# Hasimoto frames and the Gibbs measure of periodic nonlinear Schrödinger
equation

Gordon Blower, Azadeh Khaleghi, Moe Kuchemann-Scales  Department of
Mathematics and Statistics, Lancaster University, United Kingdom.

(Date: 13th May 2022)

Abstract The paper interprets the cubic nonlinear Schrödinger equation as a
Hamiltonian system with infinite dimensional phase space. There is a Gibbs
measure which is invariant under the flow associated with the canonical
equations of motion. The logarithmic Sobolev and concentration of measure
inequalities hold for the Gibbs measures, and here are extended to the
$k$-point correlation function and distributions of related empirical
measures. By Hasimoto’s theorem, NLSE gives a Lax pair of coupled ODE for
which the solutions give a system of moving frames. The paper studies the
evolution of the measure induced on the moving frames by the Gibbs measure.

AMS Classification: 35Q82 Primary; 37L55, 35Q55 Secondary.

Keywords: concentration of measure; statistical mechanics; empirical processes

##  1\. Introduction

Consider the Hamiltonian

$H_{3}={\frac{1}{2}}\int_{\mathbb{T}}\Bigl{(}\Bigl{(}{\frac{\partial
P}{\partial x}}\Bigr{)}^{2}+\Bigl{(}{\frac{\partial Q}{\partial
x}}\Bigr{)}^{2}\Bigr{)}dx+{\frac{\beta}{2\gamma}}\int_{\mathbb{T}}\bigl{(}P^{2}+Q^{2}\bigr{)}^{\gamma}dx,$
(1.1)

on ${\mathbb{T}}={\mathbb{R}}/2\pi{\mathbb{Z}}$ which gives the canonical
equations of motion

$\begin{bmatrix}0&1\\\ -1&0\end{bmatrix}{\frac{\partial}{\partial
t}}\begin{bmatrix}Q\\\ P\end{bmatrix}=-{\frac{\partial^{2}}{\partial
x^{2}}}\begin{bmatrix}Q\\\
P\end{bmatrix}+\beta(P^{2}+Q^{2})^{\gamma-1}\begin{bmatrix}Q\\\
P\end{bmatrix}$ (1.2)

so $u=P+iQ$ satisfies the nonlinear Schrödinger equation

$i{\frac{\partial u}{\partial t}}=-{\frac{\partial^{2}u}{\partial
x^{2}}}+\beta|u|^{2(\gamma-1)}u.$ (1.3)

When $\gamma=2$, we have the cubic nonlinear Schrödinger equation. The spatial
variable is $x\in{\mathbb{T}}$, and the functions are periodic, so that the
system applies to fields parametrized by a circle.

Throughout the paper, we write $(M,d,\mu)$ for a complete and separable metric
space with a Radon (inner regular) probability measure $\mu$ on the
$\sigma$-algebra generated by the Borel subsets.

The squared $L^{2}$ norm $H_{1}=\int(P^{2}+Q^{2})$ is formally invariant under
the canonical equations of motion, so we can consider possible invariant
measures on

$B_{K}=\Bigl{\\{}u=P+iQ:P,Q\in
L^{2}({\mathbb{T}};{\mathbb{R}}):\int_{\mathbb{T}}(P^{2}+Q^{2})dx\leq
K\Bigr{\\}}.$ (1.4)

The Gibbs measure on $B_{K}$ for this micro-canonical ensemble is

$\mu_{K,\beta}(du)=Z_{K}(\beta)^{-1}{\mathbb{I}}_{B_{K}}(u)\exp\Bigl{(}-{\frac{\beta}{4}}\int_{\mathbb{T}}|u|^{4}dx\Bigr{)}W(du)$
(1.5)

where $W(du)$ is Wiener loop measure, and $Z_{K}(\beta)$ is a normalizing
constant. Lebowitz, Rose and Speer [LRS] proved existence of such an invariant
measure, so that for all $K>0$ and $\beta\in{\mathbb{R}}$ there exists
$Z_{K}(\beta)>0$ such that $\mu_{K,\beta}$ is a Radon probability measure on
$B_{K}\subset L^{2}({\mathbb{T}};{\mathbb{R}}^{2})$. When $\beta=0$, we refer
to the measure as free Wiener loop measure, indicating that the dynamics are
free of potentials. For $\beta<0$, (1.3) is said to be focussing and the
Hamiltonian is unbounded below, giving the source of the technical problem.

