Introduction
In this article, we consider the parabolic Anderson model in dimension d = 1 d=1 d = 1
{ ∂ u ∂ t ( t , x ) = 1 2 ∂ 2 u ∂ x 2 ( t , x ) + u ( t , x ) X ˙ ( x ) , t > 0 , x ∈ R , u ( 0 , x ) = 1 , x ∈ R . (1) \begin{cases}
\frac{\partial u}{\partial t}(t,x)=\frac{1}{2}\frac{\partial^2u}{\partial x^2}(t,x)+u(t,x)\dot{X}(x), & t>0,\ x\in\mathbb{R},\\
u(0,x)=1, & x\in\mathbb{R}. \tag*{(1)}
\end{cases} { ∂ t ∂ u ( t , x ) = 2 1 ∂ x 2 ∂ 2 u ( t , x ) + u ( t , x ) X ˙ ( x ) , u ( 0 , x ) = 1 , t > 0 , x ∈ R , x ∈ R . (1) and the hyperbolic Anderson model in dimension d ∈ { 1 , 2 } d\in\{1,2\} d ∈ { 1 , 2 }
{ ∂ 2 u ∂ t 2 ( t , x ) = Δ u ( t , x ) + u ( t , x ) X ˙ ( x ) , t > 0 , x ∈ R d , u ( 0 , x ) = 1 , ∂ u ∂ t ( 0 , x ) = 0 , x ∈ R d . (2) \begin{cases}
\frac{\partial^2u}{\partial t^2}(t,x)=\Delta u(t,x)+u(t,x)\dot{X}(x), & t>0,\ x\in\mathbb{R}^d,\\
u(0,x)=1,\quad\frac{\partial u}{\partial t}(0,x)=0, & x\in\mathbb{R}^d. \tag*{(2)}
\end{cases} { ∂ t 2 ∂ 2 u ( t , x ) = Δ u ( t , x ) + u ( t , x ) X ˙ ( x ) , u ( 0 , x ) = 1 , ∂ t ∂ u ( 0 , x ) = 0 , t > 0 , x ∈ R d , x ∈ R d . (2) driven by the (time-independent) Lévy colored noise introduced in the recent preprint [4 ] . The goal is to prove a central limit theorem (as R → ∞ R\to\infty R → ∞ spatial average of the solution:
F R ( t ) = ∫ B R ( u ( t , x ) − 1 ) d x , (3) F_R(t)=\int_{B_R}\bigl(u(t,x)-1\bigr)\,dx, \tag*{(3)} F R ( t ) = ∫ B R ( u ( t , x ) − 1 ) d x , (3) where B R = { x ∈ R d ; ∣ x ∣ ≤ R } B_R=\{x\in\mathbb{R}^d;|x|\le R\} B R = { x ∈ R d ; ∣ x ∣ ≤ R } ∣ ⋅ ∣ |\cdot| ∣ ⋅ ∣ σ R 2 ( t ) = Var ( F R ( t ) ) \sigma_R^2(t)=\operatorname{Var}(F_R(t)) σ R 2 ( t ) = Var ( F R ( t ))
This problem has received a lot of attention in the recent years, especially in the case of the Gaussian noise, starting with the seminal article [14 ] . We refer the reader to [9 , 10 , 15 , 17 , 18 , 19 ] for a sample of relevant references. These references rely on a combination of Malliavin calculus with Stein’s method, which yields the precise rate of convergence in the central limit theorem, a result which is now called the Quantitative Central Limit Theorem (QCLT). The major tool is a key estimate for the p p p H ∈ ( 0 , 1 ) H\in(0,1) H ∈ ( 0 , 1 ) H = 1 / 2 H=1/2 H = 1/2 H = 1 H=1 H = 1
Recently, the QCLT problem has been studied also for the spatial average of the solution of a stochastic partial differential equation (SPDE) driven by a finite-variance Lévy noise, using Malliavin calculus with respect to the underlying Poisson random measure of the noise. References [7 , 8 ] studied the case of the Lévy white noise, while [3 ] treated the Lévy colored noise. In the present article, we continue these investigations in the case of the time-independent Lévy noise, which may be white or colored in space. This case introduces new challenges which we explain below.
First, defining an Itô integral with respect to the time-independent noise (Gaussian or Lévy) is a delicate issue. In the recent article [1 ] , in dimension d = 1 d=1 d = 1 L L L ( F x ) x ∈ R (\mathcal{F}_x)_{x\in\mathbb{R}} ( F x ) x ∈ R L u ( t , x ) = u ( t , x ) L ( x ) \mathcal{L}u(t,x)=u(t,x)\mathcal{L}(x) L u ( t , x ) = u ( t , x ) L ( x ) u ( 0 , x ) = 1 u(0,x)=1 u ( 0 , x ) = 1 L \mathcal{L} L
u n + 1 ( t , x ) = 1 + ∫ 0 t ∫ R G t − s ( x − y ) u n ( s , y ) L ( d y ) d s ( u 0 = 1 ) , u_{n+1}(t,x)=1+\int_0^t\int_{\mathbb{R}}G_{t-s}(x-y)u_n(s,y)L(dy)ds\quad(u_0=1), u n + 1 ( t , x ) = 1 + ∫ 0 t ∫ R G t − s ( x − y ) u n ( s , y ) L ( d y ) d s ( u 0 = 1 ) , since the integrand is not F y \mathcal{F}_y F y u 1 ( t , x ) = 1 + 1 2 L ( ( x − t + s , x + t − s ) ) u_1(t,x)=1+\frac{1}{2}L((x-t+s,x+t-s)) u 1 ( t , x ) = 1 + 2 1 L (( x − t + s , x + t − s )) F x + t − s \mathcal{F}_{x+t-s} F x + t − s
Secondly, the traditional methods for proving a QCLT in the Gaussian case do not work in the case of the time-independent Lévy noise. The major problem is that, in the absence of hypercontractivity and Burkholder-Davis-Gundy inequality, there are no techniques for handling p p p [20 ] gives a powerful version of the second-order Poincaré inequality for the Poisson case, which has been applied in [7 ] for proving the QCLT for the wave equation driven by Lévy white noise. But applying this inequality requires an estimate for the p p p
We continue now by describing our framework. We start with the definition of the noise. We let S ( R d ) \mathcal{S}(\mathbb{R}^d) S ( R d ) R d \mathbb{R}^d R d S ′ ( R d ) \mathcal{S}'(\mathbb{R}^d) S ′ ( R d ) Lévy colored noise is the collection X = { X ( φ ) ; φ ∈ S ( R d ) } X=\{X(\varphi);\varphi\in\mathcal{S}(\mathbb{R}^d)\} X = { X ( φ ) ; φ ∈ S ( R d )}
X ( φ ) = L ( φ ∗ κ ) , X(\varphi)=L(\varphi\ast\kappa), X ( φ ) = L ( φ ∗ κ ) , where κ \kappa κ L L L Lévy white noise given by:
L ( ϕ ) = ∫ R d × R 0 ϕ ( x ) z N ^ ( d x , d z ) for all ϕ ∈ L 2 ( R d ) . (4) L(\phi)=\int_{\mathbb{R}^d\times\mathbb{R}_0}\phi(x)z\widehat{N}(dx,dz)\quad\text{for all }\phi\in L^2(\mathbb{R}^d). \tag*{(4)} L ( ϕ ) = ∫ R d × R 0 ϕ ( x ) z N ( d x , d z ) for all ϕ ∈ L 2 ( R d ) . (4) Here N N N R d × R 0 \mathbb{R}^d\times\mathbb{R}_0 R d × R 0 m = Leb × ν \mathbf{m}=\operatorname{Leb}\times\nu m = Leb × ν ( Ω , F , P ) (\Omega,\mathcal{F},\mathbb{P}) ( Ω , F , P ) N ^ ( F ) = N ( F ) − m ( F ) \widehat{N}(F)=N(F)-\mathbf{m}(F) N ( F ) = N ( F ) − m ( F ) N N N R 0 : = R \ { 0 } \mathbb{R}_0:=\mathbb{R}\backslash\{0\} R 0 := R \ { 0 } d ( x , y ) = ∣ x − 1 − y − 1 ∣ d(x,y)=|x^{-1}-y^{-1}| d ( x , y ) = ∣ x − 1 − y − 1 ∣
In this article, we assume that κ ∈ L 1 ( R d ) \kappa\in L^1(\mathbb{R}^d) κ ∈ L 1 ( R d ) d = 1 d=1 d = 1 κ = δ 0 \kappa=\delta_0 κ = δ 0 X = L X=L X = L ν \nu ν
m 2 : = ∫ R 0 ∣ z ∣ 2 ν ( d z ) < ∞ . (5) m_2:=\int_{\mathbb{R}_0}|z|^2\nu(dz)<\infty. \tag*{(5)} m 2 := ∫ R 0 ∣ z ∣ 2 ν ( d z ) < ∞. (5) If κ ∈ L 1 ( R d ) \kappa\in L^1(\mathbb{R}^d) κ ∈ L 1 ( R d ) X ( φ ) X(\varphi) X ( φ ) φ ∈ L 2 ( R d ) \varphi\in L^2(\mathbb{R}^d) φ ∈ L 2 ( R d ) φ ∗ κ ∈ L 2 ( R d ) \varphi\ast\kappa\in L^2(\mathbb{R}^d) φ ∗ κ ∈ L 2 ( R d ) f = κ ∗ κ ~ f=\kappa\ast\widetilde{\kappa} f = κ ∗ κ κ ~ ( x ) = κ ( − x ) \widetilde{\kappa}(x)=\kappa(-x) κ ( x ) = κ ( − x ) x ∈ R d x\in\mathbb{R}^d x ∈ R d φ , ψ ∈ S ( R d ) \varphi,\psi\in\mathcal{S}(\mathbb{R}^d) φ , ψ ∈ S ( R d )
⟨ φ , ψ ⟩ 0 : = ∫ R d ∫ R d φ ( x ) ψ ( y ) f ( x − y ) d x d y = ∫ R d F φ ( ξ ) F ψ ( ξ ) μ ( d ξ ) , (6) \langle\varphi,\psi\rangle_0:=\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}\varphi(x)\psi(y)f(x-y)\,dx\,dy=\int_{\mathbb{R}^d}\mathcal{F}\varphi(\xi)\mathcal{F}\psi(\xi)\,\mu(d\xi), \tag*{(6)} ⟨ φ , ψ ⟩ 0 := ∫ R d ∫ R d φ ( x ) ψ ( y ) f ( x − y ) d x d y = ∫ R d F φ ( ξ ) F ψ ( ξ ) μ ( d ξ ) , (6) where μ ( d ξ ) = 1 ( 2 π ) d ∣ F κ ( ξ ) ∣ 2 d ξ \mu(d\xi)=\frac{1}{(2\pi)^d}|\mathcal{F}\kappa(\xi)|^2d\xi μ ( d ξ ) = ( 2 π ) d 1 ∣ F κ ( ξ ) ∣ 2 d ξ F φ ( x ) = ∫ R d e − i ξ ⋅ x φ ( x ) d x \mathcal{F}\varphi(x)=\int_{\mathbb{R}^d}e^{-i\xi\cdot x}\varphi(x)\,dx F φ ( x ) = ∫ R d e − i ξ ⋅ x φ ( x ) d x φ ∈ L 1 ( R d ) \varphi\in L^1(\mathbb{R}^d) φ ∈ L 1 ( R d ) ξ ⋅ x \xi\cdot x ξ ⋅ x R d \mathbb{R}^d R d φ , ψ ∈ S ( R d ) \varphi,\psi\in\mathcal{S}(\mathbb{R}^d) φ , ψ ∈ S ( R d )
E [ X ( φ ) X ( ψ ) ] = m 2 ⟨ φ ∗ κ , ψ ∗ κ ⟩ L 2 ( R d ) = m 2 ⟨ φ , ψ ⟩ 0 . \mathbb{E}[X(\varphi)X(\psi)]=m_2\langle\varphi\ast\kappa,\psi\ast\kappa\rangle_{L^2(\mathbb{R}^d)}=m_2\langle\varphi,\psi\rangle_0. E [ X ( φ ) X ( ψ )] = m 2 ⟨ φ ∗ κ , ψ ∗ κ ⟩ L 2 ( R d ) = m 2 ⟨ φ , ψ ⟩ 0 . Intuitively, the solution of (1) should satisfy:
u ( t , x ) = 1 + ∫ 0 t ∫ R d G t − s ( x − y ) u ( s , y ) X ( d y ) d s , (7) u(t,x)=1+\int_0^t\int_{\mathbb{R}^d}G_{t-s}(x-y)u(s,y)X(dy)\,ds, \tag*{(7)} u ( t , x ) = 1 + ∫ 0 t ∫ R d G t − s ( x − y ) u ( s , y ) X ( d y ) d s , (7) where G t G_t G t
G t ( x ) = 1 ( 2 π t ) d / 2 exp ( − ∣ x ∣ 2 2 t ) , G_t(x) = \frac{1}{(2\pi t)^{d/2}} \exp\left(-\frac{|x|^2}{2t}\right), G t ( x ) = ( 2 π t ) d /2 1 exp ( − 2 t ∣ x ∣ 2 ) , and ∣ ⋅ ∣ |\cdot| ∣ ⋅ ∣ (2) should satisfy (7) , where G t G_t G t
G t ( x ) = 1 2 1 { ∣ x ∣ < t } if d = 1 , G_t(x) = \frac{1}{2}\mathbf{1}_{\{|x|<t\}} \quad\text{if } d=1, G t ( x ) = 2 1 1 { ∣ x ∣ < t } if d = 1 , G t ( x ) = 1 2 π 1 t 2 − ∣ x ∣ 2 1 { ∣ x ∣ < t } if d = 2. G_t(x) = \frac{1}{2\pi}\frac{1}{\sqrt{t^2-|x|^2}}\mathbf{1}_{\{|x|<t\}} \quad\text{if } d=2. G t ( x ) = 2 π 1 t 2 − ∣ x ∣ 2 1 1 { ∣ x ∣ < t } if d = 2. The rigorous definition of the solution is given Section 2 below. From [4 ] , it is known that the solutions of (1) and (2) exist, provided that Dalang’s condition holds, i.e.
∫ R d 1 1 + ∣ ξ ∣ 2 μ ( d ξ ) < ∞ . (8) \int_{\mathbb{R}^d} \frac{1}{1+|\xi|^2}\,\mu(\mathrm{d}\xi)<\infty. \tag*{(8)} ∫ R d 1 + ∣ ξ ∣ 2 1 μ ( d ξ ) < ∞. (8) Note that condition (8) always holds in dimension d = 1 d=1 d = 1
We are now ready to state our main results.
Theorem 1.1. Suppose that either one of the following conditions hold:
(i) d = 1 d=1 d = 1 κ = δ 0 \kappa=\delta_0 κ = δ 0 X = L X=L X = L u u u
(ii) d = 1 d=1 d = 1 κ ∈ L 1 ( R ) \kappa\in L^1(\mathbb{R}) κ ∈ L 1 ( R ) u u u
(iii) d = 2 d=2 d = 2 κ ∈ L 1 ( R 2 ) \kappa\in L^1(\mathbb{R}^2) κ ∈ L 1 ( R 2 ) u u u
Then { u ( t , x ) } x ∈ R d \{u(t,x)\}_{x\in\mathbb{R}^d} { u ( t , x ) } x ∈ R d
lim R → ∞ 1 R d E [ F R ( t ) F R ( s ) ] = : c ( t , s ) is finite . (9) \lim_{R\to\infty}\frac{1}{R^d}\mathbb{E}[F_R(t)F_R(s)]=:c(t,s)\quad\text{is finite}. \tag*{(9)} R → ∞ lim R d 1 E [ F R ( t ) F R ( s )] =: c ( t , s ) is finite . (9) In particular, σ R 2 ( t ) ∼ c ( t ) R d \sigma_R^2(t)\sim c(t)R^d σ R 2 ( t ) ∼ c ( t ) R d R → ∞ R\to\infty R → ∞ c ( t ) = c ( t , t ) c(t)=c(t,t) c ( t ) = c ( t , t )
Theorem 1.2. Suppose that either one of conditions (i)-(iii) from Theorem 1.1 hold. In case (iii), we assume in addition that κ ∈ L ∞ ( R 2 ) \kappa\in L^\infty(\mathbb{R}^2) κ ∈ L ∞ ( R 2 )
F R ( t ) σ R ( t ) → d Z ∼ N ( 0 , 1 ) as R → ∞ , \frac{F_R(t)}{\sigma_R(t)}\xrightarrow{d} Z\sim\mathcal{N}(0,1)\quad\text{as }R\to\infty, σ R ( t ) F R ( t ) d Z ∼ N ( 0 , 1 ) as R → ∞ , where → d \xrightarrow{d} d
Theorem 1.2 is called a “chaotic CLT”, because it is proved by handling separately the projections of F R ( t ) F_R(t) F R ( t ) [13 ] ), we prove that the projection I n ( g n , R ) I_n(g_{n,R}) I n ( g n , R ) n n n R → ∞ R\to\infty R → ∞ I n ( g n , R ) I_n(g_{n,R}) I n ( g n , R ) g n , R ⋆ r ℓ g n , R g_{n,R}\star_r^\ell g_{n,R} g n , R ⋆ r ℓ g n , R
This article is organized as follows. In Section 2, we introduce some preliminaries. Sections 3 and 4 contain the proofs of Theorems 1.1 , respectively 1.2. The appendices contain some auxiliary results.
Preliminaries
In this section, we include some preliminaries regarding the existence of solutions of equations (1) and (2).
Using the relations between ( X , L ) (X,L) ( X , L ) ( L , N ^ ) (L,\widehat{N}) ( L , N )
u ( t , x ) = 1 + ∫ 0 t ∫ R d ( G t − s ( x − ⋅ ) u ( s , ⋅ ) ∗ κ ) ( y ) L ( d y ) d s = 1 + ∫ 0 t ∫ R d ∫ R 0 ( G t − s ( x − ⋅ ) u ( s , ⋅ ) ∗ κ ) ( y ) z N ^ ( d y , d z ) d s \begin{aligned}
u(t,x) &= 1+\int_0^t\int_{\mathbb{R}^d}(G_{t-s}(x-\cdot)u(s,\cdot)*\kappa)(y)L(\mathrm{d}y)\mathrm{d}s \\
&= 1+\int_0^t\int_{\mathbb{R}^d}\int_{\mathbb{R}_0}(G_{t-s}(x-\cdot)u(s,\cdot)*\kappa)(y)z\widehat{N}(\mathrm{d}y,\mathrm{d}z)\mathrm{d}s
\end{aligned} u ( t , x ) = 1 + ∫ 0 t ∫ R d ( G t − s ( x − ⋅ ) u ( s , ⋅ ) ∗ κ ) ( y ) L ( d y ) d s = 1 + ∫ 0 t ∫ R d ∫ R 0 ( G t − s ( x − ⋅ ) u ( s , ⋅ ) ∗ κ ) ( y ) z N ( d y , d z ) d s But there is no stochastic integral with respect to N ^ \widehat{N} N random integrands. Since N N N ( F t ) t ≥ 0 (\mathcal{F}_t)_{t\geq0} ( F t ) t ≥ 0
Malliavin calculus on the Poisson space
In this section, we include some basic material about Malliavin calculus with respect to the compensated Poisson random measure N ^ \widehat{N} N
We recall that H = L 2 ( Z , Z , m ) \mathcal{H}=L^2(Z,\mathcal{Z},\mathfrak{m}) H = L 2 ( Z , Z , m )
( Z , Z , m ) = ( R d × R 0 , B ( R d ) ⊗ B ( R 0 ) , Leb × ν ) . (Z,\mathcal{Z},\mathfrak{m})=(\mathbb{R}^d\times\mathbb{R}_0,\mathcal{B}(\mathbb{R}^d)\otimes\mathcal{B}(\mathbb{R}_0),\operatorname{Leb}\times\nu). ( Z , Z , m ) = ( R d × R 0 , B ( R d ) ⊗ B ( R 0 ) , Leb × ν ) . Then H ⊗ n = L 2 ( Z n , Z n , m n ) \mathcal{H}^{\otimes n}=L^2(Z^n,\mathcal{Z}^n,\mathfrak{m}^n) H ⊗ n = L 2 ( Z n , Z n , m n ) H ⊙ n \mathcal{H}^{\odot n} H ⊙ n H ⊗ n \mathcal{H}^{\otimes n} H ⊗ n
F = E ( F ) + ∑ n ≥ 1 I n ( f n ) , for some f n ∈ H ⊙ n , (10) F=\mathbb{E}(F)+\sum_{n\geq1}I_n(f_n),\qquad\text{for some }f_n\in\mathcal{H}^{\odot n}, \tag*{(10)} F = E ( F ) + n ≥ 1 ∑ I n ( f n ) , for some f n ∈ H ⊙ n , (10) where I n I_n I n N ^ \widehat{N} N L 2 ( Ω ) L^2(\Omega) L 2 ( Ω )
For any f ∈ H ⊗ n f\in\mathcal{H}^{\otimes n} f ∈ H ⊗ n I n ( f ) = I n ( f ~ ) I_n(f)=I_n(\widetilde{f}) I n ( f ) = I n ( f )
E [ I n ( f ) ] = 0 and E ∣ I n ( f ) ∣ 2 = n ! ∥ f ~ ∥ H ⊗ n 2 , \mathbb{E}[I_n(f)]=0\quad\text{and}\quad\mathbb{E}|I_n(f)|^2=n!\|\widetilde{f}\|^2_{\mathcal{H}^{\otimes n}}, E [ I n ( f )] = 0 and E ∣ I n ( f ) ∣ 2 = n ! ∥ f ∥ H ⊗ n 2 , where f ~ \widetilde{f} f f f f
f ~ ( ξ 1 , … , ξ n ) = 1 n ! ∑ ρ ∈ S n f ( ξ ρ ( 1 ) , … , ξ ρ ( n ) ) , for ξ 1 , … , ξ n ∈ Z , \widetilde{f}(\xi_1,\ldots,\xi_n)=\frac{1}{n!}\sum_{\rho\in S_n}f(\xi_{\rho(1)},\ldots,\xi_{\rho(n)}),\qquad\text{for }\xi_1,\ldots,\xi_n\in Z, f ( ξ 1 , … , ξ n ) = n ! 1 ρ ∈ S n ∑ f ( ξ ρ ( 1 ) , … , ξ ρ ( n ) ) , for ξ 1 , … , ξ n ∈ Z , and S n S_n S n 1 , … , n 1,\ldots,n 1 , … , n
D ξ F = ∑ n ≥ 1 n I n − 1 ( f n ( ⋅ , ξ ) ) , for all ξ ∈ Z , D_\xi F=\sum_{n\geq1}nI_{n-1}(f_n(\cdot,\xi)),\qquad\text{for all }\xi\in Z, D ξ F = n ≥ 1 ∑ n I n − 1 ( f n ( ⋅ , ξ )) , for all ξ ∈ Z , provided that
E ∥ D F ∥ H 2 = ∑ n ≥ 1 n n ! ∥ f ~ n ∥ H ⊗ n 2 < ∞ . \mathbb{E}\lVert DF\rVert_{\mathfrak{H}}^2=\sum_{n\geq1}n n!\lVert\widetilde{f}_n\rVert_{\mathfrak{H}^{\otimes n}}^2<\infty. E ∥ D F ∥ H 2 = n ≥ 1 ∑ nn ! ∥ f n ∥ H ⊗ n 2 < ∞. In this case, we write F ∈ Dom ( D ) F\in\operatorname{Dom}(D) F ∈ Dom ( D )
We denote by δ : Dom ( δ ) → L 2 ( Ω ) \delta:\operatorname{Dom}(\delta)\to L^2(\Omega) δ : Dom ( δ ) → L 2 ( Ω ) D D D Dom ( δ ) \operatorname{Dom}(\delta) Dom ( δ ) V ∈ L 2 ( Ω ; H ) V\in L^2(\Omega;\mathfrak{H}) V ∈ L 2 ( Ω ; H ) C = C V > 0 C=C_V>0 C = C V > 0 V V V
∣ E ⟨ D F , V ⟩ H ∣ ≤ C ∥ F ∥ 2 \left|\mathbb{E}\langle DF,V\rangle_{\mathfrak{H}}\right|\leq C\lVert F\rVert_2 ∣ E ⟨ D F , V ⟩ H ∣ ≤ C ∥ F ∥ 2 for any F ∈ Dom ( D ) F\in\operatorname{Dom}(D) F ∈ Dom ( D )
We say that δ ( V ) \delta(V) δ ( V ) Skorohod integral of V V V N ^ \widehat{N} N
δ ( V ) = ∫ R d ∫ R 0 V ( x , z ) N ^ ( d x , d z ) . \delta(V)=\int_{\mathbb{R}^d}\int_{\mathbb{R}_0}V(x,z)\widehat{N}(\mathrm{d}x,\mathrm{d}z). δ ( V ) = ∫ R d ∫ R 0 V ( x , z ) N ( d x , d z ) . By duality, for any V ∈ Dom ( δ ) V\in\operatorname{Dom}(\delta) V ∈ Dom ( δ )
E ⟨ D F , V ⟩ H = E [ F δ ( V ) ] \mathbb{E}\langle DF,V\rangle_{\mathfrak{H}}=\mathbb{E}[F\delta(V)] E ⟨ D F , V ⟩ H = E [ F δ ( V )] for any F ∈ Dom ( D ) F\in\operatorname{Dom}(D) F ∈ Dom ( D )
Existence of solution and chaos expansions
In this section, we introduce the definition of solution of equation (1) or (2) . We refer the reader to Section 4.2 of [4 ] for the heuristics behind this definition.
