the spectator
summation and travel with the root-output map.
For a signed family Delta (uniformly in records and sle A_i/2),
-|mathcal L_calL_{i,s}Delta|_{TV}+|mathcal E_calE_{i,s}|Delta||_{TV}
+|L_{i,s}Delta|_{TV}+|E_{i,s}|Delta||_{TV}
le e^{C_*A_iΛ_i^2log(1/ε)}max_{qle A_{i-1}/2}|Delta_q|_{TV}
Indeed relevant qle(s+1)Λ_i^2/2<A_{i-1}/2, with at most q^q
@@ -1048,8 +1048,8 @@ activity lemu^q, and at most A_iΛ_i^2 values of q; take logs.
Define the unmarked one-root measures recursively by
-g_i=mathcal U_i(g_calU_i(g_{i-1}),
-mathcal U_i(g):=calU_i(g):=mathcal L_calL_{i,1}((g^{ ⊗ q})_{qge0})
+g_i=U_i(g_{i-1}),
+U_i(g):=calU_i(g):=L_{i,1}((g^{ ⊗ q})_{qge0})
a finite signed measure using the full retained one-root block. It
approximates the Boltzmann density:
@@ -1067,7 +1067,7 @@ The bridge from the retained block to the unrestricted tree sum is the
following; it is where ``retained'' and ``DHM signed trees'' are shown to
differ only by error classes.
-Lemma (Modified one-root bridge). The difference between mathcal U_i(g)calU_i(g) and the signed unrestricted
+Lemma (Modified one-root bridge). The difference between U_i(g) and the signed unrestricted
one-root particle-tree sum with n<Λ_i/3 is dominated, with inputs
replaced by |g|, by the positive integrals of
@@ -1119,11 +1119,11 @@ Fix finitely many observation pulses. Let m bound both the number of
pulses and of record-carrying labels; record coordinates lie in a fixed
bounded interval. For a root set S and required records on R⊆ S,
let F^R_{i,S} be the pasted factorial measure at interface i with record
-coordinates on R. The operators mathcal L_icalL_i append the slab
+coordinates on R. The operators L_i append the slab
observations on the same labels; compatibility is record-independent. The
retained hierarchy is the finite composition
-widehatwidehat F^R_i=F^R_i=mathcal L_icalL_imathcal L_calL_{i-1}...mathcal L_1F^R_0calL_1F^R_0
+widehatwidehat F^R_i=F^R_i=L_i L_calL_{i-1}...L_1F^R_0
well defined order by order (finitely many previous orders per output).
@@ -1164,7 +1164,7 @@ the exact blocks; artificial links for marked lines are not connections.
Lemma (Labels and connected components). The retained expansion factors over global components. If D^R_{i,B} is
the sum of diagrams connected on roots B, then
-widehatwidehat F^R_F^R_{i,S}=sum_{piinmathcal P(S)}calP(S)⊗_{Binpi}D^{Rcap B}_{i,B}
+widehatwidehat F^R_F^R_{i,S}=sum_{piinP(S)}⊗_{Binpi}D^{Rcap B}_{i,B}
Hence factorial-cumulant inversion keeps exactly the connected diagrams,
even after resumming into g_i. At fixed root order h=|S| and bounded
@@ -1206,14 +1206,14 @@ uniformly for a partial final slab (no velocity weight claimed).
Proof.
Write Delta_i=F_i-widehatwidehat F_i.F_i. Then
-Delta_i=R_i+mathcal L_icalL_iDelta_{i-1}, |R_i|lemathcal E_iF_calE_iF_{i-1},
+Delta_i=R_i+L_i{i-1}, |R_i|leE_iF_{i-1},
Delta_0=0. Substitute F_{i-1}=widehatwidehat F_F_{i-1}+Delta_{i-1} in the
positive stopped bound and expand the retained part as in
the resummation with records; refining autonomous histories into C graphs
and picking one incoming contact per incompatibility gives exactly the
positive C/O histories of the coarse package with previous-interface
continuations. The only top-slab difference, a stopped component with
-M_ilecdot<2M_i-2<Λ_i clusters, is in the same coarse family with
+M_ile·<2M_i-2<Λ_i clusters, is in the same coarse family with
atom count ge M_i-1. Bookkeeping: the lower independent inputs are
already resummed into g; the top stopped forest is not resummed. Top
components failing continuation are paid by the coarse boundary modifications, part (i)
@@ -1225,7 +1225,7 @@ powers of the label bookkeeping. Choosing L so
Ctau^{c_0}<1 and using Λ_i=A_i^{30}, A_ige|logε| makes the
top loss absorb into the exponential size gain,
-|mathcal E_icalE_iwidehatwidehat F_F_{i-1}|_{TV}le
+|E_i F_F_{i-1}|_{TV}le
ε^{-C_*|S|-C_m}sum_{bge M_i-1}(Ctau^{c_0})^b
sum_{rhoge0}|logε|^{C_*rho}ε^{c_*rho}le e^{-cΛ_i}
@@ -1315,7 +1315,7 @@ survives at the sharp fluctuation scale mu^{1-h} for the h-th
factorial cumulant. The mechanism is a change of relative
translations: every collision that merges two clusters of particles
adjusts one relative position by the collision geometry, producing a
-factor ε^2[(v-v_*)cdotomega]_+ d t d omega. After accounting
+factor ε^2[(v-v_*)·omega]_+ d t d omega. After accounting
for the activity mu per birth line, a connected diagram with p
physical initial links scales like mu^{1-h}ε^p, which is exactly the
size sought; cycles and nontrivial initial correlations are smaller by a
@@ -1325,7 +1325,7 @@ life time tau=T/L and the base CT/sqrt L<1 for large L.