Bourgain [Bo] gave an alternative existence proof using random Fourier series,
and showed that the measure is invariant under the flow in the sense that the
Cauchy problem is well posed on the support. Further refinements include a
result of McKean [M], that the sample paths are Hölder continuous, and a
result from Theorem 1.2(iv) in [BBD] that the invariant measure of the micro-
canonical ensemble satisfies a logarithmic Sobolev inequality. Random Fourier
series fit naturally into Sturm’s theory of metric measure spaces, which we
use to reduce some of the analysis to invariant measures on finite-dimensional
Hamiltonian systems.

The focusing case for spatial variable $x\in{\mathbb{R}}$ captures soliton
solutions, and [LRS] discuss the possible transition of the system between an
ambient bounded random field and a soliton solution. For $x\in{\mathbb{T}}$,
the notion of a spatially localized solution is inapplicable, but some of the
[LRS] results are still relevant.

In section two, we consider tensor products of Hilbert space $H$ and a
$k$-point density matrix. For $\mu_{0}$ a centered Gaussian measure on $H$, we
express a specific integral

$J^{(k)}=\int_{H}|u^{\otimes k}\rangle\langle u^{\otimes k}|\,\mu_{0}(du)$

as a series of elementary tensors. This calculation involves combinatorial
results which are expressed in terms of Knuth’s odd and even decompositions of
Young diagrams. In section three, we use concentration of measure results to
show how $|u^{\otimes k}\rangle\langle u^{\otimes k}|\,$ is close to its mean
value $J^{(k)}$ on a set of large probability. This statement also holds when
we replace $\mu_{0}$ by the Gibbs measure. In section 4 we introduce metric
probability measure spaces and show how the infinite-dimensional dynamical
system (1.2) can be approximated by finite-dimensional dynamical systems,
particularly involving random Fourier series. In particular, we show that
$x\mapsto u(x,t)$ is $\gamma$-Hölder continuous $[0,2\pi]\mapsto L^{4}$ for
$0<\gamma<1/16$. We also obtain results on the empirical distributions that
arise when we sample solutions of (1.2) with respect to Gibbs measure (1.5),
which we use in the numerical experiments in section 7.

Hasimoto observed that (1.2) can be expressed as a Lax pair of coupled
ordinary differential equations with solutions in $SO(3)$, one of which is the
Serret-Frenet system for a moving frame on a curve in ${\mathbb{R}}^{3}$. In
sections 5 and 6 we consider the evolution of the dynamical system
corresponding to Hasimoto frames under the Gibbs measure. In section 7 we
present numerical experiments regarding the solutions, which illustrate the
nature of frames that arise from the solutions of (1.2) for typical elements
in the support of the Gibbs measure (1.5).

##  2\. Tensor products and $k$-point density matrices for Gaussian measure

Let $H$ be a separable complex Hilbert space, with inner product
$\langle\cdot\mid\cdot\rangle$ which is linear in the second argument. We
identify the injective tensor product $H\check{\otimes}H$ with the algebra
${\mathcal{L}}(H)$ of bounded linear operators on $H$ and the projective
tensor product $H\hat{\otimes}H$ with the ideal ${\mathcal{L}}^{1}(H)$ of
trace class operators on $H$. For $H=L^{2}$, the identification is

$f\otimes\bar{g}=\mid f\rangle\langle g\mid:h\mapsto
f(x)\int\bar{g}(y)h(y)dy.$

Let $A\in{\mathcal{L}}^{1}(H)$ be self-adjoint such that $0\leq A\leq I$, and
let $\mu_{0}$ be a Gaussian measure on $H$ of mean zero and covariance $A$. By
the spectral theorem, We can choose an orthonormal basis
$(\varphi_{j})_{j=1}^{\infty}$ of $H$ such that
$A\varphi_{j}=\alpha_{j}\varphi_{j}$ where the spectrum of $A$ is the closure
of $\\{\alpha_{j}:j=1,2,\dots\\}$. Then we introduce mutually independent
Gaussian $N(0,1)$ random variables $(\gamma_{j})_{j=1}^{\infty}$ and the
vector