We introduce some notation. For any ( t , x ) ∈ R + × R d (t,x)\in\mathbb{R}_+\times\mathbb{R}^d ( t , x ) ∈ R + × R d x 1 , … , x n ∈ R d x_1,\ldots,x_n\in\mathbb{R}^d x 1 , … , x n ∈ R d
f n ( t 1 , x 1 , … , t n , x n , t , x ) = G t − t n ( x − x n ) G t n − t n − 1 ( x n − x n − 1 ) … G t 2 − t 1 ( x 2 − x 1 ) 1 T n ( t ) ( t ) f_n(t_1,x_1,\ldots,t_n,x_n,t,x)=G_{t-t_n}(x-x_n)G_{t_n-t_{n-1}}(x_n-x_{n-1})\ldots G_{t_2-t_1}(x_2-x_1)\mathbf{1}_{T_n(t)}(t) f n ( t 1 , x 1 , … , t n , x n , t , x ) = G t − t n ( x − x n ) G t n − t n − 1 ( x n − x n − 1 ) … G t 2 − t 1 ( x 2 − x 1 ) 1 T n ( t ) ( t ) f n ( x , x ; t ) = ∫ T n ( t ) f n ( t 1 , x 1 , … , t n , x n , t , x ) d t , (11) f_n(x,x;t)=\int_{T_n(t)}f_n(t_1,x_1,\ldots,t_n,x_n,t,x)\mathrm{d}t, \tag*{(11)} f n ( x , x ; t ) = ∫ T n ( t ) f n ( t 1 , x 1 , … , t n , x n , t , x ) d t , (11) where x = ( x 1 , … , x n ) \mathbf{x}=(x_1,\ldots,x_n) x = ( x 1 , … , x n ) t = ( t 1 , … , t n ) \mathbf{t}=(t_1,\ldots,t_n) t = ( t 1 , … , t n ) T n ( t ) = { ( t 1 , … , t n ) ∈ [ 0 , t ] n ; t 1 < … < t n } T_n(t)=\{(t_1,\ldots,t_n)\in[0,t]^n;t_1<\ldots<t_n\} T n ( t ) = {( t 1 , … , t n ) ∈ [ 0 , t ] n ; t 1 < … < t n }
φ n ( x , x ; t ) = ∫ T n ( t ) ( f n ( t 1 , … , t n , ⋅ , t , x ) ∗ κ ⊗ n ) ( x ) d t = ( f n ( ⋅ , x ; t ) ∗ κ ⊗ n ) ( x ) . (12) \varphi_n(\mathbf{x},x;t)=\int_{T_n(t)}(f_n(t_1,\ldots,t_n,\cdot,t,x)*\kappa^{\otimes n})(\mathbf{x})\mathrm{d}t=(f_n(\cdot,x;t)*\kappa^{\otimes n})(\mathbf{x}). \tag*{(12)} φ n ( x , x ; t ) = ∫ T n ( t ) ( f n ( t 1 , … , t n , ⋅ , t , x ) ∗ κ ⊗ n ) ( x ) d t = ( f n ( ⋅ , x ; t ) ∗ κ ⊗ n ) ( x ) . (12) For any n ≥ 1 n\geq1 n ≥ 1 ( x 1 , z 1 ) , … , ( x n , z n ) ∈ Z (x_1,z_1),\ldots,(x_n,z_n)\in\mathbb{Z} ( x 1 , z 1 ) , … , ( x n , z n ) ∈ Z
f n ∗ ( x 1 , z 1 , … , x n , z n , x ; t ) : = ∏ i = 1 n z i φ n ( x 1 , … , x n , x ; t ) , (13) f_n^*(x_1,z_1,\ldots,x_n,z_n,x;t):=\prod_{i=1}^n z_i\varphi_n(x_1,\ldots,x_n,x;t), \tag*{(13)} f n ∗ ( x 1 , z 1 , … , x n , z n , x ; t ) := i = 1 ∏ n z i φ n ( x 1 , … , x n , x ; t ) , (13) with the convention that φ n ( ⋅ , x ; t ) = f n ( ⋅ , x ; t ) \varphi_n(\cdot,x;t)=f_n(\cdot,x;t) φ n ( ⋅ , x ; t ) = f n ( ⋅ , x ; t ) κ = δ 0 \kappa=\delta_0 κ = δ 0
We have the following definition.
Definition 2.1. The process { u ( t , x ) ; t ≥ 0 , x ∈ R d } \{u(t,x);t\geq0,x\in\mathbb{R}^d\} { u ( t , x ) ; t ≥ 0 , x ∈ R d } E ∣ u ( t , x ) ∣ 2 < ∞ \mathbb{E}|u(t,x)|^2<\infty E ∣ u ( t , x ) ∣ 2 < ∞ ( t , x ) ∈ R + × R d (t,x)\in\mathbb{R}_+\times\mathbb{R}^d ( t , x ) ∈ R + × R d
u ( t , x ) = 1 + ∑ n ≥ 1 I n ( f n ∗ ( ⋅ , x ; t ) ) , (14) u(t,x)=1+\sum_{n\geq1}I_n(f_n^*(\cdot,x;t)), \tag*{(14)} u ( t , x ) = 1 + n ≥ 1 ∑ I n ( f n ∗ ( ⋅ , x ; t )) , (14) is the (Skorohod) solution of equation (1) (or (2) ), provided that f n ∗ ( ⋅ , x ; t ) ∈ H ⊗ n f_n^*(\cdot,x;t)\in\mathcal{H}^{\otimes n} f n ∗ ( ⋅ , x ; t ) ∈ H ⊗ n ( t , x ) ∈ R + × R d (t,x)\in\mathbb{R}_+\times\mathbb{R}^d ( t , x ) ∈ R + × R d n ≥ 1 n\geq1 n ≥ 1 L 2 ( Ω ) L^2(\Omega) L 2 ( Ω )
We recall the following result; see Corollary 4.6 of [4 ] .
Theorem 2.2. If Dalang’s condition (8) holds, then equations (1) and (2) have unique solutions.
Recall the definition (3) of the spatial integral F R ( t ) F_R(t) F R ( t ) F R ( t ) F_R(t) F R ( t )
F R ( t ) = ∑ n ≥ 1 I n ( g n , R ( ⋅ , t ) ) , where g n , R ( ⋅ , t ) = ∫ B R f n ∗ ( ⋅ , x ; t ) d x . F_R(t)=\sum_{n\geq1}I_n(g_{n,R}(\cdot,t)),\quad\text{where}\quad g_{n,R}(\cdot,t)=\int_{B_R}f_n^*(\cdot,x;t)\,\mathrm{d}x. F R ( t ) = n ≥ 1 ∑ I n ( g n , R ( ⋅ , t )) , where g n , R ( ⋅ , t ) = ∫ B R f n ∗ ( ⋅ , x ; t ) d x . Using definition (13) of f n ∗ ( ⋅ , x ; t ) f_n^*(\cdot,x;t) f n ∗ ( ⋅ , x ; t )
g n , R ( x 1 , z 1 , … , x n , z n , t ) = ∏ i = 1 n z i Q n , R ( x 1 , … , x n ; t ) (15) g_{n,R}(x_1,z_1,\ldots,x_n,z_n,t)=\prod_{i=1}^{n}z_i Q_{n,R}(x_1,\ldots,x_n;t) \tag*{(15)} g n , R ( x 1 , z 1 , … , x n , z n , t ) = i = 1 ∏ n z i Q n , R ( x 1 , … , x n ; t ) (15) where
Q n , R ( x 1 , … , x n ; t ) : = ∫ B R φ n ( x 1 , … , x n , x ; t ) d x . (16) Q_{n,R}(x_1,\ldots,x_n;t):=\int_{B_R}\varphi_n(x_1,\ldots,x_n,x;t)\,\mathrm{d}x. \tag*{(16)} Q n , R ( x 1 , … , x n ; t ) := ∫ B R φ n ( x 1 , … , x n , x ; t ) d x . (16) Limiting covariance
In this section, we give the proof of Theorem 1.1.
Similarly to Lemma 4.1 of [7 ] , it can be proved that for any ( t 1 , x 1 ) , … , ( t k , x k ) ∈ R + × R d (t_1,x_1),\ldots,(t_k,x_k)\in\mathbb{R}_+\times\mathbb{R}^d ( t 1 , x 1 ) , … , ( t k , x k ) ∈ R + × R d z ∈ R d z\in\mathbb{R}^d z ∈ R d
( u ( t 1 , x 1 + z ) , … , u ( t k , x k + z ) ) = d ( u ( t 1 , x 1 ) , … , u ( t k , x k ) ) , \left(u(t_1,x_1+z),\ldots,u(t_k,x_k+z)\right)\mathrel{\overset{d}{=}}\left(u(t_1,x_1),\ldots,u(t_k,x_k)\right), ( u ( t 1 , x 1 + z ) , … , u ( t k , x k + z ) ) = d ( u ( t 1 , x 1 ) , … , u ( t k , x k ) ) , where = d \overset{d}{=} = d { u ( t , x ) } x ∈ R d \{u(t,x)\}_{x\in\mathbb{R}^d} { u ( t , x ) } x ∈ R d t , s > 0 t,s>0 t , s > 0
Cov ( u ( t , x ) , u ( s , y ) ) = : ρ t , s ( x − y ) \operatorname{Cov}(u(t,x),u(s,y))=:\rho_{t,s}(x-y) Cov ( u ( t , x ) , u ( s , y )) =: ρ t , s ( x − y ) depends only on x − y x-y x − y
Lemma 3.1. Suppose that either one of conditions ( i ) (i) ( i ) ( i i i ) (iii) ( iii )
∫ R d ρ t , s ( x ) d x < ∞ for any t , s > 0. (17) \int_{\mathbb{R}^d}\rho_{t,s}(x)\,\mathrm{d}x<\infty\quad\text{for any }t,s>0. \tag*{(17)} ∫ R d ρ t , s ( x ) d x < ∞ for any t , s > 0. (17) Proof. As in Remark 4.7 of [2 ] , we make a connection with the following Gaussian model:
{ L U ( t , x ) = m 2 U ( t , x ) W ˙ ( x ) , t > 0 , x ∈ R d ( d ≤ 2 ) , U ( 0 , x ) = 1 , ∂ U ∂ t ( 0 , x ) = 0 , x ∈ R d . (18) \begin{cases}
\mathcal{L}U(t,x)=\sqrt{m_2}\,U(t,x)\dot{W}(x),\quad t>0,\ x\in\mathbb{R}^d\quad(d\leq2),\\
U(0,x)=1,\quad\dfrac{\partial U}{\partial t}(0,x)=0,\quad x\in\mathbb{R}^d. \tag*{(18)}
\end{cases} ⎩ ⎨ ⎧ L U ( t , x ) = m 2 U ( t , x ) W ˙ ( x ) , t > 0 , x ∈ R d ( d ≤ 2 ) , U ( 0 , x ) = 1 , ∂ t ∂ U ( 0 , x ) = 0 , x ∈ R d . (18) where m 2 m_2 m 2 L \mathcal{L} L k ∈ L 1 ( R d ) k\in L^1(\mathbb{R}^d) k ∈ L 1 ( R d ) W W W { W ( φ ) ; φ ∈ P 0 ( R d ) } \{W(\varphi);\varphi\in\mathcal{P}_0(\mathbb{R}^d)\} { W ( φ ) ; φ ∈ P 0 ( R d )}
E [ W ( φ ) W ( ψ ) ] = ⟨ φ , ψ ⟩ 0 , \mathbb{E}[W(\varphi)W(\psi)]=\langle\varphi,\psi\rangle_0, E [ W ( φ ) W ( ψ )] = ⟨ φ , ψ ⟩ 0 , with inner product ⟨ ⋅ , ⋅ ⟩ 0 \langle\cdot,\cdot\rangle_0 ⟨ ⋅ , ⋅ ⟩ 0 (6) . If k = δ 0 k=\delta_0 k = δ 0 W W W { W ( φ ) ; φ ∈ L 2 ( R d ) } \{W(\varphi);\varphi\in L^2(\mathbb{R}^d)\} { W ( φ ) ; φ ∈ L 2 ( R d )}
The solution of (18) satisfies the integral equation:
U ( t , x ) = 1 + ∫ 0 t ∫ R d G t − s ( x − y ) U ( s , y ) W ( δ y ) d s U(t,x)=1+\int_0^t\int_{\mathbb{R}^d}G_{t-s}(x-y)U(s,y)W(\delta y)\,ds U ( t , x ) = 1 + ∫ 0 t ∫ R d G t − s ( x − y ) U ( s , y ) W ( δ y ) d s where the integral is interpreted in the Skorohod sense. It is known that the solution of (18) exists and has the chaos expansion
U ( t , x ) = 1 + ∑ n ≥ 1 m 2 n / 2 I n W ( f ~ n ( ⋅ , x ; t ) ) U(t,x)=1+\sum_{n\geq1}m_2^{n/2}I_n^W(\widetilde{f}_n(\cdot,x;t)) U ( t , x ) = 1 + n ≥ 1 ∑ m 2 n /2 I n W ( f n ( ⋅ , x ; t )) where I n W I_n^W I n W W W W f ~ n ( ⋅ , x ; t ) \widetilde{f}_n(\cdot,x;t) f n ( ⋅ , x ; t ) (11) . See Theorem 3.2 of [6 ] for the heat equation, respectively Theorem 2.1 of [5 ] for the wave equation.
Note that f ~ n ∗ ( x 1 , z 1 , … , x n , z n , x ; t ) = ∏ i = 1 n z i ( f ~ n ( ⋅ , x ; t ) ∗ κ ⊗ ) ( x ) \widetilde{f}_n^*(x_1,z_1,\ldots,x_n,z_n,x;t)=\prod_{i=1}^n z_i(\widetilde{f}_n(\cdot,x;t)*\kappa^\otimes)(x) f n ∗ ( x 1 , z 1 , … , x n , z n , x ; t ) = ∏ i = 1 n z i ( f n ( ⋅ , x ; t ) ∗ κ ⊗ ) ( x )
⟨ f ~ n ∗ ( ⋅ , x ; t ) , f ~ n ∗ ( ⋅ , 0 ; s ) ⟩ H ⊗ n = m 2 n ⟨ f ~ n ( ⋅ , x ; t ) , f ~ n ( ⋅ , 0 ; s ) ⟩ P 0 ( R d ) ⊗ n . \langle\widetilde{f}_n^*(\cdot,x;t),\widetilde{f}_n^*(\cdot,0;s)\rangle_{\mathcal{H}^{\otimes n}}=m_2^n\langle\widetilde{f}_n(\cdot,x;t),\widetilde{f}_n(\cdot,0;s)\rangle_{\mathcal{P}_0(\mathbb{R}^d)^{\otimes n}}. ⟨ f n ∗ ( ⋅ , x ; t ) , f n ∗ ( ⋅ , 0 ; s ) ⟩ H ⊗ n = m 2 n ⟨ f n ( ⋅ , x ; t ) , f n ( ⋅ , 0 ; s ) ⟩ P 0 ( R d ) ⊗ n . It follows that
ρ t , s ( x ) = E [ ( u ( t , x ) − 1 ) ( u ( s , 0 ) − 1 ) ] = ∑ n ≥ 1 n ! ⟨ f ~ n ∗ ( ⋅ , x ; t ) , f ~ n ∗ ( ⋅ , 0 ; s ) ⟩ H ⊗ n = ∑ n ≥ 1 n ! m 2 n ⟨ f ~ n ( ⋅ , x ; t ) , f ~ n ( ⋅ , 0 ; s ) ⟩ P 0 ( R d ) ⊗ n = E [ ( U ( t , x ) − 1 ) ( U ( s , 0 ) − 1 ) ] = : ρ t , s U ( x ) . \begin{aligned}
\rho_{t,s}(x)=\mathbb{E}[(u(t,x)-1)(u(s,0)-1)]=\sum_{n\geq1}n!\langle\widetilde{f}_n^*(\cdot,x;t),\widetilde{f}_n^*(\cdot,0;s)\rangle_{\mathcal{H}^{\otimes n}} \\
&=\sum_{n\geq1}n!m_2^n\langle\widetilde{f}_n(\cdot,x;t),\widetilde{f}_n(\cdot,0;s)\rangle_{\mathcal{P}_0(\mathbb{R}^d)^{\otimes n}}=\mathbb{E}[(U(t,x)-1)(U(s,0)-1)]=:\rho^U_{t,s}(x).
\end{aligned} ρ t , s ( x ) = E [( u ( t , x ) − 1 ) ( u ( s , 0 ) − 1 )] = n ≥ 1 ∑ n ! ⟨ f n ∗ ( ⋅ , x ; t ) , f n ∗ ( ⋅ , 0 ; s ) ⟩ H ⊗ n = n ≥ 1 ∑ n ! m 2 n ⟨ f n ( ⋅ , x ; t ) , f n ( ⋅ , 0 ; s ) ⟩ P 0 ( R d ) ⊗ n = E [( U ( t , x ) − 1 ) ( U ( s , 0 ) − 1 )] =: ρ t , s U ( x ) . The conclusion follows, since
∫ R d ρ t , s U ( x ) d x < ∞ . \int_{\mathbb{R}^d}\rho^U_{t,s}(x)\,dx<\infty. ∫ R d ρ t , s U ( x ) d x < ∞. See Theorem 1.1 of [6 ] for the heat equation, and Theorem 1.3 of [5 ] for the wave equation.
Lemma 3.2. Suppose that either one of conditions (i)-(iii) in Theorem 1.1 hold. Then,
lim R → ∞ 1 R d E [ F R ( t ) F R ( s ) ] = ω d ∫ R d ρ t , s ( x ) d x = : c ( t , s ) , for any t , s > 0 , (19) \lim_{R\to\infty}\frac{1}{R^d}\mathbb{E}[F_R(t)F_R(s)]=\omega_d\int_{\mathbb{R}^d}\rho_{t,s}(x)\,dx=:c(t,s),\qquad\text{for any }t,s>0, \tag*{(19)} R → ∞ lim R d 1 E [ F R ( t ) F R ( s )] = ω d ∫ R d ρ t , s ( x ) d x =: c ( t , s ) , for any t , s > 0 , (19) where ω d \omega_d ω d B 1 B_1 B 1 R d \mathbb{R}^d R d ω 1 = 2 \omega_1=2 ω 1 = 2 ω 2 = π \omega_2=\pi ω 2 = π
Proof. By the dominated convergence theorem,
1 R d E [ F R ( t ) F R ( s ) ] = 1 R d ∫ B R ∫ B R ρ t , s ( x − y ) d x d y = 1 R d ∫ B 2 R ρ t , s ( z ) Vol ( B R ∩ B R ( − z ) ) d z \frac{1}{R^d}\mathbb{E}[F_R(t)F_R(s)]=\frac{1}{R^d}\int_{B_R}\int_{B_R}\rho_{t,s}(x-y)\,dx\,dy=\frac{1}{R^d}\int_{B_{2R}}\rho_{t,s}(z)\operatorname{Vol}(B_R\cap B_R(-z))\,dz R d 1 E [ F R ( t ) F R ( s )] = R d 1 ∫ B R ∫ B R ρ t , s ( x − y ) d x d y = R d 1 ∫ B 2 R ρ t , s ( z ) Vol ( B R ∩ B R ( − z )) d z ⟶ ω d ∫ R d ρ t , s ( z ) d z , as R → ∞ . \longrightarrow\omega_d\int_{\mathbb{R}^d}\rho_{t,s}(z)\,dz,\qquad\text{as }R\to\infty. ⟶ ω d ∫ R d ρ t , s ( z ) d z , as R → ∞. The application of this theorem is justified by Lemma 3.1.
Proof of Theorem 1.2
In this section, we give the proof of Theorem 1.2 .
First, we study the asymptotic behaviour of the variance of the projection of F R ( t ) F_R(t) F R ( t ) n n n
σ n , R 2 ( t ) : = Var ( I n ( g n , R ( ⋅ , t ) ) ) . \sigma_{n,R}^{2}(t) := \operatorname{Var}\left(I_n(g_{n,R}(\cdot,t))\right). σ n , R 2 ( t ) := Var ( I n ( g n , R ( ⋅ , t )) ) . By orthogonality, σ R 2 ( t ) = ∑ n ≥ 1 σ n , R 2 ( t ) \sigma_R^2(t) = \sum_{n \ge1} \sigma_{n,R}^2(t) σ R 2 ( t ) = ∑ n ≥ 1 σ n , R 2 ( t )
1 R d σ n , R 2 ( t ) = 1 R d ∫ B R ∫ B R ρ t ( n ) ( x − y ) d x d y = 1 R d ∫ B 2 R ρ t ( n ) ( z ) Vol ( B R ∩ B R ( − z ) ) d z (20) \frac{1}{R^d}\sigma_{n,R}^2(t)=\frac{1}{R^d}\int_{B_R}\int_{B_R}\rho_t^{(n)}(x-y)\,\mathrm{d}x\mathrm{d}y=\frac{1}{R^d}\int_{B_{2R}}\rho_t^{(n)}(z)\operatorname{Vol}(B_R\cap B_R(-z))\,\mathrm{d}z \tag*{(20)} R d 1 σ n , R 2 ( t ) = R d 1 ∫ B R ∫ B R ρ t ( n ) ( x − y ) d x d y = R d 1 ∫ B 2 R ρ t ( n ) ( z ) Vol ( B R ∩ B R ( − z )) d z (20) ⟶ ω d ∫ R d ρ t ( n ) ( z ) d z = : c n ( t ) , as R → ∞ , (21) \longrightarrow\omega_d\int_{\mathbb{R}^d}\rho_t^{(n)}(z)\,\mathrm{d}z=:c_n(t),\qquad\text{as }R\to\infty, \tag*{(21)} ⟶ ω d ∫ R d ρ t ( n ) ( z ) d z =: c n ( t ) , as R → ∞ , (21) where
ρ t ( n ) ( x ) = E [ I n ( f ~ n ∗ ( ⋅ , x ; t ) ) I n ( f ~ n ∗ ( ⋅ , 0 ; t ) ) ] = n ! ⟨ f ~ n ∗ ( ⋅ , x ; t ) , f ~ n ∗ ( ⋅ , 0 ; t ) ⟩ H ⊗ n . \rho_t^{(n)}(x)=\mathbb{E}\left[I_n(\widetilde{f}_n^*(\cdot,x;t))I_n(\widetilde{f}_n^*(\cdot,0;t))\right]=n!\left\langle\widetilde{f}_n^*(\cdot,x;t),\widetilde{f}_n^*(\cdot,0;t)\right\rangle_{\mathcal{H}^{\otimes n}}. ρ t ( n ) ( x ) = E [ I n ( f n ∗ ( ⋅ , x ; t )) I n ( f n ∗ ( ⋅ , 0 ; t )) ] = n ! ⟨ f n ∗ ( ⋅ , x ; t ) , f n ∗ ( ⋅ , 0 ; t ) ⟩ H ⊗ n . Therefore, c ( t ) = ∑ n ≥ 1 c n ( t ) c(t)=\sum_{n\ge1}c_n(t) c ( t ) = ∑ n ≥ 1 c n ( t ) c ( t ) = c ( t , t ) c(t)=c(t,t) c ( t ) = c ( t , t )
We will use the following Fourth Moment Theorems on the Poisson Space.