Fix finitely many observation times and bounded continuous observation
functions. When observations are accumulated in a mark, update
-ymapsto y+phi_j(x,v), so |y|le M with M independent of the
+y|-> y+phi_j(x,v), so |y|le M with M independent of the
number of particles. More generally one may retain finitely many phase
points along each designated trajectory. A record is kept only along a
line on which it is required; forgetting part of a record means pushing
@@ -1358,7 +1358,7 @@ fixed bounded observation list. For each fixed h,
For h=1,2, the measures mu^{h-1}c_h^ε converge against bounded
continuous record tests to the signed sums of retained global trees with
trivial initial cumulants, with contact weight
-[(v-v_*)cdotomega]_+ d t d omega (scattering +1,
+[(v-v_*)·omega]_+ d t d omega (scattering +1,
non-scattering -1) and independent birth factors f_{itau}. The sums
of absolute retained diagram measures, after multiplication by
mu^{h-1}, are bounded uniformly with a small Gaussian weight in their
@@ -1377,7 +1377,7 @@ counting translations and activity factors.
Work with a term of complexity le K. Replace each initial cumulant by
its bound from the initial ensemble,
-C_Ksum_{mathcal LcalL forest}prod_{(p,q)inmathcal L}calL
+C_Ksum_{L forest}prod_{(p,q)inL}
( 1 _{|x_p(0)-x_q(0)|le R_{K,D}ε}+ε^D)
prod_pf_0(z_p(0))
@@ -1421,7 +1421,7 @@ absolute translation per global component, a vector in a ball of radius
R_{K,D}ε per physical initial merger, and (t,omega)in Rtimes
S^2 per dynamic merger. The positional Jacobian per dynamic merger is
-ε^2[(v-v_*)cdotomega]_+ d t d omega
+ε^2[(v-v_*)·omega]_+ d t d omega
A specified redundant contact has no free continuous parameter. The
coordinate map has multiplicity at most one on each incoming transversal
@@ -1438,16 +1438,16 @@ known up to a common translation. A merger of groups with translations
X,X_* on the indicated straight segments satisfies
X-X_*=-d(t)-εomega with d(t) the known relative displacement,
dot d=v-v_*, and omega opposite the outward normal, so the incoming
-condition is (v-v_*)cdotomega>0. The determinant of
-(t,omega)mapsto X-X_* on the sphere is
-ε^2|(v-v_*)cdotomega|, and the incoming convention selects the sign
+condition is (v-v_*)·omega>0. The determinant of
+(t,omega)|-> X-X_* on the sphere is
+ε^2|(v-v_*)·omega|, and the incoming convention selects the sign
in the coordinate formula. A scattering reflection or O-continuation
updates velocities; the two groups now share one translation.
If the next specified contact is redundant, its relative displacement and
velocities are already fixed. A straight relative flight r(t)=d+tu has
at most one incoming crossing, and
-int_Idelta(|r|-ε)[-ucdot r/ε]_+ d t= 1 _{{incoming
+int_Idelta(|r|-ε)[-u· r/ε]_+ d t= 1 _{{incoming
crossing in I}}, since the flux cancels the absolute derivative of
|r|-ε; the outgoing deltas then make a deterministic substitution.
If the crossing is absent the branch has weight zero. Segments starting
@@ -1670,15 +1670,15 @@ giving convergence to zero. Dominated convergence in space (checked first
on bounded velocities/size, then removing truncations with
the Stirling bound) defines the limiting signed tree measures.