$u=\sum_{j=1}^{\infty}\sqrt{\alpha_{j}}\gamma_{j}\varphi_{j}$ (2.1)

so that $\mu_{0}$ is the distribution of $u$ on $H$, as one easily checks by
computing the expectation

$\displaystyle{\mathbb{E}}\exp(i\langle f,u\rangle)$
$\displaystyle={\mathbb{E}}\exp\Bigl{(}\sum_{j}i\sqrt{\alpha_{j}}\langle
f,\varphi_{j}\rangle\gamma_{j}\Bigr{)}$
$\displaystyle=\exp\Bigl{(}\sum_{j}-{\frac{1}{2}}\alpha_{j}\langle
f,\varphi_{j}\rangle^{2}\Bigr{)}$
$\displaystyle=\exp\bigl{(}-{\frac{1}{2}}\langle
Af,f\rangle\bigr{)}\qquad(f\in H).$ (2.2)

Hence $A$ is the mean of rank-one tensors with respect to Gaussian measure

$A=\int_{H}\mid u\rangle\langle u\mid\,\mu_{0}(du).$

The $k$-fold tensor product $H^{\otimes k}$ can be completed to give a Hilbert
space, so that the space $H^{s\otimes k}$ of symmetric tensors gives a closed
linear subspace. We consider

$J^{(k)}=\int_{H}|u^{\otimes k}\rangle\langle u^{\otimes k}|\,\mu_{0}(du).$

An element of $H^{\otimes k}\hat{\otimes}H^{\otimes k}$ determines a linear
operator ${\mathcal{L}}(H^{\otimes k})$, commonly referred to as a matrix, so
$J^{(k)}\in(L^{2})^{s\otimes k}\hat{\otimes}(L^{2})^{\otimes k}$ gives a
$k$-point density matrix, or equivalently a trace class operator
$J^{(k)}\in{\mathcal{L}}^{1}(H^{s\otimes k})$ .

Lemma 3.3 of [LNR1] contains calculations regarding $J^{(k)}$ which we have
not been able to interpret, particularly line 8 of page 79. Here we calculate
$J^{(2)}$ directly, before addressing the case of general $k$. Evidently we
have ${\mathbb{E}}(\gamma_{j}\gamma_{\ell}\gamma_{m}\gamma_{n})=0$ if one of
the indices $j,\ell,m,n$ is distinct from all the others; otherwise, we have
all the indices equal, or two distinct pairs of equal indices. Hence we have

$\displaystyle\int_{H}\mid u\otimes u\rangle\langle u\otimes
u\mid\,\mu_{0}(du)$
$\displaystyle=\sum_{j,\ell,m,n}\sqrt{\alpha_{j}\alpha_{\ell}\alpha_{m}\alpha_{n}}\mid\varphi_{j}\otimes\varphi_{\ell}\rangle\langle\varphi_{m}\otimes\varphi_{n}\mid{\mathbb{E}}(\gamma_{j}\gamma_{\ell}\gamma_{m}\gamma_{n})$
$\displaystyle=\sum_{j}3\alpha_{j}^{2}\mid\varphi_{j}\otimes\varphi_{j}\rangle\langle\varphi_{j}\otimes\varphi_{j}\mid$
$\displaystyle\quad+\sum_{j,\ell:j\neq\ell}\alpha_{j}\alpha_{\ell}\mid\varphi_{j}\otimes\varphi_{\ell}\rangle\langle\varphi_{j}\otimes\varphi_{\ell}\mid$
$\displaystyle\quad+\sum_{j,m:j\neq
m}\alpha_{j}\alpha_{m}\mid\varphi_{j}\otimes\varphi_{j}\rangle\langle\varphi_{m}\otimes\varphi_{m}\mid$
$\displaystyle\quad+\sum_{j,\ell:j\neq\ell}\alpha_{j}\alpha_{\ell}\mid\varphi_{j}\otimes\varphi_{\ell}\rangle\langle\varphi_{\ell}\otimes\varphi_{j}\mid$
(2.3)