Theorem 4.1 (Corollary 1.3 of [13 ] ). For each n ≥ 1 n\ge1 n ≥ 1 F n = I q n ( f n ) F_n=I_{q_n}(f_n) F n = I q n ( f n ) f n ∈ H ⊗ n f_n\in\mathcal{H}^{\otimes n} f n ∈ H ⊗ n q n ≥ 1 q_n\ge1 q n ≥ 1 E ( F n 2 ) → 1 \mathbb{E}(F_n^2)\to1 E ( F n 2 ) → 1 E ( F n 4 ) → 3 \mathbb{E}(F_n^4)\to3 E ( F n 4 ) → 3
F n → d Z ∼ N ( 0 , 1 ) . F_n\xrightarrow{d}Z\sim N(0,1). F n d Z ∼ N ( 0 , 1 ) . Theorem 4.2 (Corollary 1.8 of [13 ] ). For each n ≥ 1 n\ge1 n ≥ 1 F n = ( F n ( 1 ) , … , F n ( d ) ) F_n=(F_n^{(1)},\ldots,F_n^{(d)}) F n = ( F n ( 1 ) , … , F n ( d ) ) F n ( i ) = I q i ( f n ( i ) ) F_n^{(i)}=I_{q_i}(f_n^{(i)}) F n ( i ) = I q i ( f n ( i ) ) f n ( i ) ∈ H ⊗ n f_n^{(i)}\in\mathcal{H}^{\otimes n} f n ( i ) ∈ H ⊗ n q i ≥ 1 q_i\ge1 q i ≥ 1 i = 1 , … , d i=1,\ldots,d i = 1 , … , d Σ = ( Σ i , j ) 1 ≤ i , j ≤ d \Sigma=(\Sigma_{i,j})_{1\le i,j\le d} Σ = ( Σ i , j ) 1 ≤ i , j ≤ d
(a) Cov ( F n ( i ) , F n ( j ) ) → Σ i , j \operatorname{Cov}(F_n^{(i)},F_n^{(j)})\to\Sigma_{i,j} Cov ( F n ( i ) , F n ( j ) ) → Σ i , j n → ∞ n\to\infty n → ∞ i , j = 1 , … , d i,j=1,\ldots,d i , j = 1 , … , d
(b) E [ ( F n ( i ) ) 4 ] → 3 Σ i , i 2 \mathbb{E}[(F_n^{(i)})^4]\to3\Sigma_{i,i}^2 E [( F n ( i ) ) 4 ] → 3 Σ i , i 2 i = 1 , … , d i=1,\ldots,d i = 1 , … , d
then
F n → d Z ∼ N ( 0 , Σ ) , F_n\xrightarrow{d}Z\sim N(0,\Sigma), F n d Z ∼ N ( 0 , Σ ) , where N ( 0 , Σ ) N(0,\Sigma) N ( 0 , Σ ) 0 0 0 Σ \Sigma Σ
Lemma 4.3. Suppose that either one of conditions (i)-(iii) of Theorem 1.1 hold. In case (iii), we assume in addition that κ ∈ L 2 ( R 2 ) \kappa\in L^2(\mathbb{R}^2) κ ∈ L 2 ( R 2 ) t > 0 t>0 t > 0 n ≥ 1 n\ge1 n ≥ 1
1 σ n , R ( t ) I n ( g n , R ( ⋅ , t ) ) → d N ( 0 , 1 ) as R → ∞ , (22) \frac{1}{\sigma_{n,R}(t)}I_n(g_{n,R}(\cdot,t))\xrightarrow{d}N(0,1)\quad\text{as }R\to\infty, \tag*{(22)} σ n , R ( t ) 1 I n ( g n , R ( ⋅ , t )) d N ( 0 , 1 ) as R → ∞ , (22) or equivalently,
1 R d / 2 I n ( g n , R ( ⋅ , t ) ) → d N ( 0 , c n ( t ) ) as R → ∞ . (23) \frac{1}{R^{d/2}}I_n(g_{n,R}(\cdot,t))\xrightarrow{d}N(0,c_n(t))\quad\text{as }R\to\infty. \tag*{(23)} R d /2 1 I n ( g n , R ( ⋅ , t )) d N ( 0 , c n ( t )) as R → ∞. (23) Proof. It is enough to prove (22). Then (23) will follow, since σ n , R 2 ( t ) ∼ c n ( t ) R d \sigma_{n,R}^{2}(t) \sim c_n(t)R^d σ n , R 2 ( t ) ∼ c n ( t ) R d
We drop t t t
lim R → ∞ E [ I n 4 ( g n , R ) ] σ n , R 4 = 3 , (24) \lim_{R\to\infty}\frac{\mathbb{E}\left[I_n^4(g_{n,R})\right]}{\sigma_{n,R}^4}=3, \tag*{(24)} R → ∞ lim σ n , R 4 E [ I n 4 ( g n , R ) ] = 3 , (24) that is
lim R → ∞ E [ I n 4 ( g n , R ) ] − 3 σ n , R 4 σ n , R 4 = 0. (25) \lim_{R\to\infty}\frac{\mathbb{E}\left[I_n^4(g_{n,R})\right]-3\sigma_{n,R}^4}{\sigma_{n,R}^4}=0. \tag*{(25)} R → ∞ lim σ n , R 4 E [ I n 4 ( g n , R ) ] − 3 σ n , R 4 = 0. (25) We use the fact that I n ( g n , R ) = I n ( g ~ n , R ) I_n(g_{n,R})=I_n(\widetilde{g}_{n,R}) I n ( g n , R ) = I n ( g n , R ) E [ I n 4 ( g ~ n , R ) ] − 3 σ n , R 4 \mathbb{E}[I_n^4(\widetilde{g}_{n,R})]-3\sigma_{n,R}^4 E [ I n 4 ( g n , R )] − 3 σ n , R 4
∥ g ~ n , R ⋆ r ℓ g ~ n , R ∥ H ⊗ ( 2 n − r − ℓ ) 2 \left\|\widetilde{g}_{n,R}\star_r^\ell\widetilde{g}_{n,R}\right\|_{\mathcal{H}^{\otimes(2n-r-\ell)}}^2 g n , R ⋆ r ℓ g n , R H ⊗ ( 2 n − r − ℓ ) 2 with 0 ≤ ℓ ≤ r ≤ n 0\leq\ell\leq r\leq n 0 ≤ ℓ ≤ r ≤ n ( ℓ , r ) ≠ ( n , n ) (\ell,r)\neq(n,n) ( ℓ , r ) = ( n , n )
Relation (25) follows by Theorem 4.4 below, using the fact that σ n , R 4 ∼ c n 2 R 2 d \sigma_{n,R}^4\sim c_n^2R^{2d} σ n , R 4 ∼ c n 2 R 2 d
Theorem 4.4. Suppose that either one of conditions (i)-(iii) of Theorem 1.1 hold. In case (iii), we assume in addition that κ ∈ L ∞ ( R 2 ) \kappa\in L^\infty(\mathbb{R}^2) κ ∈ L ∞ ( R 2 )
Let g n , R = g n , R ( ⋅ , t ) g_{n,R}=g_{n,R}(\cdot,t) g n , R = g n , R ( ⋅ , t ) t > 0 t>0 t > 0 n ≥ 1 n\geq1 n ≥ 1 0 ≤ ℓ ≤ r ≤ n 0\leq\ell\leq r\leq n 0 ≤ ℓ ≤ r ≤ n ( ℓ , r ) ≠ ( n , n ) (\ell,r)\neq(n,n) ( ℓ , r ) = ( n , n ) ( ℓ , r ) ≠ ( 0 , 0 ) (\ell,r)\neq(0,0) ( ℓ , r ) = ( 0 , 0 )
lim R → ∞ 1 R 2 d ∥ g ~ n , R ⋆ r ℓ g ~ n , R ∥ H ⊗ ( 2 n − r − ℓ ) 2 = 0. \lim_{R\to\infty}\frac{1}{R^{2d}}\left\|\widetilde{g}_{n,R}\star_r^\ell\widetilde{g}_{n,R}\right\|_{\mathcal{H}^{\otimes(2n-r-\ell)}}^2=0. R → ∞ lim R 2 d 1 g n , R ⋆ r ℓ g n , R H ⊗ ( 2 n − r − ℓ ) 2 = 0. Proof. We consider the variables
γ r − ℓ = ( γ 1 , … , γ r − ℓ ) with γ i = ( u i , ζ i ) ∈ R d × R 0 , for i = 1 , … , r − ℓ \gamma_{r-\ell}=(\gamma_1,\ldots,\gamma_{r-\ell})\quad\text{with}\quad\gamma_i=(u_i,\zeta_i)\in\mathbb{R}^d\times\mathbb{R}_0,\ \text{for }i=1,\ldots,r-\ell γ r − ℓ = ( γ 1 , … , γ r − ℓ ) with γ i = ( u i , ζ i ) ∈ R d × R 0 , for i = 1 , … , r − ℓ y n − r = ( y 1 , … , y n − r ) with y i = ( v i , a i ) ∈ R d × R 0 , for i = 1 , … , n − r \mathbf{y}_{n-r}=(y_1,\ldots,y_{n-r})\quad\text{with}\quad y_i=(v_i,a_i)\in\mathbb{R}^d\times\mathbb{R}_0,\ \text{for }i=1,\ldots,n-r y n − r = ( y 1 , … , y n − r ) with y i = ( v i , a i ) ∈ R d × R 0 , for i = 1 , … , n − r y n − r ′ = ( y 1 ′ , … , y n − r ′ ) with y i ′ = ( v i ′ , a i ′ ) ∈ R d × R 0 , for i = 1 , … , n − r \mathbf{y}'_{n-r}=(y'_1,\ldots,y'_{n-r})\quad\text{with}\quad y'_i=(v'_i,a'_i)\in\mathbb{R}^d\times\mathbb{R}_0,\ \text{for }i=1,\ldots,n-r y n − r ′ = ( y 1 ′ , … , y n − r ′ ) with y i ′ = ( v i ′ , a i ′ ) ∈ R d × R 0 , for i = 1 , … , n − r ξ ℓ = ( ξ 1 , … , ξ ℓ ) with ξ i = ( x i , z i ) ∈ R d × R 0 , for i = 1 , … , ℓ . \boldsymbol{\xi}_\ell=(\xi_1,\ldots,\xi_\ell)\quad\text{with}\quad\xi_i=(x_i,z_i)\in\mathbb{R}^d\times\mathbb{R}_0,\ \text{for }i=1,\ldots,\ell. ξ ℓ = ( ξ 1 , … , ξ ℓ ) with ξ i = ( x i , z i ) ∈ R d × R 0 , for i = 1 , … , ℓ . By definition of ⋆ r ℓ \star_r^\ell ⋆ r ℓ
( g ~ n , R ⋆ r ℓ g ~ n , R ) ( γ r − ℓ , y n − r , y n − r ′ ) = ∫ Z ℓ g ~ n , R ( ξ ℓ , γ r − ℓ , y n − r ) g ~ n , R ( ξ ℓ , γ r − ℓ , y n − r ′ ) m ⊗ ℓ ( d ξ ℓ ) = m 2 ℓ ∏ i = 1 r − ℓ ζ i 2 ∏ i = 1 n − r a i ∏ i = 1 n − r a i ′ ( Q ~ n , R ⋆ r ℓ Q ~ n , R ) ( u r − ℓ , v n − r , v n − r ′ ) , \begin{aligned}
(\widetilde{g}_{n,R}\star_r^\ell\widetilde{g}_{n,R})(\gamma_{r-\ell},\mathbf{y}_{n-r},\mathbf{y}'_{n-r})
&=\int_{\mathbb{Z}_\ell}\widetilde{g}_{n,R}(\boldsymbol{\xi}_\ell,\gamma_{r-\ell},\mathbf{y}_{n-r})\widetilde{g}_{n,R}(\boldsymbol{\xi}_\ell,\gamma_{r-\ell},\mathbf{y}'_{n-r})\mathfrak{m}^{\otimes\ell}(d\boldsymbol{\xi}_\ell)\\
&=m_2^\ell\prod_{i=1}^{r-\ell}\zeta_i^2\prod_{i=1}^{n-r}a_i\prod_{i=1}^{n-r}a'_i(\widetilde{Q}_{n,R}\star_r^\ell\widetilde{Q}_{n,R})(\mathbf{u}_{r-\ell},\mathbf{v}_{n-r},\mathbf{v}'_{n-r}),
\end{aligned} ( g n , R ⋆ r ℓ g n , R ) ( γ r − ℓ , y n − r , y n − r ′ ) = ∫ Z ℓ g n , R ( ξ ℓ , γ r − ℓ , y n − r ) g n , R ( ξ ℓ , γ r − ℓ , y n − r ′ ) m ⊗ ℓ ( d ξ ℓ ) = m 2 ℓ i = 1 ∏ r − ℓ ζ i 2 i = 1 ∏ n − r a i i = 1 ∏ n − r a i ′ ( Q n , R ⋆ r ℓ Q n , R ) ( u r − ℓ , v n − r , v n − r ′ ) , where u r − ℓ = ( u 1 , … , u r − ℓ ) \mathbf{u}_{r-\ell}=(u_1,\ldots,u_{r-\ell}) u r − ℓ = ( u 1 , … , u r − ℓ ) v n − r = ( v 1 , … , v n − r ) \mathbf{v}_{n-r}=(v_1,\ldots,v_{n-r}) v n − r = ( v 1 , … , v n − r ) v n − r ′ = ( v 1 ′ , … , v n − r ′ ) \mathbf{v}'_{n-r}=(v'_1,\ldots,v'_{n-r}) v n − r ′ = ( v 1 ′ , … , v n − r ′ )
∥ g ~ n , R ⋆ r ℓ g ~ n , R ∥ H ⊗ ( 2 n − r − ℓ ) 2 = ∫ Z 2 n − r − ℓ [ ( g ~ n , R ⋆ r ℓ g ~ n , R ) ( γ r − ℓ , y n − r , y n − r ′ ) ] 2 m ⊗ ( 2 n − r − ℓ ) ( d γ r − ℓ , d y n − r , d y n − r ′ ) = m 2 2 ℓ + 2 ( n − r ) m 4 r − ℓ ∥ Q ~ n , R ⋆ r ℓ Q ~ n , R ∥ L 2 ( R d ( 2 n − r − ℓ ) ) 2 . (26) \begin{aligned}
\left\|\widetilde{g}_{n,R}\star_r^\ell\widetilde{g}_{n,R}\right\|_{\mathcal{H}^{\otimes(2n-r-\ell)}}^2
&=\int_{\mathbb{Z}^{2n-r-\ell}}\left[(\widetilde{g}_{n,R}\star_r^\ell\widetilde{g}_{n,R})(\gamma_{r-\ell},\mathbf{y}_{n-r},\mathbf{y}'_{n-r})\right]^2\mathfrak{m}^{\otimes(2n-r-\ell)}(d\gamma_{r-\ell},d\mathbf{y}_{n-r},d\mathbf{y}'_{n-r})\\
&=m_2^{2\ell+2(n-r)}m_4^{r-\ell}\left\|\widetilde{Q}_{n,R}\star_r^\ell\widetilde{Q}_{n,R}\right\|_{L^2(\mathbb{R}^{d(2n-r-\ell)})}^2. \tag*{(26)}
\end{aligned} g n , R ⋆ r ℓ g n , R H ⊗ ( 2 n − r − ℓ ) 2 = ∫ Z 2 n − r − ℓ [ ( g n , R ⋆ r ℓ g n , R ) ( γ r − ℓ , y n − r , y n − r ′ ) ] 2 m ⊗ ( 2 n − r − ℓ ) ( d γ r − ℓ , d y n − r , d y n − r ′ ) = m 2 2 ℓ + 2 ( n − r ) m 4 r − ℓ Q n , R ⋆ r ℓ Q n , R L 2 ( R d ( 2 n − r − ℓ ) ) 2 . (26) We evaluate separately the last norm above. We let x ℓ = ( x 1 , … , x ℓ ) x_\ell=(x_1,\ldots,x_\ell) x ℓ = ( x 1 , … , x ℓ )
∥ Q ~ n , R ⋆ r ℓ Q ~ n , R ∥ L 2 ( R d ( 2 n − r − ℓ ) ) 2 = ∫ ( R d ) 2 n − r − ℓ [ ∫ ( R d ) ℓ Q ~ n , R ( x ℓ , u r − ℓ , v n − r ) Q ~ n , R ( x ℓ , u r − ℓ , v n − r ′ ) d x ℓ ] 2 d u r − ℓ d v n − r d v n − r ′ = ∫ ( R d ) 2 n − r − ℓ ∫ ( R d ) ℓ ∫ ( R d ) ℓ Q ~ n , R ( x ℓ , u r − ℓ , v n − r ) Q ~ n , R ( x ℓ , u r − ℓ , v n − r ′ ) Q ~ n , R ( x ℓ ′ , u r − ℓ , v n − r ) Q ~ n , R ( x ℓ ′ , u r − ℓ , v n − r ′ ) d x ℓ d x ℓ ′ d u r − ℓ d v n − r d v n − r ′ . (27) \begin{aligned}
\left\|\widetilde{Q}_{n,R}\star_r^\ell\widetilde{Q}_{n,R}\right\|_{L^2(\mathbb{R}^{d(2n-r-\ell)})}^2
&=\int_{(\mathbb{R}^d)^{2n-r-\ell}}\left[\int_{(\mathbb{R}^d)^\ell}\widetilde{Q}_{n,R}(x_\ell,u_{r-\ell},v_{n-r})\widetilde{Q}_{n,R}(x_\ell,u_{r-\ell},v'_{n-r})\,dx_\ell\right]^2du_{r-\ell}dv_{n-r}dv'_{n-r}\\
&=\int_{(\mathbb{R}^d)^{2n-r-\ell}}\int_{(\mathbb{R}^d)^\ell}\int_{(\mathbb{R}^d)^\ell}\widetilde{Q}_{n,R}(x_\ell,u_{r-\ell},v_{n-r})\widetilde{Q}_{n,R}(x_\ell,u_{r-\ell},v'_{n-r})\\
&\qquad\qquad\qquad\qquad\widetilde{Q}_{n,R}(x'_\ell,u_{r-\ell},v_{n-r})\widetilde{Q}_{n,R}(x'_\ell,u_{r-\ell},v'_{n-r})\,dx_\ell dx'_\ell du_{r-\ell}dv_{n-r}dv'_{n-r}. \tag*{(27)}
\end{aligned} Q n , R ⋆ r ℓ Q n , R L 2 ( R d ( 2 n − r − ℓ ) ) 2 = ∫ ( R d ) 2 n − r − ℓ [ ∫ ( R d ) ℓ Q n , R ( x ℓ , u r − ℓ , v n − r ) Q n , R ( x ℓ , u r − ℓ , v n − r ′ ) d x ℓ ] 2 d u r − ℓ d v n − r d v n − r ′ = ∫ ( R d ) 2 n − r − ℓ ∫ ( R d ) ℓ ∫ ( R d ) ℓ Q n , R ( x ℓ , u r − ℓ , v n − r ) Q n , R ( x ℓ , u r − ℓ , v n − r ′ ) Q n , R ( x ℓ ′ , u r − ℓ , v n − r ) Q n , R ( x ℓ ′ , u r − ℓ , v n − r ′ ) d x ℓ d x ℓ ′ d u r − ℓ d v n − r d v n − r ′ . (27) Since
Q ~ n , R ( x 1 , … , x n ; x ) = ∫ B R φ ~ n ( x 1 , … , x n , x ; t ) d x , \widetilde{Q}_{n,R}(x_1,\ldots,x_n;x)=\int_{B_R}\widetilde{\varphi}_n(x_1,\ldots,x_n,x;t)\,dx, Q n , R ( x 1 , … , x n ; x ) = ∫ B R φ n ( x 1 , … , x n , x ; t ) d x , we have:
∥ Q ~ n , R ⋆ r ℓ Q ~ n , R ∥ L 2 ( R d ( 2 n − r − ℓ ) ) 2 = ∫ ( R d ) 2 n − r + ℓ ∫ B R 4 φ ~ n ( x ℓ , u r − ℓ , v n − r , a 1 ; t ) φ ~ n ( x ℓ , u r − ℓ , v n − r ′ , a 2 ; t ) φ ~ n ( x ℓ ′ , u r − ℓ , v n − r , a 3 ; t ) φ ~ n ( x ℓ ′ , u r − ℓ , v n − r ′ , a 4 ; t ) d a 4 d x ℓ d x ℓ ′ d u r − ℓ d v n − r d v n − r ′ , \begin{aligned}
\left\|\widetilde{Q}_{n,R}\star_r^\ell\widetilde{Q}_{n,R}\right\|_{L^2(\mathbb{R}^{d(2n-r-\ell)})}^2
&=\int_{(\mathbb{R}^d)^{2n-r+\ell}}\int_{B_R^4}\widetilde{\varphi}_n(x_\ell,u_{r-\ell},v_{n-r},a_1;t)\widetilde{\varphi}_n(x_\ell,u_{r-\ell},v'_{n-r},a_2;t)\\
&\qquad\widetilde{\varphi}_n(x'_\ell,u_{r-\ell},v_{n-r},a_3;t)\widetilde{\varphi}_n(x'_\ell,u_{r-\ell},v'_{n-r},a_4;t)\,da_4\,dx_\ell\,dx'_\ell\,du_{r-\ell}\,dv_{n-r}\,dv'_{n-r},
\end{aligned} Q n , R ⋆ r ℓ Q n , R L 2 ( R d ( 2 n − r − ℓ ) ) 2 = ∫ ( R d ) 2 n − r + ℓ ∫ B R 4 φ n ( x ℓ , u r − ℓ , v n − r , a 1 ; t ) φ n ( x ℓ , u r − ℓ , v n − r ′ , a 2 ; t ) φ n ( x ℓ ′ , u r − ℓ , v n − r , a 3 ; t ) φ n ( x ℓ ′ , u r − ℓ , v n − r ′ , a 4 ; t ) d a 4 d x ℓ d x ℓ ′ d u r − ℓ d v n − r d v n − r ′ , where a 4 = ( a 1 , a 2 , a 3 , a 4 ) \mathbf{a}_4=(a_1,a_2,a_3,a_4) a 4 = ( a 1 , a 2 , a 3 , a 4 ) a ∈ R d a\in\mathbb{R}^d a ∈ R d
φ ~ n ( x 1 , … , x n , x ; t ) = φ ~ n ( x 1 − a , … , x n − a , x − a ; t ) . (28) \widetilde{\varphi}_n(x_1,\ldots,x_n,x;t)=\widetilde{\varphi}_n(x_1-a,\ldots,x_n-a,x-a;t). \tag*{(28)} φ n ( x 1 , … , x n , x ; t ) = φ n ( x 1 − a , … , x n − a , x − a ; t ) . (28) We use (28) with a = a 1 a=a_1 a = a 1
( x ‾ ℓ , x ‾ ℓ ′ , u ‾ r − ℓ , v ‾ n − r , v ‾ n − r ′ ) = ( x ℓ − a 1 , x ℓ ′ − a 1 , u r − ℓ − a 1 , v n − r − a 1 , v n − r ′ − a 1 ) , (\overline{x}_\ell,\overline{x}'_\ell,\overline{u}_{r-\ell},\overline{v}_{n-r},\overline{v}'_{n-r})=(x_\ell-a_1,x'_\ell-a_1,u_{r-\ell}-a_1,v_{n-r}-a_1,v'_{n-r}-a_1), ( x ℓ , x ℓ ′ , u r − ℓ , v n − r , v n − r ′ ) = ( x ℓ − a 1 , x ℓ ′ − a 1 , u r − ℓ − a 1 , v n − r − a 1 , v n − r ′ − a 1 ) , where x ‾ ℓ − a 1 = ( x 1 − a 1 , … , x ℓ − a 1 ) \overline{x}_\ell-a_1=(x_1-a_1,\ldots,x_\ell-a_1) x ℓ − a 1 = ( x 1 − a 1 , … , x ℓ − a 1 )
∥ Q ~ n , R ⋆ r ℓ Q ~ n , R ∥ L 2 ( R d ( 2 n − r − ℓ ) ) 2 = ∫ ( R d ) 2 n − r + ℓ ∫ B R 4 φ ~ n ( x ‾ ℓ , u ‾ r − ℓ , v ‾ n − r , 0 ; t ) φ ~ n ( x ‾ ℓ , u ‾ r − ℓ , v ‾ n − r ′ , a 2 − a 1 ; t ) φ ~ n ( x ‾ ℓ ′ , u ‾ r − ℓ , v ‾ n − r , a 3 − a 1 ; t ) φ ~ n ( x ‾ ℓ ′ , u ‾ r − ℓ , v ‾ n − r ′ , a 4 − a 1 ; t ) d a 4 d x ‾ ℓ d x ‾ ℓ ′ d u ‾ r − ℓ d v ‾ n − r d v ‾ n − r ′ . \begin{aligned}
\left\|\widetilde{Q}_{n,R}\star_r^\ell\widetilde{Q}_{n,R}\right\|_{L^2(\mathbb{R}^{d(2n-r-\ell)})}^2
&=\int_{(\mathbb{R}^d)^{2n-r+\ell}}\int_{B_R^4}\widetilde{\varphi}_n(\overline{x}_\ell,\overline{u}_{r-\ell},\overline{v}_{n-r},0;t)\widetilde{\varphi}_n(\overline{x}_\ell,\overline{u}_{r-\ell},\overline{v}'_{n-r},a_2-a_1;t)\\
&\qquad\widetilde{\varphi}_n(\overline{x}'_\ell,\overline{u}_{r-\ell},\overline{v}_{n-r},a_3-a_1;t)\widetilde{\varphi}_n(\overline{x}'_\ell,\overline{u}_{r-\ell},\overline{v}'_{n-r},a_4-a_1;t)\,da_4\,d\overline{x}_\ell\,d\overline{x}'_\ell\,d\overline{u}_{r-\ell}\,d\overline{v}_{n-r}\,d\overline{v}'_{n-r}.
\end{aligned} Q n , R ⋆ r ℓ Q n , R L 2 ( R d ( 2 n − r − ℓ ) ) 2 = ∫ ( R d ) 2 n − r + ℓ ∫ B R 4 φ n ( x ℓ , u r − ℓ , v n − r , 0 ; t ) φ n ( x ℓ , u r − ℓ , v n − r ′ , a 2 − a 1 ; t ) φ n ( x ℓ ′ , u r − ℓ , v n − r , a 3 − a 1 ; t ) φ n ( x ℓ ′ , u r − ℓ , v n − r ′ , a 4 − a 1 ; t ) d a 4 d x ℓ d x ℓ ′ d u r − ℓ d v n − r d v n − r ′ . Next, we use the change of variables b 1 = a 2 − a 1 b_1=a_2-a_1 b 1 = a 2 − a 1 b 2 = a 3 − a 1 b_2=a_3-a_1 b 2 = a 3 − a 1 b 3 = a 4 − a 1 b_3=a_4-a_1 b 3 = a 4 − a 1 B R ( a ) = { x ∈ R d ; ∣ x − a ∣ ≤ R } B_R(a)=\{x\in\mathbb{R}^d;|x-a|\le R\} B R ( a ) = { x ∈ R d ; ∣ x − a ∣ ≤ R } a ∈ R d a\in\mathbb{R}^d a ∈ R d R R R
∥ Q ~ n , R ⋆ r ℓ Q ~ n , R ∥ L 2 ( R d ( 2 n − r − ℓ ) ) 2 = ∫ ( R d ) 2 n − r + ℓ ∫ B R ∫ ( B R ( − a 1 ) ) 3 φ ~ n ( x ‾ ℓ , u ‾ r − ℓ , v ‾ n − r , 0 ; t ) φ ~ n ( x ‾ ℓ , u ‾ r − ℓ , v ‾ n − r ′ , b 1 ; t ) φ ~ n ( x ‾ ℓ ′ , u ‾ r − ℓ , v ‾ n − r , b 2 ; t ) φ ~ n ( x ‾ ℓ ′ , u ‾ r − ℓ , v ‾ n − r ′ , b 3 ; t ) d b 3 d a 1 d x ‾ ℓ d x ‾ ℓ ′ d u ‾ r − ℓ d v ‾ n − r d v ‾ n − r ′ , \begin{aligned}
\left\|\widetilde{Q}_{n,R}\star_r^\ell\widetilde{Q}_{n,R}\right\|_{L^2(\mathbb{R}^{d(2n-r-\ell)})}^2
&=\int_{(\mathbb{R}^d)^{2n-r+\ell}}\int_{B_R}\int_{(B_R(-a_1))^3}\widetilde{\varphi}_n(\overline{x}_\ell,\overline{u}_{r-\ell},\overline{v}_{n-r},0;t)\widetilde{\varphi}_n(\overline{x}_\ell,\overline{u}_{r-\ell},\overline{v}'_{n-r},b_1;t)\\
&\qquad\widetilde{\varphi}_n(\overline{x}'_\ell,\overline{u}_{r-\ell},\overline{v}_{n-r},b_2;t)\widetilde{\varphi}_n(\overline{x}'_\ell,\overline{u}_{r-\ell},\overline{v}'_{n-r},b_3;t)\,db_3\,da_1\,d\overline{x}_\ell\,d\overline{x}'_\ell\,d\overline{u}_{r-\ell}\,d\overline{v}_{n-r}\,d\overline{v}'_{n-r},
\end{aligned} Q n , R ⋆ r ℓ Q n , R L 2 ( R d ( 2 n − r − ℓ ) ) 2 = ∫ ( R d ) 2 n − r + ℓ ∫ B R ∫ ( B R ( − a 1 ) ) 3 φ n ( x ℓ , u r − ℓ , v n − r , 0 ; t ) φ n ( x ℓ , u r − ℓ , v n − r ′ , b 1 ; t ) φ n ( x ℓ ′ , u r − ℓ , v n − r , b 2 ; t ) φ n ( x ℓ ′ , u r − ℓ , v n − r ′ , b 3 ; t ) d b 3 d a 1 d x ℓ d x ℓ ′ d u r − ℓ d v n − r d v n − r ′ , where b 3 = ( b 1 , b 2 , b 3 ) \mathbf{b}_3=(b_1,b_2,b_3) b 3 = ( b 1 , b 2 , b 3 ) ( B R ( − a 1 ) ) 3 (B_R(-a_1))^3 ( B R ( − a 1 ) ) 3 ( R d ) 3 (\mathbb{R}^d)^3 ( R d ) 3 a 1 a_1 a 1 B R B_R B R
Vol ( B R ) = ω d R d where ω d = 2 π d / 2 d Γ ( d / 2 ) . \operatorname{Vol}(B_R)=\omega_dR^d\qquad\text{where}\qquad\omega_d=\frac{2\pi^{d/2}}{d\Gamma(d/2)}. Vol ( B R ) = ω d R d where ω d = d Γ ( d /2 ) 2 π d /2 . This leads to:
∥ Q ~ n , R ⋆ r ℓ Q ~ n , R ∥ L 2 ( R d ( 2 n − r − ℓ ) ) 2 ≤ ω d R d J , (29) \left\|\widetilde{Q}_{n,R}\star_r^\ell\widetilde{Q}_{n,R}\right\|_{L^2(\mathbb{R}^{d(2n-r-\ell)})}^2\le\omega_dR^dJ, \tag*{(29)} Q n , R ⋆ r ℓ Q n , R L 2 ( R d ( 2 n − r − ℓ ) ) 2 ≤ ω d R d J , (29) where
J : = ∫ ( R d ) 2 n − r + ℓ + 3 φ ~ n ( x ℓ , u r − ℓ , v n − r , 0 ; t ) φ ~ n ( x ℓ , u r − ℓ , v n − r ′ , b 1 ; t ) φ ~ n ( x ℓ ′ , u r − ℓ , v n − r , b 2 ; t ) φ ~ n ( x ℓ ′ , u r − ℓ , v n − r ′ , b 3 ; t ) d x ℓ d x ℓ ′ d u r − ℓ d v n − r d v n − r ′ d b 3 . J := \int_{(\mathbb{R}^d)^{2n-r+\ell+3}} \widetilde{\varphi}_n(\mathbf{x}_{\ell},\mathbf{u}_{r-\ell},\mathbf{v}_{n-r},0;t)\widetilde{\varphi}_n(\mathbf{x}_{\ell},\mathbf{u}_{r-\ell},\mathbf{v}'_{n-r},b_1;t)
\widetilde{\varphi}_n(\mathbf{x}'_{\ell},\mathbf{u}_{r-\ell},\mathbf{v}_{n-r},b_2;t)\widetilde{\varphi}_n(\mathbf{x}'_{\ell},\mathbf{u}_{r-\ell},\mathbf{v}'_{n-r},b_3;t)\,\mathrm{d}\mathbf{x}_{\ell}\,\mathrm{d}\mathbf{x}'_{\ell}\,\mathrm{d}\mathbf{u}_{r-\ell}\,\mathrm{d}\mathbf{v}_{n-r}\,\mathrm{d}\mathbf{v}'_{n-r}\,\mathrm{d}b_3. J := ∫ ( R d ) 2 n − r + ℓ + 3 φ n ( x ℓ , u r − ℓ , v n − r , 0 ; t ) φ n ( x ℓ , u r − ℓ , v n − r ′ , b 1 ; t ) φ n ( x ℓ ′ , u r − ℓ , v n − r , b 2 ; t ) φ n ( x ℓ ′ , u r − ℓ , v n − r ′ , b 3 ; t ) d x ℓ d x ℓ ′ d u r − ℓ d v n − r d v n − r ′ d b 3 . By (26) and (29), it follows that:
1 R 2 d ∥ g ~ n , R ⋆ g ~ n , R ∥ H ⊗ ( 2 n − r − ℓ ) 2 ≤ m 2 2 ℓ + 2 ( n − r ) m 4 r − ℓ ω d 1 R d J . \frac{1}{R^{2d}}\left\|\widetilde{g}_{n,R}\star\widetilde{g}_{n,R}\right\|_{\mathcal{H}^{\otimes(2n-r-\ell)}}^2 \le m_2^{2\ell+2(n-r)}m_4^{r-\ell}\omega_d\frac{1}{R^d}J. R 2 d 1 ∥ g n , R ⋆ g n , R ∥ H ⊗ ( 2 n − r − ℓ ) 2 ≤ m 2 2 ℓ + 2 ( n − r ) m 4 r − ℓ ω d R d 1 J . Therefore, the proof of the theorem will be complete, once we show that
J < ∞ . (30) J<\infty. \tag*{(30)} J < ∞. (30) We now start the proof of (30). This will require a major effort. We will use graph theory. The graph will be defined differently for the white noise, and the colored noise.