-Lemma (Spatial traces of tree measures). Let mathcal TcalT be one of the summable tree measures above, with a line
+Lemma (Spatial traces of tree measures). Let T be one of the summable tree measures above, with a line
and a time at which its position is required. It has a disintegration
-mathcal T=calT= d x mathcal T_xcalT_x
+T= d x T_x
-with Gaussian-weighted variation of mathcal T_xcalT_x bounded uniformly in x
+with Gaussian-weighted variation of T_x bounded uniformly in x
and finite integral over x. For every bounded continuous test of the
other coordinates (and continuous tests of smaller Gaussian growth),
-xmapstomathcal T_x(calT_x(Phi) is continuous. If two tree components are
+x|->T_x( is continuous. If two tree components are
joined at their designated positions, the product of their conditional
measures at a common position, integrated against incoming flux, is an
absolutely defined measure; it equals imposing the joining position in
@@ -1870,11 +1870,11 @@ At a collision, with xi=(x,v,y), xi_*=(x,v_*,y_*),
xi'=(x,v',y), xi_*'=(x,v_*',y_*), put
DeltaPhi=Phi(xi')+Phi(xi_*')-Phi(xi)-Phi(xi_*),
-DPhi=vcdotnabla_xPhi,
+DPhi=v·nabla_xPhi,
and
-(H_tPhi)(xi)=int_{R^3times[-M_{obs},M_{obs}]timesmathbb S^2}bbS^2
+(H_tPhi)(xi)=int_{R^3times[-M_{obs},M_{obs}]timesS^2}
B(v-v_*,omega)DeltaPhi F_t(x, d v_* d y_*) d omega
In the formulas below, int BFF_*(...) is the simultaneous position
@@ -1910,7 +1910,7 @@ Three counts enter. (a) One root: the partner chosen among j fresh labels change
to 1/(j-1)! and splitting remaining labels uses
frac1{(j_1+j_2)!}binom{j_1+j_2}{j_1}=frac1{j_1!j_2!}, giving
int BFF_*(Phi(xi')-Phi(xi)), symmetrized by particle interchange
-(including omegamapsto-omega) into the collision term of
+(including omega|->-omega) into the collision term of
the pushforward identity.
(b) Two roots: a contact between them is one
unordered pair and no fresh label; deleting it gives two one-root means and
@@ -1931,7 +1931,7 @@ concatenation is the empty-top-slab no-contact term, preserved by the
absolutely convergent tree sums. Updates are deterministic pushforwards,
giving the pushforward identity.
-The pre/post substitution with omegamapsto-omega preserves
+The pre/post substitution with omega|->-omega preserves
B(v-v_*,omega) d v d v_* d omega, as does particle interchange with
the normal reversed. Hence the normalization in the Boltzmann equation gives
@@ -1965,8 +1965,8 @@ and at an update G pushes forward in both slots.
the G evolution is needed on the spaces
-mathcal V_a=calV_a={Phi=c y+h:cin R, hin E_a^0},
-|Phi|_{mathcal V_a}calV_a=|c|+|h|_a .
+V_a={Phi=c y+h:cin R, hin E_a^0},
+|Phi|_{V_a}|c|+|h|_a .
Let alpha>0 be a common exponent in the integrated and conditional
Gaussian variation bounds for F,J, and choose 0<a<b<beta with
@@ -1982,7 +1982,7 @@ At pulse t_j, backward substitution sends c y+h to c y+h+cphi_j.
Use the backward propagator in this possibly nonsmooth class: the Gaussian
variation bounds make G_t a uniformly bounded bilinear form on
-mathcal V_b,calV_b, and the transport mild identity extends from smooth tests
+V_b, and the transport mild identity extends from smooth tests
by local uniform approximation with a small Gaussian margin; the unmarked
collision operator is K_t, so the linearized operator estimate applies. Iterating
the bilinear mild identity chooses one of two slots per collision factor;
@@ -2020,10 +2020,10 @@ and all centered moments converge to the Gaussian ones.
Proof.
Put S^ε=sum_iy_i(T). Since |S^ε|le M_{obs}N and
-mathcal Z_εcalZ_εle e^mu, S^ε has exponential moments for each fixed
+Z_εcalZ_εle e^mu, S^ε has exponential moments for each fixed
ε. The exact factorial-to-ordinary identity at the final marked time is
-kappa_k(S^ε)=sum_{piinmathcal P([k])}calP([k])mu^{|pi|}
+kappa_k(S^ε)=sum_{piinP([k])}calP([k])mu^{|pi|}
int c_{|pi|}^ε( d xi_1... d xi_{|pi|})prod_{Binpi}y_B^{|B|}
Directly: write
@@ -2206,14 +2206,14 @@ power. With no exponential growth M_{top}to1.
Lemma (Causal order and relative speeds). In a slab with q_i ordinary atoms, retain the partial order induced by the
full graph (including chains through exceptional atoms), and let
-mathcal T_icalT_i be its time-order region. On the original support with total
+T_i be its time-order region. On the original support with total
birth energy le P, an empty slab contributing one,
-|mathcal T_i|calT_i|supprod_{a ordinary in i}|v_a-v_{*a}|
+|T_i|calT_i|supprod_{a ordinary in i}|v_a-v_{*a}|
lefrac{tau^{q_i}(2q_iP)^{q_i/2}}{q_i!}
Proof.