and we combine the second and fourth of these to obtain

$\displaystyle\int_{H}\mid u\otimes u\rangle\langle u\otimes
u\mid\,\mu_{0}(du)$
$\displaystyle=\sum_{j}3\alpha_{j}^{2}\mid\varphi_{j}\otimes\varphi_{j}\rangle\langle\varphi_{j}\otimes\varphi_{j}\mid$
$\displaystyle\quad+{\frac{1}{2}}\sum_{j,\ell:j\neq\ell}\alpha_{j}\alpha_{\ell}\mid\varphi_{j}\otimes\varphi_{\ell}+\varphi_{\ell}\otimes\varphi_{j}\rangle\langle\varphi_{j}\otimes\varphi_{\ell}+\varphi_{\ell}\otimes\varphi_{j}\mid$
$\displaystyle\quad+\sum_{j,m:j\neq
m}\alpha_{j}\alpha_{m}\mid\varphi_{j}\otimes\varphi_{j}\rangle\langle\varphi_{m}\otimes\varphi_{m}\mid$
(2.4)

which exhibits the right-hand side as a symmetric tensor, in which the final
term shows the integral is not diagonal with respect to the orthonormal basis

$\Bigl{\\{}\varphi_{j}\otimes\varphi_{j},\quad(\varphi_{j}\otimes\varphi_{\ell}+\varphi_{\ell}\otimes\varphi_{j})/\sqrt{2};\quad
j,\ell\in{\mathbb{N}};j\neq\ell\Bigr{\\}}$

of the symmetric tensor product $H\otimes_{s}H$, hence $J^{(2)}$ is not a
multiple of $A\otimes A$.

For $k\in{\mathbb{N}}$, let $\Pi_{k}$ be the set of all partitions of $k$ so
that $\pi\in\Pi_{k}$ may be expressed as $k=k_{1}+k_{2}+\dots+k_{n}$ where the
row lengths $k_{j}\in{\mathbb{N}}$ have $k_{1}\geq k_{2}\geq\dots\geq k_{n}$.
Given such $\pi$ and a $n$-element subset $\\{j_{1},\dots,j_{n}\\}$ of
${\mathbb{N}}$, there is a symmetric tensor

${\frac{1}{\sqrt{n!}}}\sum_{\sigma}\varphi_{\sigma(j_{1})}^{\otimes
k_{1}}\otimes\dots\otimes\varphi_{\sigma(j_{n})}^{\otimes k_{n}}\in H^{\otimes
k}$

where the sum is over all the permutations $\sigma$ of
$\\{j_{1},\dots,j_{n}\\}$. The set of all such tensors gives an orthonormal
basis of the $k$-fold symmetric tensor product $H^{s\otimes k}$.

We express $u$ as in (2.1) and consider the expansion

$\displaystyle\mid u^{\otimes k}\rangle$ $\displaystyle\langle u^{\otimes
k}\mid$
$\displaystyle=\sum_{(m_{1},\dots,m_{2k})\in{\mathbb{N}}^{2k}}\sqrt{\alpha_{m_{1}}\dots\alpha_{m_{2k}}}\gamma_{m_{1}}\dots\gamma_{m_{2k}}\mid\varphi_{m_{1}}\otimes\dots\otimes\varphi_{m_{k}}\rangle\langle\varphi_{m_{k+1}}\otimes\dots\otimes\varphi_{m_{2k}}\mid$
(2.5)

in terms of this orthonormal basis of $H^{s\otimes k}$, and look for the terms
that do not vanish after integration with respect to $\mu_{0}$.

######  Definition 2.1.

(even decomposition) Given $\pi\in\Pi_{k}$ consider a pair
$(\lambda,\rho)\in\Pi_{k}^{2}$ with rows
$\lambda:k=\ell_{1}+\ell_{2}+\dots+\ell_{n}$ where
$\ell_{j}\in{\mathbb{N}}\cup\\{0\\}$ and $\rho:k=r_{1}+r_{2}+\dots+r_{n}$
where $r_{j}\in{\mathbb{N}}\cup\\{0\\}$ and

$2k_{j}=\ell_{j}+r_{j}\qquad(j=1,\dots,n),$

so that $\lambda$ and $\rho$ have equal numbers of odd rows; here rows may
have zero lengths, and the rows are not necessarily in decreasing order. We
refer to $(\lambda,\rho)$ as an even decomposition of $\pi$.