First, by definition,
φ ~ n ( ⋅ , x ; t ) = 1 n ! ∑ ρ ∈ S n φ n , ρ ( ⋅ , x ; t ) , \widetilde{\varphi}_n(\cdot,x;t)=\frac{1}{n!}\sum_{\rho\in S_n}\varphi_{n,\rho}(\cdot,x;t), φ n ( ⋅ , x ; t ) = n ! 1 ρ ∈ S n ∑ φ n , ρ ( ⋅ , x ; t ) , where φ n , ρ ( x 1 , … , x n , ⋅ , x ; t ) = φ n ( x ρ ( 1 ) , … , x ρ ( n ) , ⋅ , x ; t ) \varphi_{n,\rho}(x_1,\ldots,x_n,\cdot,x;t)=\varphi_n(x_{\rho(1)},\ldots,x_{\rho(n)},\cdot,x;t) φ n , ρ ( x 1 , … , x n , ⋅ , x ; t ) = φ n ( x ρ ( 1 ) , … , x ρ ( n ) , ⋅ , x ; t )
J = 1 ( n ! ) 4 ∑ ρ 1 , … , ρ 4 ∈ S n J ρ 1 , … , ρ 4 , J=\frac{1}{(n!)^4}\sum_{\rho_1,\ldots,\rho_4\in S_n}J_{\rho_1,\ldots,\rho_4}, J = ( n ! ) 4 1 ρ 1 , … , ρ 4 ∈ S n ∑ J ρ 1 , … , ρ 4 , where
J ρ 1 , … , ρ 4 = ∫ ( R d ) 2 n − r + ℓ + 3 φ n , ρ 1 ( x ℓ , u r − ℓ , v n − r , 0 ; t ) φ n , ρ 2 ( x ℓ , u r − ℓ , v n − r ′ , b 1 ; t ) φ n , ρ 3 ( x ℓ ′ , u r − ℓ , v n − r , b 2 ; t ) φ n , ρ 4 ( x ℓ ′ , u r − ℓ , v n − r ′ , b 3 ; t ) d x ℓ d x ℓ ′ d u r − ℓ d v n − r d v n − r ′ d b 3 . (31) J_{\rho_1,\ldots,\rho_4}=\int_{(\mathbb{R}^d)^{2n-r+\ell+3}}\varphi_{n,\rho_1}(\mathbf{x}_{\ell},\mathbf{u}_{r-\ell},\mathbf{v}_{n-r},0;t)\varphi_{n,\rho_2}(\mathbf{x}_{\ell},\mathbf{u}_{r-\ell},\mathbf{v}'_{n-r},b_1;t)
\varphi_{n,\rho_3}(\mathbf{x}'_{\ell},\mathbf{u}_{r-\ell},\mathbf{v}_{n-r},b_2;t)\varphi_{n,\rho_4}(\mathbf{x}'_{\ell},\mathbf{u}_{r-\ell},\mathbf{v}'_{n-r},b_3;t)\,\mathrm{d}\mathbf{x}_{\ell}\,\mathrm{d}\mathbf{x}'_{\ell}\,\mathrm{d}\mathbf{u}_{r-\ell}\,\mathrm{d}\mathbf{v}_{n-r}\,\mathrm{d}\mathbf{v}'_{n-r}\,\mathrm{d}b_3. \tag*{(31)} J ρ 1 , … , ρ 4 = ∫ ( R d ) 2 n − r + ℓ + 3 φ n , ρ 1 ( x ℓ , u r − ℓ , v n − r , 0 ; t ) φ n , ρ 2 ( x ℓ , u r − ℓ , v n − r ′ , b 1 ; t ) φ n , ρ 3 ( x ℓ ′ , u r − ℓ , v n − r , b 2 ; t ) φ n , ρ 4 ( x ℓ ′ , u r − ℓ , v n − r ′ , b 3 ; t ) d x ℓ d x ℓ ′ d u r − ℓ d v n − r d v n − r ′ d b 3 . (31) Therefore, to prove (30), it is enough to show that for any ρ 1 , … , ρ 4 ∈ S n \rho_1,\ldots,\rho_4\in S_n ρ 1 , … , ρ 4 ∈ S n
J ρ 1 , … , ρ 4 < ∞ . (32) J_{\rho_1,\ldots,\rho_4}<\infty. \tag*{(32)} J ρ 1 , … , ρ 4 < ∞. (32) We consider separately the white noise and the colored noise.
Case 1. (white noise, d = 1 d=1 d = 1 In this case, φ n ( ⋅ , x ; t ) = f n ( ⋅ , x ; t ) \varphi_n(\cdot,x;t)=f_n(\cdot,x;t) φ n ( ⋅ , x ; t ) = f n ( ⋅ , x ; t )
J ρ 1 , … , ρ 4 = ∫ ( R d ) 2 n − r + ℓ + 3 f n , ρ 1 ( x ℓ , u r − ℓ , v n − r , 0 ; t ) f n , ρ 2 ( x ℓ , u r − ℓ , v n − r ′ , b 1 ; t ) f n , ρ 3 ( x ℓ ′ , u r − ℓ , v n − r , b 2 ; t ) f n , ρ 4 ( x ℓ ′ , u r − ℓ , v n − r ′ , b 3 ; t ) d x ℓ d x ℓ ′ d u r − ℓ d v n − r d v n − r ′ d b 3 , J_{\rho_1,\ldots,\rho_4}=\int_{(\mathbb{R}^d)^{2n-r+\ell+3}}f_{n,\rho_1}(\mathbf{x}_{\ell},\mathbf{u}_{r-\ell},\mathbf{v}_{n-r},0;t)f_{n,\rho_2}(\mathbf{x}_{\ell},\mathbf{u}_{r-\ell},\mathbf{v}'_{n-r},b_1;t)
f_{n,\rho_3}(\mathbf{x}'_{\ell},\mathbf{u}_{r-\ell},\mathbf{v}_{n-r},b_2;t)f_{n,\rho_4}(\mathbf{x}'_{\ell},\mathbf{u}_{r-\ell},\mathbf{v}'_{n-r},b_3;t)\,\mathrm{d}\mathbf{x}_{\ell}\,\mathrm{d}\mathbf{x}'_{\ell}\,\mathrm{d}\mathbf{u}_{r-\ell}\,\mathrm{d}\mathbf{v}_{n-r}\,\mathrm{d}\mathbf{v}'_{n-r}\,\mathrm{d}b_3, J ρ 1 , … , ρ 4 = ∫ ( R d ) 2 n − r + ℓ + 3 f n , ρ 1 ( x ℓ , u r − ℓ , v n − r , 0 ; t ) f n , ρ 2 ( x ℓ , u r − ℓ , v n − r ′ , b 1 ; t ) f n , ρ 3 ( x ℓ ′ , u r − ℓ , v n − r , b 2 ; t ) f n , ρ 4 ( x ℓ ′ , u r − ℓ , v n − r ′ , b 3 ; t ) d x ℓ d x ℓ ′ d u r − ℓ d v n − r d v n − r ′ d b 3 , where f n , ρ ( x 1 , … , x n , ⋅ , x ; t ) = f n ( x ρ ( 1 ) , … , x ρ ( n ) , ⋅ , x ; t ) f_{n,\rho}(x_1,\ldots,x_n,\cdot,x;t)=f_n(x_{\rho(1)},\ldots,x_{\rho(n)},\cdot,x;t) f n , ρ ( x 1 , … , x n , ⋅ , x ; t ) = f n ( x ρ ( 1 ) , … , x ρ ( n ) , ⋅ , x ; t )
f n ( x 1 , … , x n , x ; t ) = ∫ T n ( t ) G t − t n ( x − x n ) ⋯ G t 2 − t 1 ( x 2 − x 1 ) d t = ∫ D n ( t ) h n ( x 1 , … , x n , δ , x ; t ) d δ , f_n(x_1,\ldots,x_n,x;t)=\int_{T_n(t)}G_{t-t_n}(x-x_n)\cdots G_{t_2-t_1}(x_2-x_1)\,\mathrm{d}t
=\int_{D_n(t)}h_n(x_1,\ldots,x_n,\delta,x;t)\,\mathrm{d}\delta, f n ( x 1 , … , x n , x ; t ) = ∫ T n ( t ) G t − t n ( x − x n ) ⋯ G t 2 − t 1 ( x 2 − x 1 ) d t = ∫ D n ( t ) h n ( x 1 , … , x n , δ , x ; t ) d δ , where δ = ( δ 1 , … , δ n ) \boldsymbol{\delta}=(\delta_1,\ldots,\delta_n) δ = ( δ 1 , … , δ n ) D n ( t ) = { δ ∈ [ 0 , t ] n ; δ 1 + ⋯ + δ n < t } D_n(t)=\{\boldsymbol{\delta}\in[0,t]^n;\delta_1+\cdots+\delta_n<t\} D n ( t ) = { δ ∈ [ 0 , t ] n ; δ 1 + ⋯ + δ n < t }
h n ( x 1 , … , x n , δ , x ; t ) : = G δ n ( x − x n ) ⋯ G δ 2 ( x 2 − x 1 ) . h_n(x_1,\ldots,x_n,\boldsymbol{\delta},x;t):=G_{\delta_n}(x-x_n)\cdots G_{\delta_2}(x_2-x_1). h n ( x 1 , … , x n , δ , x ; t ) := G δ n ( x − x n ) ⋯ G δ 2 ( x 2 − x 1 ) . Similarly, for any permutation ρ ∈ S n \rho\in S_n ρ ∈ S n
f n , ρ ( x 1 , … , x n , x ; t ) = ∫ D n ( t ) h n , ρ ( x 1 , … , x n , δ , x ; t ) d δ f_{n,\rho}(x_1,\ldots,x_n,x;t)=\int_{D_n(t)}h_{n,\rho}(x_1,\ldots,x_n,\boldsymbol{\delta},x;t)\,d\boldsymbol{\delta} f n , ρ ( x 1 , … , x n , x ; t ) = ∫ D n ( t ) h n , ρ ( x 1 , … , x n , δ , x ; t ) d δ where h n , ρ ( x 1 , … , x n , δ , x ; t ) = h n ( x ρ ( 1 ) , … , x ρ ( n ) , δ , x ; t ) h_{n,\rho}(x_1,\ldots,x_n,\boldsymbol{\delta},x;t)=h_n(x_{\rho(1)},\ldots,x_{\rho(n)},\boldsymbol{\delta},x;t) h n , ρ ( x 1 , … , x n , δ , x ; t ) = h n ( x ρ ( 1 ) , … , x ρ ( n ) , δ , x ; t )
J ρ 1 , … , ρ 4 = ∫ D n 4 ( t ) J ρ 1 , … , ρ 4 ( δ ( 1 ) , … , δ ( 4 ) ) d δ ( 1 ) ⋯ d δ ( 4 ) , (33) J_{\rho_1,\ldots,\rho_4}=\int_{D_n^4(t)}J_{\rho_1,\ldots,\rho_4}(\boldsymbol{\delta}^{(1)},\ldots,\boldsymbol{\delta}^{(4)})\,d\boldsymbol{\delta}^{(1)}\cdots d\boldsymbol{\delta}^{(4)}, \tag*{(33)} J ρ 1 , … , ρ 4 = ∫ D n 4 ( t ) J ρ 1 , … , ρ 4 ( δ ( 1 ) , … , δ ( 4 ) ) d δ ( 1 ) ⋯ d δ ( 4 ) , (33) where
J ρ 1 , … , ρ 4 ( δ ( 1 ) , … , δ ( 4 ) ) = ∫ ( R d ) 2 n − r + ℓ + 3 h n , ρ 1 ( x ℓ , u r − ℓ , v n − r , δ ( 1 ) , 0 ; t ) h n , ρ 2 ( x ℓ ′ , u r − ℓ , v n − r ′ , δ ( 2 ) , b 1 ; t ) h n , ρ 3 ( x ℓ ′ , u r − ℓ , v n − r , δ ( 3 ) , b 2 ; t ) h n , ρ 4 ( x ℓ ′ , u r − ℓ , v n − r ′ , δ ( 4 ) , b 3 ; t ) d x ℓ d x ℓ ′ d u r − ℓ d v n − r d v n − r ′ d b 3 . \begin{aligned}
J_{\rho_1,\ldots,\rho_4}(\boldsymbol{\delta}^{(1)},\ldots,\boldsymbol{\delta}^{(4)})
={}&\int_{(\mathbb{R}^d)^{2n-r+\ell+3}}
h_{n,\rho_1}(x_\ell,u_{r-\ell},v_{n-r},\boldsymbol{\delta}^{(1)},0;t)\\
&\quad h_{n,\rho_2}(x'_\ell,u_{r-\ell},v'_{n-r},\boldsymbol{\delta}^{(2)},b_1;t)
h_{n,\rho_3}(x'_\ell,u_{r-\ell},v_{n-r},\boldsymbol{\delta}^{(3)},b_2;t)\\
&\quad h_{n,\rho_4}(x'_\ell,u_{r-\ell},v'_{n-r},\boldsymbol{\delta}^{(4)},b_3;t)\,dx_\ell\,dx'_\ell\,du_{r-\ell}\,dv_{n-r}\,dv'_{n-r}\,db_3.
\end{aligned} J ρ 1 , … , ρ 4 ( δ ( 1 ) , … , δ ( 4 ) ) = ∫ ( R d ) 2 n − r + ℓ + 3 h n , ρ 1 ( x ℓ , u r − ℓ , v n − r , δ ( 1 ) , 0 ; t ) h n , ρ 2 ( x ℓ ′ , u r − ℓ , v n − r ′ , δ ( 2 ) , b 1 ; t ) h n , ρ 3 ( x ℓ ′ , u r − ℓ , v n − r , δ ( 3 ) , b 2 ; t ) h n , ρ 4 ( x ℓ ′ , u r − ℓ , v n − r ′ , δ ( 4 ) , b 3 ; t ) d x ℓ d x ℓ ′ d u r − ℓ d v n − r d v n − r ′ d b 3 . Consider first the case when all permutations ρ i \rho_i ρ i h n , ρ i = h n h_{n,\rho_i}=h_n h n , ρ i = h n i = 1 , … , 4 i=1,\ldots,4 i = 1 , … , 4 J 0 ( δ ( 1 ) , … , δ ( 4 ) ) J_0(\boldsymbol{\delta}^{(1)},\ldots,\boldsymbol{\delta}^{(4)}) J 0 ( δ ( 1 ) , … , δ ( 4 ) )
J 0 : = ∫ D n 4 ( t ) J 0 ( δ ( 1 ) , … , δ ( 4 ) ) d δ ( 1 ) ⋯ d δ ( 4 ) . J_0:=\int_{D_n^4(t)}J_0(\boldsymbol{\delta}^{(1)},\ldots,\boldsymbol{\delta}^{(4)})\,d\boldsymbol{\delta}^{(1)}\cdots d\boldsymbol{\delta}^{(4)}. J 0 := ∫ D n 4 ( t ) J 0 ( δ ( 1 ) , … , δ ( 4 ) ) d δ ( 1 ) ⋯ d δ ( 4 ) . We will prove below that J 0 < ∞ J_0<\infty J 0 < ∞ ρ i \rho_i ρ i
To see what J 0 ( δ ( 1 ) , … , δ ( 4 ) ) J_0(\boldsymbol{\delta}^{(1)},\ldots,\boldsymbol{\delta}^{(4)}) J 0 ( δ ( 1 ) , … , δ ( 4 ) ) n = 7 n=7 n = 7 ℓ = 2 \ell=2 ℓ = 2 r = 5 r=5 r = 5
J 0 ( δ ( 1 ) , … , δ ( 4 ) ) = ∫ R 14 d x 2 d x 2 ′ d u 3 d v 2 d v 2 ′ d b 3 G δ 7 ( 1 ) ( 0 − v 2 ) G δ 6 ( 1 ) ( v 2 − v 1 ) G δ 5 ( 1 ) ( v 1 − u 3 ) G δ 4 ( 1 ) ( u 3 − u 2 ) G δ 3 ( 1 ) ( u 2 − u 1 ) G δ 2 ( 1 ) ( u 1 − x 2 ) G δ 1 ( 1 ) ( x 2 − x 1 ) G δ 7 ( 2 ) ( b 1 − v 2 ′ ) G δ 6 ( 2 ) ( v 2 ′ − v 1 ′ ) G δ 5 ( 2 ) ( v 1 ′ − u 3 ) G δ 4 ( 2 ) ( u 3 − u 2 ) G δ 3 ( 2 ) ( u 2 − u 1 ) G δ 2 ( 2 ) ( u 1 − x 2 ) G δ 1 ( 2 ) ( x 2 − x 1 ) G δ 7 ( 3 ) ( b 2 − v 2 ) G δ 6 ( 3 ) ( v 2 − v 1 ) G δ 5 ( 3 ) ( v 1 − u 3 ) G δ 4 ( 3 ) ( u 3 − u 2 ) G δ 3 ( 3 ) ( u 2 − u 1 ) G δ 2 ( 3 ) ( u 1 − x 2 ′ ) G δ 1 ( 3 ) ( x 2 ′ − x 1 ′ ) G δ 7 ( 4 ) ( b 3 − v 2 ′ ) G δ 6 ( 4 ) ( v 2 ′ − v 1 ′ ) G δ 5 ( 4 ) ( v 1 ′ − u 3 ) G δ 4 ( 4 ) ( u 3 − u 2 ) G δ 3 ( 4 ) ( u 2 − u 1 ) G δ 2 ( 4 ) ( u 1 − x 2 ′ ) G δ 1 ( 4 ) ( x 2 ′ − x 1 ′ ) . \begin{aligned}
J_0(\boldsymbol{\delta}^{(1)},\ldots,\boldsymbol{\delta}^{(4)})
={}&\int_{\mathbb{R}^{14}}dx_2\,dx'_2\,du_3\,dv_2\,dv'_2\,db_3\\
& G_{\delta^{(1)}_7}(0-v_2)G_{\delta^{(1)}_6}(v_2-v_1)G_{\delta^{(1)}_5}(v_1-u_3)G_{\delta^{(1)}_4}(u_3-u_2)G_{\delta^{(1)}_3}(u_2-u_1)G_{\delta^{(1)}_2}(u_1-x_2)G_{\delta^{(1)}_1}(x_2-x_1)\\
& G_{\delta^{(2)}_7}(b_1-v'_2)G_{\delta^{(2)}_6}(v'_2-v'_1)G_{\delta^{(2)}_5}(v'_1-u_3)G_{\delta^{(2)}_4}(u_3-u_2)G_{\delta^{(2)}_3}(u_2-u_1)G_{\delta^{(2)}_2}(u_1-x_2)G_{\delta^{(2)}_1}(x_2-x_1)\\
& G_{\delta^{(3)}_7}(b_2-v_2)G_{\delta^{(3)}_6}(v_2-v_1)G_{\delta^{(3)}_5}(v_1-u_3)G_{\delta^{(3)}_4}(u_3-u_2)G_{\delta^{(3)}_3}(u_2-u_1)G_{\delta^{(3)}_2}(u_1-x'_2)G_{\delta^{(3)}_1}(x'_2-x'_1)\\
& G_{\delta^{(4)}_7}(b_3-v'_2)G_{\delta^{(4)}_6}(v'_2-v'_1)G_{\delta^{(4)}_5}(v'_1-u_3)G_{\delta^{(4)}_4}(u_3-u_2)G_{\delta^{(4)}_3}(u_2-u_1)G_{\delta^{(4)}_2}(u_1-x'_2)G_{\delta^{(4)}_1}(x'_2-x'_1).