-With e_i the number of linear extensions, |mathcal T_i|=e_icalT_i|=e_itau^{q_i}/q_i!.
+With e_i the number of linear extensions, |T_i|=e_icalT_i|=e_itau^{q_i}/q_i!.
An antichain of ordinary vertices occupies disjoint incoming pairs on a
common causal slice: place all strict ancestors first in a total extension;
no two vertices share a line; energy conservation bounds the sum of their
@@ -2231,11 +2231,11 @@ Suppose every exceptional kernel K_ell(h, d z) has mass at most
H_ell, uniformly in earlier-cut data (every restriction needed for that
bound included, zero outside the admissibility domain). Remove the a_j
from the ordinary measures to get a positive measure nu. If E is a set
-of complete histories and mathcal DcalD a measurable set containing its
+of complete histories and D a measurable set containing its
projection onto θ, then
nu(E)le(prod_ell H_ell)
-int_{mathcal D}calDprod_{j=1}^q d t_j d omega_j d w_j
+int_{D}{j=1}^q d t_j d omega_j d w_j
The same holds if the density after division by a_j is merely bounded by
one.
@@ -2248,7 +2248,7 @@ the surviving mass by prod_ell H_ell, and intervening deterministic
substitutions have norm at most one on bounded functions. Nonnegative
Fubini represents nu as the product in θ times these
kernels, and majorizing the indicator of E by
- 1 _{mathcal D}calD(θ) proves the extraction inequality.
+ 1 _{D}θ) proves the extraction inequality.
Exhaustion removes the boxes. Equivalently, restrict ordinary relative
speeds away from zero and bound a rectangle by its product volume; cover a
compact containing set by products of rectangles and take the infimum;
@@ -2343,7 +2343,7 @@ strongly close: |t_2-t_1|leε^{9/10},
(i) a primitive strongly close pair cut freely as one piece has mass at most
-CR^Cmuε^{9/10+3cdot4/5}=CR^Cε^{13/10};
+CR^Cmuε^{9/10+3·4/5}=CR^Cε^{13/10};
(ii) without any strongly close primitive pair, each type-A pair used by a
sequence has conditional mass le CR^Cε^{1/20}, or splits into a
@@ -2380,9 +2380,9 @@ b the total atoms and q the ordinary ones, q=b-O_K(1).
Lemma (Total birth energy bounds the velocity domain). Fix a cutoff E_{cut}=P_ε and, as in
the energy tail, use the local kernels at the scopes where
the cutting alternatives and the amplified cut retain them. Then
-for a witness with ordinary projection within the containing set mathcal DcalD
+for a witness with ordinary projection within the containing set D
(energy le P_ε), there is a compact measurable set inside
-mathcal D,calD, of volume
+D, of volume
leprod_i(C(L)/(q_i))^{q_i}(1+P_ε/q_i)^{3q_i/2}
@@ -2408,12 +2408,12 @@ zeta in a changes one chronological velocity by lesqrt szeta and
the observations by le C√(qs)zeta. Choose c_s so simultaneous
error is le1. The allowed W lie in ∪_{a_0}
(d(omega,a_0)+A(omega)B_{3n}(√(P_ε))+B_{3q}(1)), and the Minkowski
-sum is in sqrt2(P_ε AA^{mathsf T}sfT+Id_m)^{1/2}B_m(1) for m=3q
-(support function: √(P_ε)|A^{mathsf T}sfTxi|+|xi|le
-sqrt{2(P_ε|A^{mathsf T}sfTxi|^2+|xi|^2)}). Combining the trace bound
+sum is in sqrt2(P_ε AA^{T}+Id_m)^{1/2}B_m(1) for m=3q
+(support function: √(P_ε)|A^{T}|+|xi|le
+sqrt{2(P_ε|A^{T}|^2+|xi|^2)}). Combining the trace bound
with AM-GM on eigenvalues and vol(B_m(1))le(C/sqrt m)^m,
-2^{m/2}vol(B_m(1))det(Id_m+P_ε AA^{mathsf T}sfT)^{1/2}
+2^{m/2}vol(B_m(1))det(Id_m+P_ε AA^{T}{1/2}
le(C√((1+P_ε)/q))^{3q}.
The identity enlargement handles small singular values without an inverse
@@ -2573,7 +2573,7 @@ one, then the moment control of the difference off the compact). Therefore
the true centered fields converge in distribution to the same Gaussian
vector. Sending the finite list over a countable dense family and using the
variance bound the field bound upgrades this to the map
-tmapstozeta_t(phi) in joint law.
+t|->zeta_t(phi) in joint law.
Weak equation. The limiting field inherits the fluctuating equation from the
isonormal construction, so its law is the one prescribed. The