\end{aligned} J 0 ( δ ( 1 ) , … , δ ( 4 ) ) = ∫ R 14 d x 2 d x 2 ′ d u 3 d v 2 d v 2 ′ d b 3 G δ 7 ( 1 ) ( 0 − v 2 ) G δ 6 ( 1 ) ( v 2 − v 1 ) G δ 5 ( 1 ) ( v 1 − u 3 ) G δ 4 ( 1 ) ( u 3 − u 2 ) G δ 3 ( 1 ) ( u 2 − u 1 ) G δ 2 ( 1 ) ( u 1 − x 2 ) G δ 1 ( 1 ) ( x 2 − x 1 ) G δ 7 ( 2 ) ( b 1 − v 2 ′ ) G δ 6 ( 2 ) ( v 2 ′ − v 1 ′ ) G δ 5 ( 2 ) ( v 1 ′ − u 3 ) G δ 4 ( 2 ) ( u 3 − u 2 ) G δ 3 ( 2 ) ( u 2 − u 1 ) G δ 2 ( 2 ) ( u 1 − x 2 ) G δ 1 ( 2 ) ( x 2 − x 1 ) G δ 7 ( 3 ) ( b 2 − v 2 ) G δ 6 ( 3 ) ( v 2 − v 1 ) G δ 5 ( 3 ) ( v 1 − u 3 ) G δ 4 ( 3 ) ( u 3 − u 2 ) G δ 3 ( 3 ) ( u 2 − u 1 ) G δ 2 ( 3 ) ( u 1 − x 2 ′ ) G δ 1 ( 3 ) ( x 2 ′ − x 1 ′ ) G δ 7 ( 4 ) ( b 3 − v 2 ′ ) G δ 6 ( 4 ) ( v 2 ′ − v 1 ′ ) G δ 5 ( 4 ) ( v 1 ′ − u 3 ) G δ 4 ( 4 ) ( u 3 − u 2 ) G δ 3 ( 4 ) ( u 2 − u 1 ) G δ 2 ( 4 ) ( u 1 − x 2 ′ ) G δ 1 ( 4 ) ( x 2 ′ − x 1 ′ ) . To represent the product of the 4 n 4n 4 n G δ i ( j ) G_{\delta_i^{(j)}} G δ i ( j ) i = 1 , … , n i=1,\ldots,n i = 1 , … , n j = 1 , … , 4 j=1,\ldots,4 j = 1 , … , 4 G w h i t e = ( V w h i t e , E w h i t e ) G_{\mathrm{white}}=(V_{\mathrm{white}},E_{\mathrm{white}}) G white = ( V white , E white )
V w h i t e = { x 1 , … , x ℓ , x 1 ′ , … , x ℓ ′ , u 1 , … , u r − ℓ , v 1 , … , v n − r , v 1 ′ , … , v n − r ′ , 0 , b 1 , b 2 , b 3 } . V_{\mathrm{white}}=\{x_1,\ldots,x_\ell,x'_1,\ldots,x'_\ell,u_1,\ldots,u_{r-\ell},v_1,\ldots,v_{n-r},v'_1,\ldots,v'_{n-r},0,b_1,b_2,b_3\}. V white = { x 1 , … , x ℓ , x 1 ′ , … , x ℓ ′ , u 1 , … , u r − ℓ , v 1 , … , v n − r , v 1 ′ , … , v n − r ′ , 0 , b 1 , b 2 , b 3 } . We connect two vertices u u u v v v G δ i ( j ) G_{\delta_i^{(j)}} G δ i ( j ) i = 1 , … , n i=1,\ldots,n i = 1 , … , n j = 1 , … , 4 j=1,\ldots,4 j = 1 , … , 4 Figure 1 shows this graph for n = 7 n=7 n = 7 ℓ = 2 \ell=2 ℓ = 2 r = 5 r=5 r = 5
Figure 1. Graph for representing J 0 J_0 J 0 n = 7 n=7 n = 7 ℓ = 2 \ell=2 ℓ = 2 r = 5 r=5 r = 5
We apply Lemma B.1 with g t = G t g_t=G_t g t = G t G t G_t G t G t ∈ L ∞ ( R ) ∩ L 1 ( R ) G_t\in L^\infty(\mathbb{R})\cap L^1(\mathbb{R}) G t ∈ L ∞ ( R ) ∩ L 1 ( R )
∥ G t ∥ L ∞ ( R ) = { ( 2 π t ) − 1 / 2 for heat equation 1 / 2 for wave equation ∥ G t ∥ L 1 ( R ) = { 1 for heat equation t for wave equation (34) \begin{aligned}
\lVert G_t\rVert_{L^\infty(\mathbb{R})} &=
\begin{cases}
(2\pi t)^{-1/2} & \text{for heat equation}\\
1/2 & \text{for wave equation}
\end{cases}
&
\lVert G_t\rVert_{L^1(\mathbb{R})} &=
\begin{cases}
1 & \text{for heat equation}\\
t & \text{for wave equation}
\end{cases} \tag*{(34)}
\end{aligned} ∥ G t ∥ L ∞ ( R ) = { ( 2 π t ) − 1/2 1/2 for heat equation for wave equation ∥ G t ∥ L 1 ( R ) = { 1 t for heat equation for wave equation (34) The fixed vertex (for which there is no integration variable in J 0 J_0 J 0 p = 0 p=0 p = 0 T T T G w h i t e G_{\mathrm{white}} G white
J 0 ( δ ( 1 ) , … , δ ( 4 ) ) ≤ ∏ i = 1 4 ( ∏ j ∈ J i ∥ G δ j ( i ) ∥ L ∞ ( R ) ∏ j ∈ I i ∥ G δ j ( i ) ∥ L 1 ( R ) ) , (35) J_0(\delta^{(1)},\ldots,\delta^{(4)})\leq\prod_{i=1}^{4}\left(\prod_{j\in J_i}\lVert G_{\delta_j^{(i)}}\rVert_{L^\infty(\mathbb{R})}\prod_{j\in I_i}\lVert G_{\delta_j^{(i)}}\rVert_{L^1(\mathbb{R})}\right), \tag*{(35)} J 0 ( δ ( 1 ) , … , δ ( 4 ) ) ≤ i = 1 ∏ 4 j ∈ J i ∏ ∥ G δ j ( i ) ∥ L ∞ ( R ) j ∈ I i ∏ ∥ G δ j ( i ) ∥ L 1 ( R ) , (35) where I i I_i I i j ≤ n j\leq n j ≤ n J i = { 1 , … , n } ∖ I i J_i=\{1,\ldots,n\}\setminus I_i J i = { 1 , … , n } ∖ I i
For instance, in the example with n = 7 n=7 n = 7 ℓ = 2 \ell=2 ℓ = 2 r = 5 r=5 r = 5 T T T E T E_T E T Figure 1 :
from the blue path: { x 1 , x 2 } \{x_1,x_2\} { x 1 , x 2 } { x 2 , u 1 } \{x_2,u_1\} { x 2 , u 1 } { u 1 , u 2 } \{u_1,u_2\} { u 1 , u 2 } { v 1 , v 2 } \{v_1,v_2\} { v 1 , v 2 } { v 2 , 0 } \{v_2,0\} { v 2 , 0 }
from the pink path: { v 1 ′ , v 2 ′ } \{v'_1,v'_2\} { v 1 ′ , v 2 ′ } { v 2 ′ , b 1 } \{v'_2,b_1\} { v 2 ′ , b 1 }
from the green path: { x 2 ′ , u 1 } \{x'_2,u_1\} { x 2 ′ , u 1 } { u 2 , u 3 } \{u_2,u_3\} { u 2 , u 3 } { u 3 , v 1 } \{u_3,v_1\} { u 3 , v 1 } { v 2 , b 2 } \{v_2,b_2\} { v 2 , b 2 }
from the purple path: { x 1 ′ , x 2 ′ } \{x'_1,x'_2\} { x 1 ′ , x 2 ′ } { u 3 , v 1 ′ } \{u_3,v'_1\} { u 3 , v 1 ′ } { v 2 ′ , b 3 } \{v'_2,b_3\} { v 2 ′ , b 3 }
Figure 2. The multigraph in Figure 1: edges of spanning tree T T T
In J 0 J_0 J 0 T T T
J 0 ( δ ( 1 ) , … , δ ( 4 ) ) = ∫ R 14 d x 2 d x 2 ′ d u 3 d v 2 d v 2 ′ d b 3 G δ 7 ( 1 ) ( 0 − v 2 ) G δ 6 ( 1 ) ( v 2 − v 1 ) G δ 5 ( 1 ) ( v 1 − u 3 ) G δ 4 ( 1 ) ( u 3 − u 2 ) G δ 3 ( 1 ) ( u 2 − u 1 ) G δ 2 ( 1 ) ( u 1 − x 2 ) G δ 1 ( 1 ) ( x 2 − x 1 ) G δ 7 ( 2 ) ( b 1 − v 2 ′ ) G δ 6 ( 2 ) ( v 2 ′ − v 1 ′ ) G δ 5 ( 2 ) ( v 1 ′ − u 3 ) G δ 4 ( 2 ) ( u 3 − u 2 ) G δ 3 ( 2 ) ( u 2 − u 1 ) G δ 2 ( 2 ) ( u 1 − x 2 ) G δ 1 ( 2 ) ( x 2 − x 1 ) G δ 7 ( 3 ) ( b 2 − v 2 ) G δ 6 ( 3 ) ( v 2 − v 1 ) G δ 5 ( 3 ) ( v 1 − u 3 ) G δ 4 ( 3 ) ( u 3 − u 2 ) G δ 3 ( 3 ) ( u 2 − u 1 ) G δ 2 ( 3 ) ( u 1 − x 2 ′ ) G δ 1 ( 3 ) ( x 2 ′ − x 1 ′ ) G δ 7 ( 4 ) ( b 3 − v 2 ′ ) G δ 6 ( 4 ) ( v 2 ′ − v 1 ′ ) G δ 5 ( 4 ) ( v 1 ′ − u 3 ) G δ 4 ( 4 ) ( u 3 − u 2 ) G δ 3 ( 4 ) ( u 2 − u 1 ) G δ 2 ( 4 ) ( u 1 − x 2 ′ ) G δ 1 ( 4 ) ( x 2 ′ − x 1 ′ ) . J_0(\delta^{(1)},\ldots,\delta^{(4)})=\int_{\mathbb{R}^{14}} dx_2\,dx'_2\,du_3\,dv_2\,dv'_2\,db_3
\begin{aligned}
& G_{\delta^{(1)}_7}(0-v_2)\,G_{\delta^{(1)}_6}(v_2-v_1)\,G_{\delta^{(1)}_5}(v_1-u_3)\,G_{\delta^{(1)}_4}(u_3-u_2)\,G_{\delta^{(1)}_3}(u_2-u_1)\,G_{\delta^{(1)}_2}(u_1-x_2)\,G_{\delta^{(1)}_1}(x_2-x_1)\\
& G_{\delta^{(2)}_7}(b_1-v'_2)\,G_{\delta^{(2)}_6}(v'_2-v'_1)\,G_{\delta^{(2)}_5}(v'_1-u_3)\,G_{\delta^{(2)}_4}(u_3-u_2)\,G_{\delta^{(2)}_3}(u_2-u_1)\,G_{\delta^{(2)}_2}(u_1-x_2)\,G_{\delta^{(2)}_1}(x_2-x_1)\\
& G_{\delta^{(3)}_7}(b_2-v_2)\,G_{\delta^{(3)}_6}(v_2-v_1)\,G_{\delta^{(3)}_5}(v_1-u_3)\,G_{\delta^{(3)}_4}(u_3-u_2)\,G_{\delta^{(3)}_3}(u_2-u_1)\,G_{\delta^{(3)}_2}(u_1-x'_2)\,G_{\delta^{(3)}_1}(x'_2-x'_1)\\
& G_{\delta^{(4)}_7}(b_3-v'_2)\,G_{\delta^{(4)}_6}(v'_2-v'_1)\,G_{\delta^{(4)}_5}(v'_1-u_3)\,G_{\delta^{(4)}_4}(u_3-u_2)\,G_{\delta^{(4)}_3}(u_2-u_1)\,G_{\delta^{(4)}_2}(u_1-x'_2)\,G_{\delta^{(4)}_1}(x'_2-x'_1).
\end{aligned} J 0 ( δ ( 1 ) , … , δ ( 4 ) ) = ∫ R 14 d x 2 d x 2 ′ d u 3 d v 2 d v 2 ′ d b 3 G δ 7 ( 1 ) ( 0 − v 2 ) G δ 6 ( 1 ) ( v 2 − v 1 ) G δ 5 ( 1 ) ( v 1 − u 3 ) G δ 4 ( 1 ) ( u 3 − u 2 ) G δ 3 ( 1 ) ( u 2 − u 1 ) G δ 2 ( 1 ) ( u 1 − x 2 ) G δ 1 ( 1 ) ( x 2 − x 1 ) G δ 7 ( 2 ) ( b 1 − v 2 ′ ) G δ 6 ( 2 ) ( v 2 ′ − v 1 ′ ) G δ 5 ( 2 ) ( v 1 ′ − u 3 ) G δ 4 ( 2 ) ( u 3 − u 2 ) G δ 3 ( 2 ) ( u 2 − u 1 ) G δ 2 ( 2 ) ( u 1 − x 2 ) G δ 1 ( 2 ) ( x 2 − x 1 ) G δ 7 ( 3 ) ( b 2 − v 2 ) G δ 6 ( 3 ) ( v 2 − v 1 ) G δ 5 ( 3 ) ( v 1 − u 3 ) G δ 4 ( 3 ) ( u 3 − u 2 ) G δ 3 ( 3 ) ( u 2 − u 1 ) G δ 2 ( 3 ) ( u 1 − x 2 ′ ) G δ 1 ( 3 ) ( x 2 ′ − x 1 ′ ) G δ 7 ( 4 ) ( b 3 − v 2 ′ ) G δ 6 ( 4 ) ( v 2 ′ − v 1 ′ ) G δ 5 ( 4 ) ( v 1 ′ − u 3 ) G δ 4 ( 4 ) ( u 3 − u 2 ) G δ 3 ( 4 ) ( u 2 − u 1 ) G δ 2 ( 4 ) ( u 1 − x 2 ′ ) G δ 1 ( 4 ) ( x 2 ′ − x 1 ′ ) . We drop the factors which are not in a box, using the fact that G t ∈ L ∞ ( R ) G_t\in L^\infty(\mathbb{R}) G t ∈ L ∞ ( R ) G t ∈ L 1 ( R ) G_t\in L^1(\mathbb{R}) G t ∈ L 1 ( R ) v n − r v_{n-r} v n − r G δ n + 1 ( 1 ) ( 0 − v n − r ) G_{\delta^{(1)}_{n+1}}(0-v_{n-r}) G δ n + 1 ( 1 ) ( 0 − v n − r )
For this particular example and spanning tree T T T
I 1 = { 1 , 2 , 3 , 6 , 7 } , I 2 = { 6 , 7 } , I 3 = { 2 , 4 , 5 , 7 } , I 4 = { 1 , 5 , 7 } . I_1=\{1,2,3,6,7\},\qquad I_2=\{6,7\},\qquad I_3=\{2,4,5,7\},\qquad I_4=\{1,5,7\}. I 1 = { 1 , 2 , 3 , 6 , 7 } , I 2 = { 6 , 7 } , I 3 = { 2 , 4 , 5 , 7 } , I 4 = { 1 , 5 , 7 } . We leave the example now, and return to the general argument. By (35) and (34),
J 0 ( δ ( 1 ) , … , δ ( 4 ) ) ≤ { ∏ i = 1 4 ∏ j ∈ J i ( 2 π δ i ( j ) ) − 1 / 2 for the heat equation , ∏ i = 1 4 ( 1 / 2 ) ∣ J i ∣ ∏ j ∈ I i δ i ( j ) for the wave equation . J_0(\delta^{(1)},\ldots,\delta^{(4)})\le
\begin{cases}
\prod_{i=1}^{4}\prod_{j\in J_i}(2\pi\delta^{(j)}_i)^{-1/2} & \text{for the heat equation},\\
\prod_{i=1}^{4}(1/2)^{|J_i|}\prod_{j\in I_i}\delta^{(j)}_i & \text{for the wave equation}.
\end{cases} J 0 ( δ ( 1 ) , … , δ ( 4 ) ) ≤ { ∏ i = 1 4 ∏ j ∈ J i ( 2 π δ i ( j ) ) − 1/2 ∏ i = 1 4 ( 1/2 ) ∣ J i ∣ ∏ j ∈ I i δ i ( j ) for the heat equation , for the wave equation . Hence,
J 0 ≤ ∏ i = 1 4 ∫ D n ( t ) ∏ j = 1 n ( δ j ( i ) ) β j ( i ) d δ ( i ) . J_0\le\prod_{i=1}^{4}\int_{D_n(t)}\prod_{j=1}^{n}(\delta^{(i)}_j)^{\beta^{(i)}_j}\,d\delta^{(i)}. J 0 ≤ i = 1 ∏ 4 ∫ D n ( t ) j = 1 ∏ n ( δ j ( i ) ) β j ( i ) d δ ( i ) . where
β j ( i ) = { − 1 / 2 if i ∈ J i , 0 if j ∈ I i for the heat equation, \beta^{(i)}_j=
\begin{cases}
-1/2 & \text{if }i\in J_i,\\
0 & \text{if }j\in I_i
\end{cases}
\qquad\text{for the heat equation,} β j ( i ) = { − 1/2 0 if i ∈ J i , if j ∈ I i for the heat equation, and
β j ( i ) = { 0 if i ∈ J i , 1 if j ∈ I i for the wave equation. \beta^{(i)}_j=
\begin{cases}
0 & \text{if }i\in J_i,\\
1 & \text{if }j\in I_i
\end{cases}
\qquad\text{for the wave equation.} β j ( i ) = { 0 1 if i ∈ J i , if j ∈ I i for the wave equation. For both equations, it follows that J 0 < ∞ J_0<\infty J 0 < ∞ β 1 , … , β n > − 1 \beta_1,\ldots,\beta_n>-1 β 1 , … , β n > − 1
∫ D n ( t ) ∏ j = 1 n δ j β j d δ 1 … d δ n = ∏ j = 1 n Γ ( β j + 1 ) t ∣ β ∣ + n Γ ( ∣ β ∣ + n + 1 ) with ∣ β ∣ = ∑ j = 1 n β j . (36) \int_{D_n(t)}\prod_{j=1}^{n}\delta_j^{\beta_j}\,d\delta_1\ldots d\delta_n
=
\frac{\prod_{j=1}^{n}\Gamma(\beta_j+1)t^{|\beta|+n}}{\Gamma(|\beta|+n+1)}
\qquad\text{with}\qquad|\beta|=\sum_{j=1}^{n}\beta_j. \tag*{(36)} ∫ D n ( t ) j = 1 ∏ n δ j β j d δ 1 … d δ n = Γ ( ∣ β ∣ + n + 1 ) ∏ j = 1 n Γ ( β j + 1 ) t ∣ β ∣ + n with ∣ β ∣ = j = 1 ∑ n β j . (36) We now return to the case of general permutations. The same method that was used above for showing that J 0 < ∞ J_0<\infty J 0 < ∞ ρ 1 , … , ρ 4 ∈ S n \rho_1,\ldots,\rho_4 \in S_n ρ 1 , … , ρ 4 ∈ S n n = 7 n=7 n = 7 ℓ = 2 \ell=2 ℓ = 2 r = 5 r=5 r = 5 Figure 3 gives this multigraph, which correspond to permutations ρ 1 , … , ρ 4 ∈ S 7 \rho_1,\ldots,\rho_4 \in S_7 ρ 1 , … , ρ 4 ∈ S 7
Figure 3. The multigraph for some general permutations ρ 1 , … , ρ 4 \rho_1,\ldots,\rho_4 ρ 1 , … , ρ 4
blue path: x 1 → v 1 → u 3 → u 2 → x 2 → v 2 → u 1 → 0 x_1 \to v_1 \to u_3 \to u_2 \to x_2 \to v_2 \to u_1 \to0 x 1 → v 1 → u 3 → u 2 → x 2 → v 2 → u 1 → 0
pink path: u 3 → v 2 ′ → x 1 → u 2 → v 1 ′ → u 1 → x 2 → b 1 u_3 \to v'_2 \to x_1 \to u_2 \to v'_1 \to u_1 \to x_2 \to b_1 u 3 → v 2 ′ → x 1 → u 2 → v 1 ′ → u 1 → x 2 → b 1
green path: v 1 → u 2 → x 1 ′ → u 3 → x 2 ′ → u 1 → v 2 → b 2 v_1 \to u_2 \to x'_1 \to u_3 \to x'_2 \to u_1 \to v_2 \to b_2 v 1 → u 2 → x 1 ′ → u 3 → x 2 ′ → u 1 → v 2 → b 2
purple path: u 3 → x 1 ′ → u 2 → v 2 ′ → u 1 → x 2 ′ → v 1 ′ → b 3 u_3 \to x'_1 \to u_2 \to v'_2 \to u_1 \to x'_2 \to v'_1 \to b_3 u 3 → x 1 ′ → u 2 → v 2 ′ → u 1 → x 2 ′ → v 1 ′ → b 3
We omit the details. This concludes the proof of the theorem in Case 1.
Case 2. (colored noise) We fix some permutations ρ 1 , … , ρ 4 ∈ S n \rho_1,\ldots,\rho_4 \in S_n ρ 1 , … , ρ 4 ∈ S n J ρ 1 , … , ρ 4 J_{\rho_1,\ldots,\rho_4} J ρ 1 , … , ρ 4
( x ℓ , u r − ℓ , v n − r ) = ( w 1 ( 1 ) , … , w n ( 1 ) ) = : w ( 1 ) , w n + 1 ( 1 ) = 0 (x_\ell,u_{r-\ell},v_{n-r})=(w_1^{(1)},\ldots,w_n^{(1)})=:w^{(1)},\qquad w_{n+1}^{(1)}=0 ( x ℓ , u r − ℓ , v n − r ) = ( w 1 ( 1 ) , … , w n ( 1 ) ) =: w ( 1 ) , w n + 1 ( 1 ) = 0 ( x ℓ , u r − ℓ , v n − r ′ ) = ( w 1 ( 2 ) , … , w n ( 2 ) ) = : w ( 2 ) , w n + 1 ( 2 ) = b 1 (x_\ell,u_{r-\ell},v'_{n-r})=(w_1^{(2)},\ldots,w_n^{(2)})=:w^{(2)},\qquad w_{n+1}^{(2)}=b_1 ( x ℓ , u r − ℓ , v n − r ′ ) = ( w 1 ( 2 ) , … , w n ( 2 ) ) =: w ( 2 ) , w n + 1 ( 2 ) = b 1 ( x ℓ ′ , u r − ℓ , v n − r ) = ( w 1 ( 3 ) , … , w n ( 3 ) ) = : w ( 3 ) , w n + 1 ( 3 ) = b 2 (x'_\ell,u_{r-\ell},v_{n-r})=(w_1^{(3)},\ldots,w_n^{(3)})=:w^{(3)},\qquad w_{n+1}^{(3)}=b_2 ( x ℓ ′ , u r − ℓ , v n − r ) = ( w 1 ( 3 ) , … , w n ( 3 ) ) =: w ( 3 ) , w n + 1 ( 3 ) = b 2 ( x ℓ ′ , u r − ℓ , v n − r ′ ) = ( w 1 ( 4 ) , … , w n ( 4 ) ) = : w ( 4 ) , w n + 1 ( 4 ) = b 3 . (x'_\ell,u_{r-\ell},v'_{n-r})=(w_1^{(4)},\ldots,w_n^{(4)})=:w^{(4)},\qquad w_{n+1}^{(4)}=b_3. ( x ℓ ′ , u r − ℓ , v n − r ′ ) = ( w 1 ( 4 ) , … , w n ( 4 ) ) =: w ( 4 ) , w n + 1 ( 4 ) = b 3 . Hence,
J ρ 1 , … , ρ 4 = ∫ ( R d ) 4 n + 3 ∏ i = 1 4 φ n , ρ i ( w ( i ) , w n + 1 ( i ) ; t ) 1 S ( w ( 1 ) , w ( 2 ) , w ( 3 ) , w ( 4 ) ) ∏ i = 1 4 d w ( i ) ∏ i = 2 4 d w n + 1 ( i ) , J_{\rho_1,\ldots,\rho_4}=\int_{(\mathbb{R}^d)^{4n+3}}\prod_{i=1}^{4}\varphi_{n,\rho_i}(w^{(i)},w_{n+1}^{(i)};t)\,1_S(w^{(1)},w^{(2)},w^{(3)},w^{(4)})\prod_{i=1}^{4}dw^{(i)}\prod_{i=2}^{4}dw_{n+1}^{(i)}, J ρ 1 , … , ρ 4 = ∫ ( R d ) 4 n + 3 i = 1 ∏ 4 φ n , ρ i ( w ( i ) , w n + 1 ( i ) ; t ) 1 S ( w ( 1 ) , w ( 2 ) , w ( 3 ) , w ( 4 ) ) i = 1 ∏ 4 d w ( i ) i = 2 ∏ 4 d w n + 1 ( i ) , where S S S
w j ( 1 ) = w j ( 2 ) = : x j and w j ( 3 ) = w j ( 4 ) = : x j ′ for all j = 1 , … , ℓ , w j ( 1 ) = w j ( 2 ) = w j ( 3 ) = w j ( 4 ) = : u j − ℓ for all j = ℓ + 1 , … , r , w j ( 1 ) = w j ( 3 ) = : v j − r and w j ( 2 ) = w j ( 4 ) = : v j − r ′ for all j = r + 1 , … , n . (37) \begin{aligned}
w_j^{(1)}=w_j^{(2)}=:x_j\quad&\text{and}\quad w_j^{(3)}=w_j^{(4)}=:x'_j &&\text{for all }j=1,\ldots,\ell,\\
w_j^{(1)}=w_j^{(2)}=w_j^{(3)}=w_j^{(4)}=:u_{j-\ell} &&\text{for all }j=\ell+1,\ldots,r,\\
w_j^{(1)}=w_j^{(3)}=:v_{j-r}\quad&\text{and}\quad w_j^{(2)}=w_j^{(4)}=:v'_{j-r} &&\text{for all }j=r+1,\ldots,n. \tag*{(37)}
\end{aligned} w j ( 1 ) = w j ( 2 ) =: x j w j ( 1 ) = w j ( 2 ) = w j ( 3 ) = w j ( 4 ) =: u j − ℓ w j ( 1 ) = w j ( 3 ) =: v j − r and w j ( 3 ) = w j ( 4 ) =: x j ′ and w j ( 2 ) = w j ( 4 ) =: v j − r ′ for all j = ℓ + 1 , … , r , for all j = 1 , … , ℓ , for all j = r + 1 , … , n . (37) By definition (12) and the change of variable δ j = t j + 1 − t j \delta_j = t_{j+1} - t_j δ j = t j + 1 − t j j ≤ n − 1 j \le n - 1 j ≤ n − 1 δ n = t − t n \delta_n = t - t_n δ n = t − t n
φ n ( x , x ~ ; t ) = ∫ D n ( t ) ∫ ( R d ) n ∏ j = 1 n − 1 G δ j ( x ~ j + 1 − x ~ j ) G δ n ( x − x ~ n ) ∏ j = 1 n κ ( x j − x ~ j ) d x ~ d δ , \varphi_n(x,\widetilde{x};t)=\int_{D_n(t)}\int_{(\mathbb{R}^{d})^n}\prod_{j=1}^{n-1}G_{\delta_j}(\widetilde{x}_{j+1}-\widetilde{x}_j)G_{\delta_n}(x-\widetilde{x}_n)\prod_{j=1}^{n}\kappa(x_j-\widetilde{x}_j)d\widetilde{x}d\delta, φ n ( x , x ; t ) = ∫ D n ( t ) ∫ ( R d ) n j = 1 ∏ n − 1 G δ j ( x j + 1 − x j ) G δ n ( x − x n ) j = 1 ∏ n κ ( x j − x j ) d x d δ , where x = ( x 1 , … , x n ) x=(x_1,\ldots,x_n) x = ( x 1 , … , x n ) x ~ = ( x ~ 1 , … , x ~ n ) \widetilde{x}=(\widetilde{x}_1,\ldots,\widetilde{x}_n) x = ( x 1 , … , x n ) δ = ( δ 1 , … , δ n ) \delta=(\delta_1,\ldots,\delta_n) δ = ( δ 1 , … , δ n )
φ n , ρ ( w , w n + 1 ; t ) = φ n ( w ρ ( 1 ) , … , w ρ ( n + 1 ) , w n + 1 ; t ) = ∫ D n ( t ) ∫ ( R d ) n ∏ j = 1 n − 1 G δ j ( w ~ j + 1 − w ~ j ) G δ n ( w n + 1 − w ~ n ) ∏ j = 1 n κ ( w ρ ( j ) − w ~ j ) d w ~ d δ , \begin{aligned}
\varphi_{n,\rho}(w,w_{n+1};t)&=\varphi_n(w_{\rho(1)},\ldots,w_{\rho(n+1)},w_{n+1};t)\\
&=\int_{D_n(t)}\int_{(\mathbb{R}^{d})^n}\prod_{j=1}^{n-1}G_{\delta_j}(\widetilde{w}_{j+1}-\widetilde{w}_j)G_{\delta_n}(w_{n+1}-\widetilde{w}_n)\prod_{j=1}^{n}\kappa(w_{\rho(j)}-\widetilde{w}_j)d\widetilde{w}d\delta,
\end{aligned} φ n , ρ ( w , w n + 1 ; t ) = φ n ( w ρ ( 1 ) , … , w ρ ( n + 1 ) , w n + 1 ; t ) = ∫ D n ( t ) ∫ ( R d ) n j = 1 ∏ n − 1 G δ j ( w j + 1 − w j ) G δ n ( w n + 1 − w n ) j = 1 ∏ n κ ( w ρ ( j ) − w j ) d w d δ , where w = ( w 1 , … , w n ) w=(w_1,\ldots,w_n) w = ( w 1 , … , w n ) w ~ = ( w ~ 1 , … , w ~ n ) \widetilde{w}=(\widetilde{w}_1,\ldots,\widetilde{w}_n) w = ( w 1 , … , w n ) (33) holds with
J ρ 1 , … , ρ 4 ( δ ( 1 ) , … , δ ( 4 ) ) = ∫ ( R d ) 8 n + 3 ∏ i = 1 4 d w ( i ) ∏ i = 1 4 d w ~ ( i ) ∏ i = 2 4 d w n + 1 ( i ) 1 S ( w ( 1 ) , w ( 2 ) , w ( 3 ) , w ( 4 ) ) × ∏ i = 1 4 ( ∏ j = 1 n − 1 G δ j ( i ) ( w ~ j + 1 ( i ) − w ~ j ( i ) ) G δ n ( i ) ( w n + 1 ( i ) − w ~ n ( i ) ) ∏ j = 1 n κ ( w ρ i ( j ) ( i ) − w ~ j ( i ) ) ) , (38) \begin{aligned}
J_{\rho_1,\ldots,\rho_4}(\delta^{(1)},\ldots,\delta^{(4)})={}&\int_{(\mathbb{R}^{d})^{8n+3}}\prod_{i=1}^{4}dw^{(i)}\prod_{i=1}^{4}d\widetilde{w}^{(i)}\prod_{i=2}^{4}dw_{n+1}^{(i)}\mathbf{1}_{S}(w^{(1)},w^{(2)},w^{(3)},w^{(4)})\\
&\quad{}\times\prod_{i=1}^{4}\left(\prod_{j=1}^{n-1}G_{\delta_j^{(i)}}(\widetilde{w}_{j+1}^{(i)}-\widetilde{w}_{j}^{(i)})G_{\delta_n^{(i)}}(w_{n+1}^{(i)}-\widetilde{w}_{n}^{(i)})\prod_{j=1}^{n}\kappa(w_{\rho_i(j)}^{(i)}-\widetilde{w}_{j}^{(i)})\right), \tag*{(38)}
\end{aligned} J ρ 1 , … , ρ 4 ( δ ( 1 ) , … , δ ( 4 ) ) = ∫ ( R d ) 8 n + 3 i = 1 ∏ 4 d w ( i ) i = 1 ∏ 4 d w ( i ) i = 2 ∏ 4 d w n + 1 ( i ) 1 S ( w ( 1 ) , w ( 2 ) , w ( 3 ) , w ( 4 ) ) × i = 1 ∏ 4 ( j = 1 ∏ n − 1 G δ j ( i ) ( w j + 1 ( i ) − w j ( i ) ) G δ n ( i ) ( w n + 1 ( i ) − w n ( i ) ) j = 1 ∏ n κ ( w ρ i ( j ) ( i ) − w j ( i ) ) ) , (38) where w ~ ( i ) = ( w ~ 1 ( i ) , … , w ~ n ( i ) ) \widetilde{w}^{(i)}=(\widetilde{w}_1^{(i)},\ldots,\widetilde{w}_n^{(i)}) w ( i ) = ( w 1 ( i ) , … , w n ( i ) ) i = 1 , … , 4 i=1,\ldots,4 i = 1 , … , 4 J ρ 1 , … , ρ 4 ( δ ( 1 ) , … , δ ( 4 ) ) J_{\rho_1,\ldots,\rho_4}(\delta^{(1)},\ldots,\delta^{(4)}) J ρ 1 , … , ρ 4 ( δ ( 1 ) , … , δ ( 4 ) )
G c o l o r e d = ( V c o l o r e d , E c o l o r e d ) with V c o l o r e d = ⋃ i = 1 4 V i , E c o l o r e d = ⋃ i = 1 4 E i , (39) G_{\mathrm{colored}}=(V_{\mathrm{colored}},E_{\mathrm{colored}})\quad\text{with}\quad V_{\mathrm{colored}}=\bigcup_{i=1}^{4}V_i,\quad E_{\mathrm{colored}}=\bigcup_{i=1}^{4}E_i, \tag*{(39)} G colored = ( V colored , E colored ) with V colored = i = 1 ⋃ 4 V i , E colored = i = 1 ⋃ 4 E i , (39) where
V i = { w 1 ( i ) , … , w n ( i ) , w n + 1 ( i ) , w ~ 1 ( i ) , … , w ~ n ( i ) } V_i=\left\{w_1^{(i)},\ldots,w_n^{(i)},w_{n+1}^{(i)},\widetilde{w}_1^{(i)},\ldots,\widetilde{w}_n^{(i)}\right\} V i = { w 1 ( i ) , … , w n ( i ) , w n + 1 ( i ) , w 1 ( i ) , … , w n ( i ) } E i = ⋃ j = 1 n − 1 { w ~ j ( i ) , w ~ j + 1 ( i ) } ⋃ { w ~ n ( i ) , w n + 1 ( i ) } ⋃ ⋃ j = 1 n { w ρ i ( j ) ( i ) , w ~ j ( i ) } . E_i=\bigcup_{j=1}^{n-1}\left\{\widetilde{w}_j^{(i)},\widetilde{w}_{j+1}^{(i)}\right\}\bigcup\left\{\widetilde{w}_n^{(i)},w_{n+1}^{(i)}\right\}\bigcup\bigcup_{j=1}^{n}\left\{w_{\rho_i(j)}^{(i)},\widetilde{w}_j^{(i)}\right\}. E i = j = 1 ⋃ n − 1 { w j ( i ) , w j + 1 ( i ) } ⋃ { w n ( i ) , w n + 1 ( i ) } ⋃ j = 1 ⋃ n { w ρ i ( j ) ( i ) , w j ( i ) } . The vertices of G c o l o r e d G_{\mathrm{colored}} G colored 8 n + 3 8n+3 8 n + 3 J ρ 1 , … , ρ 4 ( δ ( 1 ) , … , δ ( 4 ) ) J_{\rho_1,\ldots,\rho_4}(\delta^{(1)},\ldots,\delta^{(4)}) J ρ 1 , … , ρ 4 ( δ ( 1 ) , … , δ ( 4 ) ) w n + 1 ( 1 ) = 0 w_{n+1}^{(1)}=0 w n + 1 ( 1 ) = 0 u u u v v v G δ j ( i ) ( u − v ) G_{\delta_j^{(i)}}(u-v) G δ j ( i ) ( u − v ) κ ( u − v ) \kappa(u-v) κ ( u − v ) J ρ 1 , … , ρ 4 ( δ ( 1 ) , … , δ ( 4 ) ) J_{\rho_1,\ldots,\rho_4}(\delta^{(1)},\ldots,\delta^{(4)}) J ρ 1 , … , ρ 4 ( δ ( 1 ) , … , δ ( 4 ) )
Note that the vertices w ~ ( i ) \widetilde{w}^{(i)} w ( i ) w ( i ) w^{(i)} w ( i ) (37) . More precisely, card ( V ) = 4 ( 2 n + 1 ) − 2 ℓ − 3 ( r − ℓ ) − 2 ( n − r ) \operatorname{card}(V)=4(2n+1)-2\ell-3(r-\ell)-2(n-r) card ( V ) = 4 ( 2 n + 1 ) − 2 ℓ − 3 ( r − ℓ ) − 2 ( n − r )
We consider separately Cases (ii) and (iii). We will select a spanning tree of G c o l o r e d G_{\mathrm{colored}} G colored (38) which correspond to an edge in the tree, and we drop the factors which are not in a box. Then we pick leaves from this tree, and we integrate with respect to them one by one. (A leaf is a variable which appears only once.)
In Case (ii), we use Lemma 4.6 to select a spanning tree which contains all edges of the form { w ~ ρ i ( j ) ( i ) , w ~ j ( i ) } \{\widetilde{w}^{(i)}_{\rho_i(j)}, \widetilde{w}^{(i)}_j\} { w ρ i ( j ) ( i ) , w j ( i ) } { w ~ n ( 1 ) , w n + 1 ( 1 ) } \{\widetilde{w}^{(1)}_n,w^{(1)}_{n+1}\} { w n ( 1 ) , w n + 1 ( 1 ) } k k k G δ n ( 1 ) ( w ~ n ( 1 ) , w n + 1 ( 1 ) ) G_{\delta^{(1)}_n}(\widetilde{w}^{(1)}_n,w^{(1)}_{n+1}) G δ n ( 1 ) ( w n ( 1 ) , w n + 1 ( 1 ) )
Step 1. The factors which are not in a box are all G G G G t ∈ L ∞ ( R ) G_t \in L^\infty(\mathbb{R}) G t ∈ L ∞ ( R ) d = 1 d=1 d = 1 ( 2 π δ j ( i ) ) − 1 / 2 (2\pi\delta^{(i)}_j)^{-1/2} ( 2 π δ j ( i ) ) − 1/2 1 / 2 1/2 1/2 κ \kappa κ G G G
Step 2. We pick leaves from the tree, and we integrate with respect to them one by one, using the fact that κ ∈ L 1 ( R ) \kappa\in L^1(\mathbb{R}) κ ∈ L 1 ( R ) G t ∈ L 1 ( R ) G_t \in L^1(\mathbb{R}) G t ∈ L 1 ( R ) w ~ n + 1 ( 1 ) \widetilde{w}^{(1)}_{n+1} w n + 1 ( 1 ) w n + 1 ( 1 ) = 0 w^{(1)}_{n+1}=0 w n + 1 ( 1 ) = 0 G δ n ( 1 ) ( 0 − w n + 1 ( 1 ) ) G_{\delta^{(1)}_n}(0-w^{(1)}_{n+1}) G δ n ( 1 ) ( 0 − w n + 1 ( 1 ) ) w n + 1 ( 1 ) = 0 w^{(1)}_{n+1}=0 w n + 1 ( 1 ) = 0 δ j ( i ) \delta^{(i)}_j δ j ( i )
Step 3. The integral with respect to δ j ( i ) \delta^{(i)}_j δ j ( i ) J ρ 1 , … , ρ 4 < ∞ J_{\rho_1,\ldots,\rho_4}<\infty J ρ 1 , … , ρ 4 < ∞
In Case (iii), we use Lemma 4.8 to select a spanning tree which contains all edges of the form { w ~ j ( i ) , w ~ j + 1 ( i ) } \{\widetilde{w}^{(i)}_j,\widetilde{w}^{(i)}_{j+1}\} { w j ( i ) , w j + 1 ( i ) } { w ~ n ( i ) , w n + 1 ( i ) } \{\widetilde{w}^{(i)}_n,w^{(i)}_{n+1}\} { w n ( i ) , w n + 1 ( i ) } G G G
Step 1. The factors which are not in a box are all κ \kappa κ κ ∈ L ∞ ( R 2 ) \kappa\in L^\infty(\mathbb{R}^2) κ ∈ L ∞ ( R 2 ) G G G κ \kappa κ
Steps 2-3. These steps are exactly as in Case (ii).
Lemma 4.5. The graph G c o l o r e d G_{\mathrm{colored}} G colored
Proof. For i i i G i = ( V i , E i ) G_i=(V_i,E_i) G i = ( V i , E i ) G G G w ~ j ( i ) \widetilde{w}^{(i)}_j w j ( i ) w ρ i ( j ) ( i ) w^{(i)}_{\rho_i(j)} w ρ i ( j ) ( i ) Figure 4 below). For each i = 1 , 2 , 3 , 4 i=1,2,3,4 i = 1 , 2 , 3 , 4
Figure 4. The subgraph G i G_i G i n = 7 n=7 n = 7 G G G κ \kappa κ
We consider different cases to show that G G G ℓ = 0 \ell= 0 ℓ = 0 a , b , c , d ∈ { 1 , 2 , … , n } a,b,c,d \in\{1,2,\ldots,n\} a , b , c , d ∈ { 1 , 2 , … , n }
ρ 1 ( a ) = ρ 2 ( b ) = ρ 3 ( c ) = ρ 4 ( d ) = 1 , \rho_1(a)=\rho_2(b)=\rho_3(c)=\rho_4(d)=1, ρ 1 ( a ) = ρ 2 ( b ) = ρ 3 ( c ) = ρ 4 ( d ) = 1 , and
w ρ 1 ( a ) ( 1 ) = w ρ 2 ( b ) ( 2 ) = w ρ 3 ( c ) ( 3 ) = w ρ 4 ( d ) ( 4 ) = u 1 . w^{(1)}_{\rho_1(a)}=w^{(2)}_{\rho_2(b)}=w^{(3)}_{\rho_3(c)}=w^{(4)}_{\rho_4(d)}=u_1. w ρ 1 ( a ) ( 1 ) = w ρ 2 ( b ) ( 2 ) = w ρ 3 ( c ) ( 3 ) = w ρ 4 ( d ) ( 4 ) = u 1 . Since the subgraphs G i G_i G i G G G
If 0 < ℓ < r 0<\ell<r 0 < ℓ < r ℓ = 0 \ell=0 ℓ = 0 ℓ + 1 \ell+1 ℓ + 1
If 0 < ℓ = r < n 0<\ell=r<n 0 < ℓ = r < n G 1 ∪ G 2 G_1\cup G_2 G 1 ∪ G 2 a , b ∈ { 1 , 2 , … , n } a,b\in\{1,2,\ldots,n\} a , b ∈ { 1 , 2 , … , n }
1 = ρ 1 ( a ) = ρ 2 ( b ) 1=\rho_1(a)=\rho_2(b) 1 = ρ 1 ( a ) = ρ 2 ( b ) and therefore
w ρ 1 ( a ) ( 1 ) = w ρ 2 ( b ) ( 2 ) = x 1 . w^{(1)}_{\rho_1(a)}=w^{(2)}_{\rho_2(b)}=x_1. w ρ 1 ( a ) ( 1 ) = w ρ 2 ( b ) ( 2 ) = x 1 . In the exact same way we can also show that G 3 ∪ G 4 G_3\cup G_4 G 3 ∪ G 4 G 1 ∪ G 3 G_1\cup G_3 G 1 ∪ G 3 c , d ∈ { 1 , 2 , … , n } c,d\in\{1,2,\ldots,n\} c , d ∈ { 1 , 2 , … , n }
ℓ + 1 = ρ 1 ( c ) = ρ 3 ( d ) \ell+1=\rho_1(c)=\rho_3(d) ℓ + 1 = ρ 1 ( c ) = ρ 3 ( d ) and therefore
x ρ 1 ( c ) ( 1 ) = x ρ 3 ( d ) ( 3 ) = x 1 ′ . x^{(1)}_{\rho_1(c)}=x^{(3)}_{\rho_3(d)}=x'_1. x ρ 1 ( c ) ( 1 ) = x ρ 3 ( d ) ( 3 ) = x 1 ′ . Therefore, we can conclude that G = G 1 ∪ G 2 ∪ G 3 ∪ G 4 G=G_1\cup G_2\cup G_3\cup G_4 G = G 1 ∪ G 2 ∪ G 3 ∪ G 4
Lemma 4.6. The graph G c o l o r e d G_{\mathrm{colored}} G colored { w ~ ρ i ( j ) ( i ) , w ~ j ( i ) } \{\widetilde{w}^{(i)}_{\rho_i(j)},\widetilde{w}^{(i)}_j\} { w ρ i ( j ) ( i ) , w j ( i ) } i = 1 , … , 4 i=1,\ldots,4 i = 1 , … , 4 j = 1 , … , n j=1,\ldots,n j = 1 , … , n { w ~ n ( 1 ) , w ~ n + 1 ( 2 ) } \{\widetilde{w}^{(1)}_n,\widetilde{w}^{(2)}_{n+1}\} { w n ( 1 ) , w n + 1 ( 2 ) }
Proof. First, we explain the representation of the graph G c o l o r e d G_{\mathrm{colored}} G colored { w ~ j ( i ) , w ρ i ( j ) ( i ) } \{\widetilde{w}^{(i)}_j,w^{(i)}_{\rho_i(j)}\} { w j ( i ) , w ρ i ( j ) ( i ) } j = 1 , … , n j=1,\ldots,n j = 1 , … , n { w j ( i ) , w ~ ρ i − 1 ( j ) ( i ) } \{w^{(i)}_j,\widetilde{w}^{(i)}_{\rho_i^{-1}(j)}\} { w j ( i ) , w ρ i − 1 ( j ) ( i ) } j = 1 , … , n j=1,\ldots,n j = 1 , … , n
-subgraph T j T_j T j w j ( 1 ) = w j ( 2 ) = x j w^{(1)}_j=w^{(2)}_j=x_j w j ( 1 ) = w j ( 2 ) = x j j = 1 , … , ℓ j=1,\ldots,\ell j = 1 , … , ℓ
-subgraph T j ′ T'_j T j ′ w j ( 3 ) = w j ( 4 ) = x j ′ w^{(3)}_j=w^{(4)}_j=x'_j w j ( 3 ) = w j ( 4 ) = x j ′ j = 1 , … , ℓ j=1,\ldots,\ell j = 1 , … , ℓ
-subgraph S j S_j S j w j ( 1 ) = w j ( 2 ) = w j ( 3 ) = w j ( 4 ) = u j − ℓ w^{(1)}_j=w^{(2)}_j=w^{(3)}_j=w^{(4)}_j=u_{j-\ell} w j ( 1 ) = w j ( 2 ) = w j ( 3 ) = w j ( 4 ) = u j − ℓ j = ℓ + 1 , … , r j=\ell+1,\ldots,r j = ℓ + 1 , … , r
-subgraph Q j Q_j Q j w j ( 1 ) = w j ( 3 ) = v j − r w^{(1)}_j=w^{(3)}_j=v_{j-r} w j ( 1 ) = w j ( 3 ) = v j − r j = r + 1 , … , n j=r+1,\ldots,n j = r + 1 , … , n
-subgraph Q j ′ Q'_j Q j ′ w j ( 2 ) = w j ( 4 ) = v j − r ′ w^{(2)}_j=w^{(4)}_j=v'_{j-r} w j ( 2 ) = w j ( 4 ) = v j − r ′ j = r + 1 , … , n j=r+1,\ldots,n j = r + 1 , … , n
Figure 5 and Figure 6 show the subgraphs T j T_j T j S j S_j S j T j ′ T'_j T j ′ Q j Q_j Q j Q j ′ Q'_j Q j ′ T j T_j T j
Figure 5. Subgraph T j T_j T j j = 1 , … , ℓ j=1,\ldots,\ell j = 1 , … , ℓ
Figure 6. Subgraph S j S_j S j j = ℓ + 1 , … , r j=\ell+1,\ldots,r j = ℓ + 1 , … , r
To represent G c o l o r e d G_{\mathrm{colored}} G colored 2 ℓ + ( r − ℓ ) + 2 ( n − r ) 2\ell+(r-\ell)+2(n-r) 2 ℓ + ( r − ℓ ) + 2 ( n − r ) κ \kappa κ (38) , then we draw the edges connecting w ~ 1 ( i ) , … , w ~ n ( i ) , w n + 1 ( i ) \widetilde{w}^{(i)}_1,\ldots,\widetilde{w}^{(i)}_n,w^{(i)}_{n+1} w 1 ( i ) , … , w n ( i ) , w n + 1 ( i ) i = 1 , … , 4 i=1,\ldots,4 i = 1 , … , 4 G G G (38) .
We now explain the construction of the tree.
Step 0. The tree includes all the subgraphs T j , T j ′ , S j , Q j , Q j ′ T_j,T'_j,S_j,Q_j,Q'_j T j , T j ′ , S j , Q j , Q j ′
Step 1. We connect the vertices w ~ 1 ( 1 ) , … , w ~ n ( 1 ) , w n + 1 ( 1 ) \widetilde{w}^{(1)}_1,\ldots,\widetilde{w}^{(1)}_n,w^{(1)}_{n+1} w 1 ( 1 ) , … , w n ( 1 ) , w n + 1 ( 1 ) C 1 C_1 C 1
the subgraphs T 1 , … , T ℓ T_1,\ldots,T_\ell T 1 , … , T ℓ x ℓ = ( x 1 , … , x ℓ ) \boldsymbol{x}_\ell=(x_1,\ldots,x_\ell) x ℓ = ( x 1 , … , x ℓ )
the subgraphs S ℓ + 1 , … , S r S_{\ell+1},\ldots,S_r S ℓ + 1 , … , S r u r − ℓ = ( u 1 , … , u r − ℓ ) \boldsymbol{u}_{r-\ell}=(u_1,\ldots,u_{r-\ell}) u r − ℓ = ( u 1 , … , u r − ℓ )
the subgraphs Q r + 1 , … , Q ℓ Q_{r+1},\ldots,Q_\ell Q r + 1 , … , Q ℓ v n − r = ( v 1 , … , v n − r ) \boldsymbol{v}_{n-r}=(v_1,\ldots,v_{n-r}) v n − r = ( v 1 , … , v n − r ) T 1 ′ , … , T ℓ ′ , Q r + 1 ′ , … , Q n ′ T'_1,\ldots,T'_\ell,Q'_{r+1},\ldots,Q'_n T 1 ′ , … , T ℓ ′ , Q r + 1 ′ , … , Q n ′ w n + 1 ( 2 ) , w n + 1 ( 3 ) , w n + 1 ( 4 ) w^{(2)}_{n+1},w^{(3)}_{n+1},w^{(4)}_{n+1} w n + 1 ( 2 ) , w n + 1 ( 3 ) , w n + 1 ( 4 ) C 1 C_1 C 1
Step 2. Pick one vertex w ~ j ( i ) \widetilde{w}^{(i)}_j w j ( i ) T 1 ′ T'_1 T 1 ′ C 1 C_1 C 1 G c o l o r e d G_{\mathrm{colored}} G colored T 2 ′ , … , T ℓ ′ T'_2,\ldots,T'_\ell T 2 ′ , … , T ℓ ′ C 2 C_2 C 2 T 1 ′ , … , T ℓ ′ T'_1,\ldots,T'_\ell T 1 ′ , … , T ℓ ′ x ℓ ′ = ( x 1 ′ , … , x ℓ ′ ) \boldsymbol{x}'_\ell=(x'_1,\ldots,x'_\ell) x ℓ ′ = ( x 1 ′ , … , x ℓ ′ ) Q r + 1 ′ , … , Q n ′ Q'_{r+1},\ldots,Q'_n Q r + 1 ′ , … , Q n ′ w n + 1 ( 2 ) , w n + 1 ( 3 ) , w n + 1 ( 4 ) w^{(2)}_{n+1},w^{(3)}_{n+1},w^{(4)}_{n+1} w n + 1 ( 2 ) , w n + 1 ( 3 ) , w n + 1 ( 4 ) C 2 C_2 C 2
Step 3. Pick one vertex w ~ j ( i ) \widetilde{w}^{(i)}_j w j ( i ) Q r + 1 ′ Q'_{r+1} Q r + 1 ′ C 2 C_2 C 2 Q r + 2 ′ , … , Q n ′ Q'_{r+2},\ldots,Q'_n Q r + 2 ′ , … , Q n ′ C 3 C_3 C 3 Q r + 1 ′ , … , Q n ′ Q'_{r+1},\ldots,Q'_n Q r + 1 ′ , … , Q n ′ v n − r ′ = ( v 1 ′ , … , v n − r ′ ) \boldsymbol{v}'_{n-r}=(v'_1,\ldots,v'_{n-r}) v n − r ′ = ( v 1 ′ , … , v n − r ′ ) w n + 1 ( 2 ) , w n + 1 ( 3 ) , w n + 1 ( 4 ) w^{(2)}_{n+1},w^{(3)}_{n+1},w^{(4)}_{n+1} w n + 1 ( 2 ) , w n + 1 ( 3 ) , w n + 1 ( 4 ) C 3 C_3 C 3
Step 4. Connect w n + 1 ( 2 ) , w n + 1 ( 3 ) , w n + 1 ( 4 ) w^{(2)}_{n+1},w^{(3)}_{n+1},w^{(4)}_{n+1} w n + 1 ( 2 ) , w n + 1 ( 3 ) , w n + 1 ( 4 ) C 3 C_3 C 3
Example 4.7. Let n = 7 n=7 n = 7 ℓ = 2 \ell=2 ℓ = 2 r = 5 r=5 r = 5 Figure 7 gives a representation of graph G c o l o r e d G_{\mathrm{colored}} G colored
Figure 7. The graph G c o l o r e d G_{\mathrm{colored}} G colored n = 7 n=7 n = 7 ℓ = 2 \ell=2 ℓ = 2 r = 5 r=5 r = 5 ρ i \rho_i ρ i
ρ 1 = ( 1 2 3 4 5 6 7 3 7 2 1 4 5 6 ) ρ 2 = ( 1 2 3 4 5 6 7 2 3 5 7 1 4 6 ) (40) \rho_1=\begin{pmatrix}1&2&3&4&5&6&7\\3&7&2&1&4&5&6\end{pmatrix}
\qquad
\rho_2=\begin{pmatrix}1&2&3&4&5&6&7\\2&3&5&7&1&4&6\end{pmatrix} \tag*{(40)} ρ 1 = ( 1 3 2 7 3 2 4 1 5 4 6 5 7 6 ) ρ 2 = ( 1 2 2 3 3 5 4 7 5 1 6 4 7 6 ) (40) ρ 3 = ( 1 2 3 4 5 6 7 7 4 5 2 6 1 3 ) , ρ 4 = ( 1 2 3 4 5 6 7 5 6 1 3 7 4 2 ) (41) \rho_3=\begin{pmatrix}1&2&3&4&5&6&7\\7&4&5&2&6&1&3\end{pmatrix},
\qquad
\rho_4=\begin{pmatrix}1&2&3&4&5&6&7\\5&6&1&3&7&4&2\end{pmatrix} \tag*{(41)} ρ 3 = ( 1 7 2 4 3 5 4 2 5 6 6 1 7 3 ) , ρ 4 = ( 1 5 2 6 3 1 4 3 5 7 6 4 7 2 ) (41) with inverses:
ρ 1 − 1 = ( 1 2 3 4 5 6 7 4 3 1 5 6 7 2 ) , ρ 2 − 1 = ( 1 2 3 4 5 6 7 5 1 2 6 3 7 4 ) , ρ 3 − 1 = ( 1 2 3 4 5 6 7 6 4 7 2 3 5 1 ) , ρ 4 − 1 = ( 1 2 3 4 5 6 7 3 7 4 6 1 2 5 ) . \begin{aligned}
\rho_1^{-1} &= \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 4 & 3 & 1 & 5 & 6 & 7 & 2 \end{pmatrix}, \qquad
\rho_2^{-1} = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 5 & 1 & 2 & 6 & 3 & 7 & 4 \end{pmatrix}, \\
\rho_3^{-1} &= \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 6 & 4 & 7 & 2 & 3 & 5 & 1 \end{pmatrix}, \qquad
\rho_4^{-1} = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 7 & 4 & 6 & 1 & 2 & 5 \end{pmatrix}.
\end{aligned} ρ 1 − 1 ρ 3 − 1 = ( 1 4 2 3 3 1 4 5 5 6 6 7 7 2 ) , ρ 2 − 1 = ( 1 5 2 1 3 2 4 6 5 3 6 7 7 4 ) , = ( 1 6 2 4 3 7 4 2 5 3 6 5 7 1 ) , ρ 4 − 1 = ( 1 3 2 7 3 4 4 6 5 1 6 2 7 5 ) . The graph in Figure 7 is constructed as follows. We first draw the subgraphs T 1 T_1 T 1 T 2 T_2 T 2 T 1 ′ T'_1 T 1 ′ T 2 ′ T'_2 T 2 ′ S 1 S_1 S 1 S 2 S_2 S 2 S 3 S_3 S 3 Q 1 Q_1 Q 1 Q 2 Q_2 Q 2 Q 1 ′ Q'_1 Q 1 ′ Q 2 ′ Q'_2 Q 2 ′ T 1 T_1 T 1 w 1 ( 1 ) = w 1 ( 2 ) w_1^{(1)}=w_1^{(2)} w 1 ( 1 ) = w 1 ( 2 ) w ~ 4 ( 1 ) \widetilde{w}_4^{(1)} w 4 ( 1 ) ρ 2 − 1 ( 1 ) = 4 \rho_2^{-1}(1)=4 ρ 2 − 1 ( 1 ) = 4 w ~ 5 ( 2 ) \widetilde{w}_5^{(2)} w 5 ( 2 ) ρ 2 − 1 ( 1 ) = 5 \rho_2^{-1}(1)=5 ρ 2 − 1 ( 1 ) = 5 { w j ( i ) , w ~ ρ i − 1 ( j ) ( i ) } \{w_j^{(i)},\widetilde{w}_{\rho_{i}^{-1}(j)}^{(i)}\} { w j ( i ) , w ρ i − 1 ( j ) ( i ) } j = 1 , … , 7 j=1,\ldots,7 j = 1 , … , 7 { w ~ j ( i ) , w ρ i ( j ) ( i ) } \{\widetilde{w}_j^{(i)},w_{\rho_i(j)}^{(i)}\} { w j ( i ) , w ρ i ( j ) ( i ) } j = 1 , … , 7 j=1,\ldots,7 j = 1 , … , 7 k k k w ~ 1 ( i ) , … , w ~ 7 ( i ) \widetilde{w}_1^{(i)},\ldots,\widetilde{w}_7^{(i)} w 1 ( i ) , … , w 7 ( i ) w 8 ( i ) w_8^{(i)} w 8 ( i ) i = 1 i=1 i = 1 i = 2 i=2 i = 2 i = 3 i=3 i = 3 i = 4 i=4 i = 4 G G G
Figure 8 gives the tree from Lemma 4.6 with the edges constructed in each step.
Figure 8. The spanning tree from Lemma 4.6.
Lemma 4.8. The graph G c o l o r e d G_{\mathrm{colored}} G colored { w ~ j ( i ) , w ~ j + 1 ( i ) } \{\widetilde{w}^{(i)}_j,\widetilde{w}^{(i)}_{j+1}\} { w j ( i ) , w j + 1 ( i ) } i = 1 , … , 4 i=1,\ldots,4 i = 1 , … , 4 j = 1 , … , n − 1 j=1,\ldots,n-1 j = 1 , … , n − 1 { w ~ n ( i ) , w n + 1 ( i ) } \{\widetilde{w}^{(i)}_n,w^{(i)}_{n+1}\} { w n ( i ) , w n + 1 ( i ) } i = 1 , … , 4 i=1,\ldots,4 i = 1 , … , 4
Proof. We draw the graph G c o l o r e d G_{\mathrm{colored}} G colored w ~ 1 ( i ) , … , w ~ n ( i ) , w n + 1 ( i ) \widetilde{w}^{(i)}_1,\ldots,\widetilde{w}^{(i)}_n,w^{(i)}_{n+1} w 1 ( i ) , … , w n ( i ) , w n + 1 ( i ) i = 1 , … , 4 i=1,\ldots,4 i = 1 , … , 4 G G G (38) . Recalling property (37) , we put the nodes x ℓ x_\ell x ℓ x ℓ ′ x'_\ell x ℓ ′ u r − ℓ u_{r-\ell} u r − ℓ v n − r v_{n-r} v n − r v n − r ′ v'_{n-r} v n − r ′ i i i w ~ j ( i ) \widetilde{w}^{(i)}_j w j ( i ) w ρ i ( j ) ( i ) w^{(i)}_{\rho_i(j)} w ρ i ( j ) ( i ) j = 1 , … , n j=1,\ldots,n j = 1 , … , n κ \kappa κ (38) .
We now explain the construction of the tree.
Step 0. The tree includes the four parallel lines mentioned above.
Step 1. We connect each vertex w ~ j ( 1 ) \widetilde{w}^{(1)}_j w j ( 1 ) w ρ 1 ( j ) ( 1 ) w^{(1)}_{\rho_1(j)} w ρ 1 ( j ) ( 1 ) j = 1 , … , n j=1,\ldots,n j = 1 , … , n x ℓ x_\ell x ℓ u r − ℓ u_{r-\ell} u r − ℓ v n − r v_{n-r} v n − r
Step 2. Pick one node from x ℓ x_\ell x ℓ w ~ j ( 2 ) \widetilde{w}^{(2)}_j w j ( 2 ) v n − r ′ v'_{n-r} v n − r ′ w ~ j ( 2 ) \widetilde{w}^{(2)}_j w j ( 2 ) C 1 C_1 C 1 x ℓ x_\ell x ℓ u r − ℓ u_{r-\ell} u r − ℓ v n − r v_{n-r} v n − r v n − r ′ v'_{n-r} v n − r ′
Step 3. Connect each node from x ℓ ′ x'_\ell x ℓ ′ w ~ j ( 3 ) \widetilde{w}^{(3)}_j w j ( 3 ) C 2 C_2 C 2
Step 4. Connect C 1 C_1 C 1 C 2 C_2 C 2 v n − r ′ v'_{n-r} v n − r ′ u r − ℓ u_{r-\ell} u r − ℓ w ~ j ( 3 ) \widetilde{w}^{(3)}_j w j ( 3 ) C 3 C_3 C 3 x ℓ x_\ell x ℓ x ℓ ′ x'_\ell x ℓ ′ u r − ℓ u_{r-\ell} u r − ℓ v n − r v_{n-r} v n − r v n − r ′ v'_{n-r} v n − r ′
Step 5. Connect C 4 C_4 C 4 x ℓ ′ x'_\ell x ℓ ′ u r − ℓ u_{r-\ell} u r − ℓ v n − r ′ v'_{n-r} v n − r ′ w ~ j ( 4 ) \widetilde{w}^{(4)}_j w j ( 4 )
:::
Example 4.9. Let n = 7 n=7 n = 7 ℓ = 2 \ell=2 ℓ = 2 r = 5 r=5 r = 5 (40) and (41) . Figure 9 gives the graph G c o l o r e d G_{\mathrm{colored}} G colored Figure 10 gives the spanning tree from Lemma 4.8.
Figure 9. The graph G c o l o r e d G_{\mathrm{colored}} G colored n = 7 n=7 n = 7 ℓ = 2 \ell=2 ℓ = 2 r = 5 r=5 r = 5 ρ i \rho_i ρ i (40) and (41) .
Figure 10. The spanning tree from Lemma 4.8
Proof of Theorem 1.2: Since σ R 2 ( t ) ∼ c ( t ) R d \sigma_R^2(t) \sim c(t)R^d σ R 2 ( t ) ∼ c ( t ) R d
F R ( t ) R d / 2 ⟶ d X ∼ N ( 0 , c ( t ) ) as R → ∞ . \frac{F_R(t)}{R^{d/2}} \mathrel{\mathop{\longrightarrow}\limits^{d}} X \sim\mathcal{N}(0,c(t)) \quad\text{as } R \to\infty. R d /2 F R ( t ) ⟶ d X ∼ N ( 0 , c ( t )) as R → ∞. We write
F R ( t ) R d / 2 = 1 R d / 2 ∑ n = 1 N I n ( g n , R ( ⋅ , t ) ) + 1 R d / 2 ∑ n > N I n ( g n , R ( ⋅ , t ) ) . \frac{F_R(t)}{R^{d/2}} = \frac{1}{R^{d/2}} \sum_{n=1}^{N} I_n(g_{n,R}(\cdot,t)) + \frac{1}{R^{d/2}} \sum_{n>N} I_n(g_{n,R}(\cdot,t)). R d /2 F R ( t ) = R d /2 1 n = 1 ∑ N I n ( g n , R ( ⋅ , t )) + R d /2 1 n > N ∑ I n ( g n , R ( ⋅ , t )) . Let X N X_N X N N ( 0 , ∑ n = 1 N c n ( t ) ) \mathcal{N}(0,\sum_{n=1}^{N} c_n(t)) N ( 0 , ∑ n = 1 N c n ( t )) R → ∞ R \to\infty R → ∞ X N X_N X N N ≥ 1 N \ge1 N ≥ 1 0 0 0 L 2 ( Ω ) L^2(\Omega) L 2 ( Ω ) N → ∞ N \to\infty N → ∞ R > 0 R > 0 R > 0
Moreover, X N ⟶ d X X_N \mathrel{\mathop{\longrightarrow}\limits^{d}} X X N ⟶ d X N → ∞ N \to\infty N → ∞ c ( t ) = ∑ n ≥ 1 c n ( t ) c(t) = \sum_{n\ge1} c_n(t) c ( t ) = ∑ n ≥ 1 c n ( t )
1 R d / 2 ∑ n = 1 N I n ( g n , R ( ⋅ , t ) ) ⟶ R → ∞ d X N ↓ in L 2 ( Ω ) , uniformly in R ↓ d , N → ∞ 1 R d / 2 F R ( t ) ⇢ X \begin{array}{c}
\frac{1}{R^{d/2}}\sum_{n=1}^{N} I_n(g_{n,R}(\cdot,t)) \mathrel{\mathop{\longrightarrow}\limits^{d}_{R\to\infty}} X_N\\
\underset{\text{in }L^2(\Omega),\ \text{uniformly in }R}{\Big\downarrow}\quad\quad\quad\quad\quad\quad\quad\quad\quad\Big\downarrow\scriptstyle{d,\ N\to\infty}\\
\frac{1}{R^{d/2}}F_R(t)\dashrightarrow X
\end{array} R d /2 1 ∑ n = 1 N I n ( g n , R ( ⋅ , t )) R → ∞ ⟶ d X N in L 2 ( Ω ) , uniformly in R ↓ ⏐ ↓ ⏐ d , N → ∞ R d /2 1 F R ( t ) ⇢ X To prove (i), it is enough to show that for any N ≥ 1 N \ge1 N ≥ 1
1 R d / 2 ( I 1 ( g 1 , R ( ⋅ , t ) ) , … , I N ( g N , R ( ⋅ , t ) ) ) ⟶ d N ( 0 , Σ N ( t ) ) as R → ∞ , \frac{1}{R^{d/2}}\left(I_1(g_{1,R}(\cdot,t)),\ldots,I_N(g_{N,R}(\cdot,t))\right) \mathrel{\mathop{\longrightarrow}\limits^{d}} \mathcal{N}(0,\Sigma_N(t)) \quad\text{as } R \to\infty, R d /2 1 ( I 1 ( g 1 , R ( ⋅ , t )) , … , I N ( g N , R ( ⋅ , t )) ) ⟶ d N ( 0 , Σ N ( t )) as R → ∞ , where Σ N ( t ) \Sigma_N(t) Σ N ( t ) c 1 ( t ) , … , c N ( t ) c_1(t),\ldots,c_N(t) c 1 ( t ) , … , c N ( t ) Theorem 4.2 . Condition (a) clearly holds, since I 1 ( g 1 , R ( ⋅ , t ) ) , … , I N ( g N , R ( ⋅ , t ) ) I_1(g_{1,R}(\cdot,t)),\ldots,I_N(g_{N,R}(\cdot,t)) I 1 ( g 1 , R ( ⋅ , t )) , … , I N ( g N , R ( ⋅ , t ))
1 R d Var ( I n ( g n , R ( ⋅ , t ) ) ) ⟶ c n ( t ) as R → ∞ , \frac{1}{R^d}\operatorname{Var}(I_n(g_{n,R}(\cdot,t))) \longrightarrow c_n(t) \quad\text{as } R \to\infty, R d 1 Var ( I n ( g n , R ( ⋅ , t ))) ⟶ c n ( t ) as R → ∞ , for any n = 1 , … , N n=1,\ldots,N n = 1 , … , N (21) . Condition (b) is also clear, since for any n = 1 , … , N n=1,\ldots,N n = 1 , … , N
1 R 2 d E [ I n , R 4 ( g n , R ( ⋅ , t ) ) ] ⟶ 3 c n 2 ( t ) as R → ∞ , \frac{1}{R^{2d}}\mathbb{E}\left[I_{n,R}^4(g_{n,R}(\cdot,t))\right] \longrightarrow3c_n^2(t) \quad\text{as } R \to\infty, R 2 d 1 E [ I n , R 4 ( g n , R ( ⋅ , t )) ] ⟶ 3 c n 2 ( t ) as R → ∞ , due to (24) and the fact that σ n , R 4 ( t ) ∼ c n 2 ( t ) R 2 d \sigma_{n,R}^4(t) \sim c_n^2(t)R^{2d} σ n , R 4 ( t ) ∼ c n 2 ( t ) R 2 d R → ∞ R \to\infty R → ∞
To prove (ii), we will show that
sup R > 0 1 R d E ∣ ∑ n > N I n ( g n , R ( ⋅ , t ) ) ∣ 2 ⟶ 0 as N → ∞ . (42) \sup_{R>0}\frac{1}{R^d}\mathbb{E}\left|\sum_{n>N} I_n(g_{n,R}(\cdot,t))\right|^2 \longrightarrow0 \quad\text{as } N \to\infty. \tag*{(42)} R > 0 sup R d 1 E n > N ∑ I n ( g n , R ( ⋅ , t )) 2 ⟶ 0 as N → ∞. (42) By orthogonality and (20) ,
sup R > 0 1 R d E ∣ ∑ n > N I n ( g n , R ( ⋅ , t ) ) ∣ 2 = sup R > 0 1 R d ∑ n > N σ n , R 2 ( t ) = sup R > 0 ∑ n > N ∫ B 2 R ρ t ( n ) ( z ) Vol ( B R ∩ B R ( − z ) ) R d d z ≤ ω d ∑ n > N ∫ R d ρ t , n ( z ) d z = ∑ n > N c n ( t ) . \begin{aligned}
\sup_{R>0}\frac{1}{R^d}\mathbb{E}\left|\sum_{n>N} I_n(g_{n,R}(\cdot,t))\right|^2
= \sup_{R>0}\frac{1}{R^d}\sum_{n>N}\sigma_{n,R}^2(t)
= \sup_{R>0}\sum_{n>N}\int_{B_{2R}}\rho_t^{(n)}(z)\frac{\operatorname{Vol}(B_R\cap B_R(-z))}{R^d}\,dz \\
&\le\omega_d\sum_{n>N}\int_{\mathbb{R}^d}\rho_{t,n}(z)\,dz
= \sum_{n>N}c_n(t).
\end{aligned} R > 0 sup R d 1 E n > N ∑ I n ( g n , R ( ⋅ , t )) 2 = R > 0 sup R d 1 n > N ∑ σ n , R 2 ( t ) = R > 0 sup n > N ∑ ∫ B 2 R ρ t ( n ) ( z ) R d Vol ( B R ∩ B R ( − z )) d z ≤ ω d n > N ∑ ∫ R d ρ t , n ( z ) d z = n > N ∑ c n ( t ) . Since ∑ n ≥ 1 c n ( t ) < ∞ \sum_{n\ge1}c_n(t)<\infty ∑ n ≥ 1 c n ( t ) < ∞ 0 0 0 N → ∞ N \to\infty N → ∞ (42) , and concludes the proof of Theorem 1.2.
Fourth cumulant estimate
In this section, we give an estimate for the fourth cumulant of a random variable which lies in a fixed Poisson chaos space. We assume that N N N ( Z , Z , m ) (Z,\mathcal{Z},\mathbf{m}) ( Z , Z , m ) H = L 2 ( Z , Z , m ) \mathcal{H}=L^2(Z,\mathcal{Z},\mathbf{m}) H = L 2 ( Z , Z , m ) H ⊙ p \mathcal{H}^{\odot p} H ⊙ p H ⊗ p \mathcal{H}^{\otimes p} H ⊗ p
We recall some definitions. For any f ∈ H ⊗ p f\in\mathcal{H}^{\otimes p} f ∈ H ⊗ p g ∈ H ⊗ q g\in\mathcal{H}^{\otimes q} g ∈ H ⊗ q modified contractions , as follows:
(i) f ⋆ 0 0 g = f ⊗ g f\star_0^0g=f\otimes g f ⋆ 0 0 g = f ⊗ g f f f g g g
(ii) for 1 ≤ r ≤ p ∧ q 1\le r\le p\wedge q 1 ≤ r ≤ p ∧ q f ⋆ r 0 g f\star_r^0g f ⋆ r 0 g Z p + q − r Z^{p+q-r} Z p + q − r
( f ⋆ r 0 g ) ( γ 1 , … , γ r , x 1 , … , x p − r , x 1 ′ , … , x q − r ′ ) = f ( γ 1 , … , γ r , x 1 , … , x p − r ) g ( γ 1 , … , γ r , x 1 ′ , … , x q − r ′ ) , (f\star_r^0g)(\gamma_1,\ldots,\gamma_r,x_1,\ldots,x_{p-r},x'_1,\ldots,x'_{q-r})=
f(\gamma_1,\ldots,\gamma_r,x_1,\ldots,x_{p-r})g(\gamma_1,\ldots,\gamma_r,x'_1,\ldots,x'_{q-r}), ( f ⋆ r 0 g ) ( γ 1 , … , γ r , x 1 , … , x p − r , x 1 ′ , … , x q − r ′ ) = f ( γ 1 , … , γ r , x 1 , … , x p − r ) g ( γ 1 , … , γ r , x 1 ′ , … , x q − r ′ ) , (iii) for 1 ≤ ℓ ≤ r ≤ p ∧ q 1\le\ell\le r\le p\wedge q 1 ≤ ℓ ≤ r ≤ p ∧ q f ⋆ r ℓ g f\star_r^\ell g f ⋆ r ℓ g Z m + n − k − ℓ Z^{m+n-k-\ell} Z m + n − k − ℓ
( f ⋆ r ℓ g ) ( γ 1 , … , γ r − ℓ , x 1 , … , x p − r , x 1 ′ , … , x q − r ′ ) = ∫ Z ℓ f ( z 1 , … , z ℓ , γ 1 , … , γ r − ℓ , x 1 , … , x p − r ) g ( z 1 , … , z ℓ , γ 1 , … , γ r − ℓ , x 1 ′ , … , x q − r ′ ) m ( d z 1 ) … m ( d z ℓ ) , (f\star_r^\ell g)(\gamma_1,\ldots,\gamma_{r-\ell},x_1,\ldots,x_{p-r},x'_1,\ldots,x'_{q-r})=
\int_{Z^\ell}f(z_1,\ldots,z_\ell,\gamma_1,\ldots,\gamma_{r-\ell},x_1,\ldots,x_{p-r})g(z_1,\ldots,z_\ell,\gamma_1,\ldots,\gamma_{r-\ell},x'_1,\ldots,x'_{q-r})\mathbf{m}(dz_1)\ldots\mathbf{m}(dz_\ell), ( f ⋆ r ℓ g ) ( γ 1 , … , γ r − ℓ , x 1 , … , x p − r , x 1 ′ , … , x q − r ′ ) = ∫ Z ℓ f ( z 1 , … , z ℓ , γ 1 , … , γ r − ℓ , x 1 , … , x p − r ) g ( z 1 , … , z ℓ , γ 1 , … , γ r − ℓ , x 1 ′ , … , x q − r ′ ) m ( d z 1 ) … m ( d z ℓ ) , provided that the integral is finite.
If 1 ≤ ℓ < r 1\le\ell<r 1 ≤ ℓ < r f ⋆ r ℓ g f\star_r^\ell g f ⋆ r ℓ g f ⋆ r r g = f ⊗ r g f\star_r^r g=f\otimes_r g f ⋆ r r g = f ⊗ r g 1 ≤ r ≤ p ∧ q 1\le r\le p\wedge q 1 ≤ r ≤ p ∧ q
( f ⊗ r g ) ( x 1 , … , x p − r , x 1 ′ , … , x q − r ′ ) = ∫ Z r f ( z 1 , … , z r , x 1 , … , x p − r ) g ( z 1 , … , z r , x 1 ′ , … , x q − r ′ ) m ( d z 1 ) … m ( d z r ) . (f\otimes_r g)(x_1,\ldots,x_{p-r},x'_1,\ldots,x'_{q-r})=
\int_{Z^r}f(z_1,\ldots,z_r,x_1,\ldots,x_{p-r})g(z_1,\ldots,z_r,x'_1,\ldots,x'_{q-r})\mathbf{m}(dz_1)\ldots\mathbf{m}(dz_r). ( f ⊗ r g ) ( x 1 , … , x p − r , x 1 ′ , … , x q − r ′ ) = ∫ Z r f ( z 1 , … , z r , x 1 , … , x p − r ) g ( z 1 , … , z r , x 1 ′ , … , x q − r ′ ) m ( d z 1 ) … m ( d z r ) . If p = q p=q p = q f ⋆ p p g = ⟨ f , g ⟩ H ⊗ p f\star_p^p g=\langle f,g\rangle_{\mathcal{H}^{\otimes p}} f ⋆ p p g = ⟨ f , g ⟩ H ⊗ p f ⋆ p 0 f = f 2 f\star_p^0f=f^2 f ⋆ p 0 f = f 2
We let f ⋆ r ℓ ~ g f\widetilde{\star_r^\ell}g f ⋆ r ℓ g f ⋆ r ℓ g f\star_r^\ell g f ⋆ r ℓ g f ⊗ ~ g f\widetilde{\otimes}g f ⊗ g f ⊗ g f\otimes g f ⊗ g
Lemma A.1. Let F = I p ( f ) F=I_p(f) F = I p ( f ) f ∈ H ⊙ p f\in\mathcal{H}^{\odot p} f ∈ H ⊙ p p ≥ 1 p\ge1 p ≥ 1 E ( F 4 ) < ∞ \mathbb{E}(F^4)<\infty E ( F 4 ) < ∞ σ 2 = E ( F 2 ) > 0 \sigma^2=\mathbb{E}(F^2)>0 σ 2 = E ( F 2 ) > 0 E ( F 4 ) − 3 σ 4 \mathbb{E}(F^4)-3\sigma^4 E ( F 4 ) − 3 σ 4
∥ f ⋆ r ℓ f ∥ H ⊗ ( 2 p − r − ℓ ) \|f\star_r^\ell f\|_{\mathcal{H}^{\otimes(2p-r-\ell)}} ∥ f ⋆ r ℓ f ∥ H ⊗ ( 2 p − r − ℓ ) with 0 ≤ ℓ ≤ r ≤ p 0\le\ell\le r\le p 0 ≤ ℓ ≤ r ≤ p ( ℓ , r ) ≠ ( p , p ) (\ell,r)\ne(p,p) ( ℓ , r ) = ( p , p ) ( ℓ , r ) ≠ ( 0 , 0 ) (\ell,r)\ne(0,0) ( ℓ , r ) = ( 0 , 0 )
Proof. By Theorem 2.2 of [12 ] , F 2 = ∑ k = 0 2 p I k ( h k ) = ∑ m = 0 2 p I 2 p − m ( h 2 p − m ) F^2=\sum_{k=0}^{2p}I_k(h_k)=\sum_{m=0}^{2p}I_{2p-m}(h_{2p-m}) F 2 = ∑ k = 0 2 p I k ( h k ) = ∑ m = 0 2 p I 2 p − m ( h 2 p − m )
h 2 p − m = ∑ r = ⌈ m / 2 ⌉ m ∧ p r ! ( p r ) 2 ( r m − r ) f ⋆ r m − r ~ f , (43) h_{2p-m}=\sum_{r=\lceil m/2\rceil}^{m\wedge p}r!\binom{p}{r}^{2}\binom{r}{m-r}f\widetilde{\star_r^{m-r}}f, \tag*{(43)} h 2 p − m = r = ⌈ m /2 ⌉ ∑ m ∧ p r ! ( r p ) 2 ( m − r r ) f ⋆ r m − r f , (43) i.e. I k ( h k ) I_k(h_k) I k ( h k ) F 2 F^2 F 2 k k k ⌈ ⋅ ⌉ \lceil\cdot\rceil ⌈ ⋅ ⌉ x ∈ R x\in\mathbb{R} x ∈ R ⌈ x ⌉ ∈ Z \lceil x\rceil\in\mathbb{Z} ⌈ x ⌉ ∈ Z ⌈ x ⌉ − 1 < x ≤ ⌈ x ⌉ \lceil x\rceil-1<x\le\lceil x\rceil ⌈ x ⌉ − 1 < x ≤ ⌈ x ⌉ ℓ : = m − r \ell:= m-r ℓ := m − r r r r r ≥ ⌈ m 2 ⌉ ≥ m 2 r \ge\left\lceil\frac{m}{2} \right\rceil\ge\frac{m}{2} r ≥ ⌈ 2 m ⌉ ≥ 2 m ℓ \ell ℓ 0 0 0 m ∧ p = m m \wedge p = m m ∧ p = m m ≤ p m \le p m ≤ p r = m r=m r = m ( ℓ , r ) (\ell,r) ( ℓ , r ) ( 0 , 0 ) (0,0) ( 0 , 0 )
By relation (43) of [11 ] ,
E ( F 4 ) − σ 4 = ∑ k = 1 2 p − 1 Var ( I k ( h k ) ) + ( 2 p ) ! ∥ f ⊗ ~ f ∥ H ⊗ 2 p 2 = ∑ k = 1 2 p − 1 k ! ∥ h k ∥ H ⊗ k 2 + ( 2 p ) ! ∥ f ⊗ ~ f ∥ H ⊗ 2 p 2 . \mathbb{E}(F^4)-\sigma^4=\sum_{k=1}^{2p-1}\operatorname{Var}(I_k(h_k))+(2p)!\left\|f\widetilde{\otimes}f\right\|_{\mathcal{H}^{\otimes2p}}^2=\sum_{k=1}^{2p-1}k!\left\|h_k\right\|_{\mathcal{H}^{\otimes k}}^2+(2p)!\left\|f\widetilde{\otimes}f\right\|_{\mathcal{H}^{\otimes2p}}^2. E ( F 4 ) − σ 4 = k = 1 ∑ 2 p − 1 Var ( I k ( h k )) + ( 2 p )! f ⊗ f H ⊗ 2 p 2 = k = 1 ∑ 2 p − 1 k ! ∥ h k ∥ H ⊗ k 2 + ( 2 p )! f ⊗ f H ⊗ 2 p 2 . We now invoke the highly non-trivial relation (5.2.12) of [16 ] , which says that
( 2 p ) ! ∥ f ⊗ ~ f ∥ H ⊗ 2 p 2 = 2 ( p ! ) 2 ∥ f ∥ H ⊗ p 4 + ( p ! ) 2 ∑ r = 1 p − 1 ( p r ) 2 ∥ f ⊗ r f ∥ H ⊗ ( 2 p − 2 r ) 2 . (2p)!\left\|f\widetilde{\otimes}f\right\|_{\mathcal{H}^{\otimes2p}}^2=2(p!)^2\left\|f\right\|_{\mathcal{H}^{\otimes p}}^4+(p!)^2\sum_{r=1}^{p-1}\binom{p}{r}^2\left\|f\otimes_r f\right\|_{\mathcal{H}^{\otimes(2p-2r)}}^2. ( 2 p )! f ⊗ f H ⊗ 2 p 2 = 2 ( p ! ) 2 ∥ f ∥ H ⊗ p 4 + ( p ! ) 2 r = 1 ∑ p − 1 ( r p ) 2 ∥ f ⊗ r f ∥ H ⊗ ( 2 p − 2 r ) 2 . We obtain that:
E [ F 4 ] − σ 4 = 2 ( p ! ) 2 ∥ f ∥ H ⊗ p 4 + ∑ m = 1 2 p − 1 ( 2 p − m ) ! ∥ h 2 p − m ∥ H ⊗ ( 2 p − m ) 2 + ( p ! ) 2 ∑ r = 1 p − 1 ( p r ) 2 ∥ f ⊗ r f ∥ H ⊗ ( 2 p − 2 r ) 2 . \mathbb{E}[F^4]-\sigma^4=2(p!)^2\left\|f\right\|_{\mathcal{H}^{\otimes p}}^4+\sum_{m=1}^{2p-1}(2p-m)!\left\|h_{2p-m}\right\|_{\mathcal{H}^{\otimes(2p-m)}}^2+(p!)^2\sum_{r=1}^{p-1}\binom{p}{r}^2\left\|f\otimes_r f\right\|_{\mathcal{H}^{\otimes(2p-2r)}}^2. E [ F 4 ] − σ 4 = 2 ( p ! ) 2 ∥ f ∥ H ⊗ p 4 + m = 1 ∑ 2 p − 1 ( 2 p − m )! ∥ h 2 p − m ∥ H ⊗ ( 2 p − m ) 2 + ( p ! ) 2 r = 1 ∑ p − 1 ( r p ) 2 ∥ f ⊗ r f ∥ H ⊗ ( 2 p − 2 r ) 2 . (This relation is similar to (3.5) of [12 ] , in which h 2 p − m h_{2p-m} h 2 p − m h 2 p − m ( n ) h_{2p-m}^{(n)} h 2 p − m ( n ) σ 4 = ( E [ F 2 ] ) 2 = ( p ! ) 2 ∥ f ∥ H ⊗ p 4 \sigma^4=(\mathbb{E}[F^2])^2=(p!)^2\left\|f\right\|_{\mathcal{H}^{\otimes p}}^4 σ 4 = ( E [ F 2 ] ) 2 = ( p ! ) 2 ∥ f ∥ H ⊗ p 4
E [ F 4 ] − 3 σ 4 = ∑ m = 1 2 p − 1 ( 2 p − m ) ! ∥ h 2 p − m ∥ H ⊗ ( 2 p − m ) 2 + ( p ! ) 2 ∑ r = 1 p − 1 ( p r ) 2 ∥ f ⊗ r f ∥ H ⊗ ( 2 p − 2 r ) 2 . \mathbb{E}[F^4]-3\sigma^4=\sum_{m=1}^{2p-1}(2p-m)!\left\|h_{2p-m}\right\|_{\mathcal{H}^{\otimes(2p-m)}}^2+(p!)^2\sum_{r=1}^{p-1}\binom{p}{r}^2\left\|f\otimes_r f\right\|_{\mathcal{H}^{\otimes(2p-2r)}}^2. E [ F 4 ] − 3 σ 4 = m = 1 ∑ 2 p − 1 ( 2 p − m )! ∥ h 2 p − m ∥ H ⊗ ( 2 p − m ) 2 + ( p ! ) 2 r = 1 ∑ p − 1 ( r p ) 2 ∥ f ⊗ r f ∥ H ⊗ ( 2 p − 2 r ) 2 . Using definition (43) of h 2 p − m h_{2p-m} h 2 p − m ( ∑ i = 1 N a i ) 2 ≤ N ∑ i = 1 N a i 2 \left(\sum_{i=1}^{N}a_i\right)^2\le N\sum_{i=1}^{N}a_i^2 ( ∑ i = 1 N a i ) 2 ≤ N ∑ i = 1 N a i 2 ∥ g ~ ∥ H ⊗ k ≤ ∥ g ∥ H ⊗ k \left\|\widetilde{g}\right\|_{\mathcal{H}^{\otimes k}}\le\left\|g\right\|_{\mathcal{H}^{\otimes k}} ∥ g ∥ H ⊗ k ≤ ∥ g ∥ H ⊗ k g ∈ H ⊗ k g\in\mathcal{H}^{\otimes k} g ∈ H ⊗ k
∥ h 2 p − m ∥ H ⊗ ( 2 p − m ) 2 ≤ p ∑ r = ⌈ m 2 ⌉ m ∧ p ( r ! ) 2 ( p r ) 4 ( r m − r ) 2 ∥ f ⋆ r m − r f ∥ H ⊗ ( 2 p − m ) 2 . \left\|h_{2p-m}\right\|_{\mathcal{H}^{\otimes(2p-m)}}^2\le p\sum_{r=\left\lceil\frac{m}{2}\right\rceil}^{m\wedge p}(r!)^2\binom{p}{r}^4\binom{r}{m-r}^2\left\|f\star_r^{m-r}f\right\|_{\mathcal{H}^{\otimes(2p-m)}}^2. ∥ h 2 p − m ∥ H ⊗ ( 2 p − m ) 2 ≤ p r = ⌈ 2 m ⌉ ∑ m ∧ p ( r ! ) 2 ( r p ) 4 ( m − r r ) 2 f ⋆ r m − r f H ⊗ ( 2 p − m ) 2 . Hence,
E [ F 4 ] − 3 σ 4 ≤ p ∑ m = 1 2 p − 1 ( 2 p − m ) ! ∑ r = ⌈ m 2 ⌉ m ∧ p ( r ! ) 2 ( p r ) 4 ( r m − r ) 2 ∥ f ⋆ r m − r f ∥ H ⊗ ( 2 p − m ) 2 + ( p ! ) 2 ∑ r = 1 p − 1 ( p r ) 2 ∥ f ⋆ r r f ∥ H ⊗ ( 2 p − 2 r ) 2 . \begin{aligned}
\mathbb{E}[F^4]-3\sigma^4
&\le p\sum_{m=1}^{2p-1}(2p-m)!\sum_{r=\left\lceil\frac{m}{2}\right\rceil}^{m\wedge p}(r!)^2\binom{p}{r}^4\binom{r}{m-r}^2\left\|f\star_r^{m-r}f\right\|_{\mathcal{H}^{\otimes(2p-m)}}^2\\
&\quad+(p!)^2\sum_{r=1}^{p-1}\binom{p}{r}^2\left\|f\star_r^r f\right\|_{\mathcal{H}^{\otimes(2p-2r)}}^2.
\end{aligned} E [ F 4 ] − 3 σ 4 ≤ p m = 1 ∑ 2 p − 1 ( 2 p − m )! r = ⌈ 2 m ⌉ ∑ m ∧ p ( r ! ) 2 ( r p ) 4 ( m − r r ) 2 f ⋆ r m − r f H ⊗ ( 2 p − m ) 2 + ( p ! ) 2 r = 1 ∑ p − 1 ( r p ) 2 ∥ f ⋆ r r f ∥ H ⊗ ( 2 p − 2 r ) 2 . Finally, we note that ( m − r , r ) ≠ ( p , p ) (m-r,r)\ne(p,p) ( m − r , r ) = ( p , p ) m = 1 , … , 2 p − 1 m=1,\ldots,2p-1 m = 1 , … , 2 p − 1
A graph-related integral
In this section, we give an estimate for an integral associated with a multigraph, which is used in the proof of Theorem 4.4 . We first recall some basic elements of graph theory.
An (undirected) graph is a pair G = ( V , E ) G=(V,E) G = ( V , E ) V V V vertices and E E E { u , v } \{u,v\} { u , v } edges . A multigraph is graph which is permitted to have multiple edges, i.e. edges which have the same end points. A graph is connected is for any two vertices u , v ∈ V u,v\in V u , v ∈ V u u u v v v finite if it has a finite number of vertices and a finite number of edges.
A tree is an (undirected) graph in which every pair of distinct vertices is connected by exactly one path. (A tree is a connected graph which does not have any cycles.)
A spanning tree of a graph G G G G G G G = ( V , E ) G=(V,E) G = ( V , E ) T = ( V , E T ) T=(V,E_T) T = ( V , E T ) G G G
∣ E T ∣ = ∣ V ∣ − 1 , |E_T|=|V|-1, ∣ E T ∣ = ∣ V ∣ − 1 , where ∣ S ∣ |S| ∣ S ∣ S S S
A leaf is a vertex that has degree 1. (The degree of a vertex is the number of edges incident to that vertex.) Every finite tree with at least two vertices has at least two leaves.
Lemma B.1. Let G = ( V , E ) G=(V,E) G = ( V , E ) finite (non-empty ) connected multigraph with ∣ V ∣ ≥ 2 |V|\ge2 ∣ V ∣ ≥ 2 edge e ∈ E e\in E e ∈ E e = { u e , v e } e=\{u_e,v_e\} e = { u e , v e } u e , v e ∈ V u_e,v_e\in V u e , v e ∈ V e ∈ E e\in E e ∈ E s e > 0 s_e>0 s e > 0 p ∈ V p\in V p ∈ V T = ( V , E T ) T=(V,E_T) T = ( V , E T ) G G G
For any t > 0 t>0 t > 0 g t : R d → R + g_t:\mathbb{R}^{d}\to\mathbb{R}_+ g t : R d → R + symmetric measurable function such that g t ∈ L ∞ ( R d ) ∩ L 1 ( R d ) g_t\in L^\infty(\mathbb{R}^{d})\cap L^1(\mathbb{R}^{d}) g t ∈ L ∞ ( R d ) ∩ L 1 ( R d )
I : = ∫ ( R d ) ∣ V ∣ − 1 ∏ e ∈ E g s e ( x v e − x u e ) ∏ v ∈ V ∖ { p } d x v ≤ ∏ e ∈ E ∖ E T ∥ g s e ∥ L ∞ ( R d ) ∏ e ∈ E T ∥ g s e ∥ L 1 ( R d ) . I:=\int_{(\mathbb{R}^{d})^{|V|-1}}\prod_{e\in E}g_{s_e}(x_{v_e}-x_{u_e})\prod_{v\in V\setminus\{p\}}dx_v\le\prod_{e\in E\setminus E_T}\|g_{s_e}\|_{L^\infty(\mathbb{R}^{d})}\prod_{e\in E_T}\|g_{s_e}\|_{L^1(\mathbb{R}^{d})}. I := ∫ ( R d ) ∣ V ∣ − 1 e ∈ E ∏ g s e ( x v e − x u e ) v ∈ V ∖ { p } ∏ d x v ≤ e ∈ E ∖ E T ∏ ∥ g s e ∥ L ∞ ( R d ) e ∈ E T ∏ ∥ g s e ∥ L 1 ( R d ) . Proof. Step 1. We bound the factors which correspond to a non-tree edge by the L ∞ L^\infty L ∞
∏ e ∈ E g s e ( x v e − x u e ) ≤ ∏ e ∈ E ∖ E T ∥ g s e ∥ L ∞ ( R d ) ∏ e ∈ E T g s e ( x v e − x u e ) . \prod_{e\in E}g_{s_e}(x_{v_e}-x_{u_e})\le\prod_{e\in E\setminus E_T}\|g_{s_e}\|_{L^\infty(\mathbb{R}^{d})}\prod_{e\in E_T}g_{s_e}(x_{v_e}-x_{u_e}). e ∈ E ∏ g s e ( x v e − x u e ) ≤ e ∈ E ∖ E T ∏ ∥ g s e ∥ L ∞ ( R d ) e ∈ E T ∏ g s e ( x v e − x u e ) . Hence, I ≤ ∏ e ∈ E ∖ E T ∥ g s e ∥ L ∞ ( R d ) I 1 I\le\prod_{e\in E\setminus E_T}\|g_{s_e}\|_{L^\infty(\mathbb{R}^{d})}I_1 I ≤ ∏ e ∈ E ∖ E T ∥ g s e ∥ L ∞ ( R d ) I 1
I 1 : = ∫ ( R d ) ∣ V ∣ − 1 ∏ e ∈ E T g s e ( x v e − x u e ) ∏ v ∈ V ∖ { p } d x v . I_1:=\int_{(\mathbb{R}^{d})^{|V|-1}}\prod_{e\in E_T}g_{s_e}(x_{v_e}-x_{u_e})\prod_{v\in V\setminus\{p\}}dx_v. I 1 := ∫ ( R d ) ∣ V ∣ − 1 e ∈ E T ∏ g s e ( x v e − x u e ) v ∈ V ∖ { p } ∏ d x v . Step 2. It remains to prove that
I 1 ≤ ∏ e ∈ E T ∥ g s e ∥ L 1 ( R d ) . I_1\le\prod_{e\in E_T}\|g_{s_e}\|_{L^1(\mathbb{R}^{d})}. I 1 ≤ e ∈ E T ∏ ∥ g s e ∥ L 1 ( R d ) . We apply Fubini theorem to calculate the integral I 1 I_1 I 1 w 1 ≠ p w_1\ne p w 1 = p e 1 = { u e 1 , v e 1 } e_1=\{u_{e_1},v_{e_1}\} e 1 = { u e 1 , v e 1 } w 1 w_1 w 1 w 1 = u e 1 w_1=u_{e_1} w 1 = u e 1 d x w 1 dx_{w_1} d x w 1
∫ R d g s e 1 ( x v e 1 − x w 1 ) d x w 1 = ∥ g s e 1 ∥ L 1 ( R d ) . \int_{\mathbb{R}^{d}}g_{s_{e_1}}(x_{v_{e_1}}-x_{w_1})\,dx_{w_1}=\|g_{s_{e_1}}\|_{L^1(\mathbb{R}^{d})}. ∫ R d g s e 1 ( x v e 1 − x w 1 ) d x w 1 = ∥ g s e 1 ∥ L 1 ( R d ) . Hence,
I 1 = ∥ g s e 1 ∥ L 1 ( R d ) ∫ ( R d ) ∣ V ∣ − 2 ∏ e ∈ E T ∖ { e 1 } g s e ( x v e − x u e ) ∏ v ∈ V ∖ { p , w 1 } d x v . I_1 = \lVert g_{s_{e_1}}\rVert_{L^1(\mathbb{R}^d)} \int_{(\mathbb{R}^d)^{|V|-2}} \prod_{e\in E_T\setminus\{e_1\}} g_{s_e}(x_{v_e}-x_{u_e}) \prod_{v\in V\setminus\{p,w_1\}} dx_v. I 1 = ∥ g s e 1 ∥ L 1 ( R d ) ∫ ( R d ) ∣ V ∣ − 2 e ∈ E T ∖ { e 1 } ∏ g s e ( x v e − x u e ) v ∈ V ∖ { p , w 1 } ∏ d x v . Pick another leaf w 1 ∉ { p , w 1 } w_1\notin\{p,w_1\} w 1 ∈ / { p , w 1 } e 2 = { u e 2 , v e 2 } e_2=\{u_{e_2},v_{e_2}\} e 2 = { u e 2 , v e 2 } w 2 w_2 w 2 w 2 = u e 2 w_2=u_{e_2} w 2 = u e 2 d x w 2 dx_{w_2} d x w 2
∫ R d g s e 2 ( x v e 2 − x w 2 ) d x w 2 = ∥ g s e 2 ∥ L 1 ( R d ) . \int_{\mathbb{R}^d} g_{s_{e_2}}(x_{v_{e_2}}-x_{w_2})dx_{w_2}=\lVert g_{s_{e_2}}\rVert_{L^1(\mathbb{R}^d)}. ∫ R d g s e 2 ( x v e 2 − x w 2 ) d x w 2 = ∥ g s e 2 ∥ L 1 ( R d ) . We obtain:
I 1 = ∥ g s e 1 ∥ L 1 ( R d ) ∥ g s e 2 ∥ L 1 ( R d ) ∫ ( R d ) ∣ V ∣ − 3 ∏ e ∈ E T ∖ { e 1 , e 2 } g s e ( x v e − x u e ) ∏ v ∈ V ∖ { p , w 1 , w 2 } d x v . I_1 = \lVert g_{s_{e_1}}\rVert_{L^1(\mathbb{R}^d)}\lVert g_{s_{e_2}}\rVert_{L^1(\mathbb{R}^d)} \int_{(\mathbb{R}^d)^{|V|-3}} \prod_{e\in E_T\setminus\{e_1,e_2\}} g_{s_e}(x_{v_e}-x_{u_e}) \prod_{v\in V\setminus\{p,w_1,w_2\}} dx_v. I 1 = ∥ g s e 1 ∥ L 1 ( R d ) ∥ g s e 2 ∥ L 1 ( R d ) ∫ ( R d ) ∣ V ∣ − 3 e ∈ E T ∖ { e 1 , e 2 } ∏ g s e ( x v e − x u e ) v ∈ V ∖ { p , w 1 , w 2 } ∏ d x v . We continue until we eliminate all leaves. If ∣ V ∣ = n |V|=n ∣ V ∣ = n
∫ R d g s e n − 1 ( x p − x w n − 1 ) d x w n − 1 = ∥ g s e n − 1 ∥ L 1 ( R d ) , \int_{\mathbb{R}^d} g_{s_{e_{n-1}}}(x_p-x_{w_{n-1}})dx_{w_{n-1}}=\lVert g_{s_{e_{n-1}}}\rVert_{L^1(\mathbb{R}^d)}, ∫ R d g s e n − 1 ( x p − x w n − 1 ) d x w n − 1 = ∥ g s e n − 1 ∥ L 1 ( R d ) , where e n − 1 = { p , w n − 1 } ∈ E T e_{n-1}=\{p,w_{n-1}\}\in E_T e n − 1 = { p , w n − 1 } ∈ E T p p p w n − 1 ≠ p w_{n-1}\ne p w n − 1 = p
We consider an example to illustrate Lemma B.1.
Example B.2. Consider the multigraph G = ( V , E ) G=(V,E) G = ( V , E ) V = { 1 , 2 , … , 7 } V=\{1,2,\ldots,7\} V = { 1 , 2 , … , 7 } Figure 11 , with p = 6 p=6 p = 6 T T T E T = { e 1 , e 2 , e 6 , e 8 , e 10 , e 11 } E_T=\{e_1,e_2,e_6,e_8,e_{10},e_{11}\} E T = { e 1 , e 2 , e 6 , e 8 , e 10 , e 11 }
Figure 11. Multigraph with V = { 1 , … , 7 } V=\{1,\ldots,7\} V = { 1 , … , 7 } p = 6 p=6 p = 6 E T = { e 1 , e 2 , e 6 , e 8 , e 10 , e 11 } E_T=\{e_1,e_2,e_6,e_8,e_{10},e_{11}\} E T = { e 1 , e 2 , e 6 , e 8 , e 10 , e 11 }
We use boxes to exhibit the factors corresponding to the edges of T T T
I = ∫ ( R d ) 6 g s e 1 ( x 1 − x 2 ) g s e 2 ( x 2 − x 3 ) g s e 3 ( x 2 − x 3 ) g s e 4 ( x 2 − x 3 ) g s e 5 ( x 3 − x 4 ) g s e 6 ( x 2 − x 4 ) × g s e 7 ( x 2 − x 5 ) g s e 8 ( x 2 − x 6 ) g s e 9 ( x 2 − x 7 ) g s e 10 ( x 5 − x 6 ) g s e 11 ( x 6 − x 7 ) g s e 12 ( x 6 − x 7 ) × d x 1 d x 2 d x 3 d x 4 d x 5 d x 7 ≤ ∏ i ∈ { 3 , 4 , 5 , 7 , 9 , 12 } ∥ g s e i ∥ L ∞ ( R d ) I 1 , \begin{aligned}
I=\int_{(\mathbb{R}^{d})^{6}} \boxed{g_{se_1}(x_1-x_2)}\boxed{g_{se_2}(x_2-x_3)}g_{se_3}(x_2-x_3)g_{se_4}(x_2-x_3)g_{se_5}(x_3-x_4)\boxed{g_{se_6}(x_2-x_4)}
\times g_{se_7}(x_2-x_5)\boxed{g_{se_8}(x_2-x_6)}g_{se_9}(x_2-x_7)\boxed{g_{se_{10}}(x_5-x_6)}\boxed{g_{se_{11}}(x_6-x_7)}g_{se_{12}}(x_6-x_7)
\times dx_1\,dx_2\,dx_3\,dx_4\,dx_5\,dx_7 \\
&\leq\prod_{i\in\{3,4,5,7,9,12\}}\lVert g_{se_i}\rVert_{L^\infty(\mathbb{R}^{d})}I_1,
\end{aligned} I = ∫ ( R d ) 6 g s e 1 ( x 1 − x 2 ) g s e 2 ( x 2 − x 3 ) g s e 3 ( x 2 − x 3 ) g s e 4 ( x 2 − x 3 ) g s e 5 ( x 3 − x 4 ) g s e 6 ( x 2 − x 4 ) × g s e 7 ( x 2 − x 5 ) g s e 8 ( x 2 − x 6 ) g s e 9 ( x 2 − x 7 ) g s e 10 ( x 5 − x 6 ) g s e 11 ( x 6 − x 7 ) g s e 12 ( x 6 − x 7 ) × d x 1 d x 2 d x 3 d x 4 d x 5 d x 7 ≤ i ∈ { 3 , 4 , 5 , 7 , 9 , 12 } ∏ ∥ g s e i ∥ L ∞ ( R d ) I 1 , where I 1 I_1 I 1 d x 1 d x 2 d x 3 d x 4 d x 5 d x 7 dx_1\,dx_2\,dx_3\,dx_4\,dx_5\,dx_7 d x 1 d x 2 d x 3 d x 4 d x 5 d x 7
g s e 1 ( x 1 − x 2 ) g s e 2 ( x 2 − x 3 ) g s e 6 ( x 2 − x 4 ) g s e 8 ( x 2 − x 6 ) g s e 10 ( x 5 − x 6 ) g s e 11 ( x 6 − x 7 ) . g_{se_1}(x_1-x_2)g_{se_2}(x_2-x_3)g_{se_6}(x_2-x_4)g_{se_8}(x_2-x_6)g_{se_{10}}(x_5-x_6)g_{se_{11}}(x_6-x_7). g s e 1 ( x 1 − x 2 ) g s e 2 ( x 2 − x 3 ) g s e 6 ( x 2 − x 4 ) g s e 8 ( x 2 − x 6 ) g s e 10 ( x 5 − x 6 ) g s e 11 ( x 6 − x 7 ) . In Step 2, to compute I 1 I_1 I 1 x 1 , x 2 , x 3 , x 4 , x 5 , x 7 x_1,x_2,x_3,x_4,x_5,x_7 x 1 , x 2 , x 3 , x 4 , x 5 , x 7 only once in this product. This corresponds to choosing the first leaf. Our choices are 1 , 3 , 5 , 7 1,3,5,7 1 , 3 , 5 , 7 x 5 x_5 x 5
∫ R d g s e 10 ( x 5 − x 6 ) d x 5 = ∥ g s e 10 ∥ L 1 ( R d ) . \int_{\mathbb{R}^{d}}g_{se_{10}}(x_5-x_6)\,dx_5=\lVert g_{se_{10}}\rVert_{L^1(\mathbb{R}^{d})}. ∫ R d g s e 10 ( x 5 − x 6 ) d x 5 = ∥ g s e 10 ∥ L 1 ( R d ) . We continue in this manner. Say that the order that we pick for the integration is x 5 , x 7 , x 1 , x 3 , x 4 x_5,x_7,x_1,x_3,x_4 x 5 , x 7 , x 1 , x 3 , x 4 5 , 7 , 1 , 3 , 4 5,7,1,3,4 5 , 7 , 1 , 3 , 4
∫ R d g s e 8 ( x 2 − x 6 ) d x 2 = ∥ g s e 8 ∥ L ∞ ( R d ) . \int_{\mathbb{R}^{d}}g_{se_8}(x_2-x_6)\,dx_2=\lVert g_{se_8}\rVert_{L^\infty(\mathbb{R}^{d})}. ∫ R d g s e 8 ( x 2 − x 6 ) d x 2 = ∥ g s e 8 ∥ L ∞ ( R d ) . Although the order in which we pick the leaves is not unique, at the end we obtain:
I 1 = ∏ i ∈ { 1 , 2 , 6 , 8 , 10 , 11 } ∥ g s e i ∥ L 1 ( R d ) . I_1=\prod_{i\in\{1,2,6,8,10,11\}}\lVert g_{se_i}\rVert_{L^1(\mathbb{R}^{d})}. I 1 = i ∈ { 1 , 2 , 6 , 8 , 10 , 11 } ∏ ∥ g s e i ∥ L 1 ( R d ) . Acknowledgement. The authors would like to thank Guangqu Zheng for useful comments.
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