######  Remark 2.2.

There are various alternative descriptions of even decompositions. We write
$\lambda\sim\rho$ if $\lambda$ and $\rho$ are partitions that have equal
numbers of boxes and equal numbers of odd rows; evidently $\sim$ is an
equivalence relation on the set of partitions. By [K] Theorem 4 there is a
bijection between symmetric matrices $A$ that have entries in
${\mathbb{N}}\cup\\{0\\}$ with column sums $c_{1},\dots,c_{n}$ and Young
tableaux $P$ such that have $c_{j}$ occurrences of $j$ as entries and number
of columns of $P$ of odd length equals the trace of $A$. Given symmetric
matrices $A$ and $B$ with entries in ${\mathbb{N}}\cup\\{0\\}$ such that $A$
and $B$ have equal traces and equal totals of entries, then the $RSK$
correspondence takes $A$ to $P$ and $B$ to $Q$ where $P$ and $Q$ are Young
tableaux with an equal number of boxes, and their transposed diagrams
$P^{\prime}$ and $Q^{\prime}$ have an equal number of odd rows, so
$P^{\prime}\sim Q^{\prime}$.

For notational convenience, we also regard $\varphi_{j}^{\otimes 0}$ as a
factor which may be omitted in tensor products. Then given such a triple
$(\pi,\lambda,\rho)$ and an $n$-subset $\\{j_{1},\dots,j_{n}\\}$ of
${\mathbb{N}}$,

$\alpha_{j_{1}}^{k_{1}}\dots\alpha_{j_{n}}^{k_{n}}\mid\varphi_{j_{1}}^{\otimes\ell_{1}}\otimes\dots\otimes\varphi_{j_{n}}^{\otimes\ell_{n}}\rangle\langle\varphi_{j_{1}}^{\otimes
r_{1}}\otimes\dots\otimes\varphi_{j_{n}}^{\otimes
r_{n}}\mid{\mathbb{E}}\bigl{(}\gamma_{j_{1}}^{2k_{1}}\gamma_{j_{2}}^{2k_{2}}\dots\gamma_{j_{n}}^{2k_{n}}\bigr{)}$
(2.6)

where

${\mathbb{E}}\bigl{(}\gamma_{j_{1}}^{2k_{1}}\gamma_{j_{2}}^{2k_{2}}\dots\gamma_{j_{n}}^{2k_{n}}\bigr{)}=\prod_{j=1}^{n}{\frac{(2k_{j})!}{2^{k_{j}}k_{j}!}}$
(2.7)

gives a nonzero summand in $J^{(k)}$.

Conversely, let $(\lambda,\rho)\in\Pi_{k}^{2}$ and suppose that $\lambda$ and
$\rho$ have equal numbers of odd rows, so that after adding zero rows and
reordering the rows we have $r_{j}+\ell_{j}$ even for all $j$. Then we
introduce $2k_{j}=\ell_{j}+r_{j}$ and after a further reordering write
$k=k_{1}+k_{2}+\dots+k_{n}$ where $k_{j}\in{\mathbb{N}}$ have $k_{1}\geq
k_{2}\geq\dots\geq k_{n}$, and we have $\pi\in\Pi_{k}$ as above. Given a
$n$-subset $\\{j_{1},\dots,j_{n}\\}$ of ${\mathbb{N}}$, we take $2k_{m}$
copies of $j_{m}$ and split them as $\ell_{m}$ on the bra side and $r_{m}$ on
the ket side of the tensor for $m=1,\dots,n$, making a contribution as in
(2.6). We summarize these results as follows.

######  Proposition 2.3.

The integral $J^{(k)}$ is the sum over the summands (2.6) that arise from a
$\pi\in\Pi_{k}$ with $n$ nonzero rows, a $n$-subset of ${\mathbb{N}}$, and an
even decomposition of $\pi$ into a pair $(\lambda,\rho)\in\Pi_{k}^{2}$ where
$\lambda$ and $\rho$ have equal numbers of odd rows.

##  3\. Concentration of $k$-point matrices for Gibbs measure

Let $H=L^{2}({\mathbb{T}};{\mathbb{R}})$ and let
$(\gamma_{j})_{j\in{\mathbb{Z}}}$ be mutually independent Gaussian $N(0,1)$
random variables, where $z_{j}=(\gamma_{j}+i\gamma_{-j})/\sqrt{2})$ and
$z_{-j}=(\gamma_{j}-i\gamma_{-j})/\sqrt{2}$ for $j\in{\mathbb{N}}$. Then we
take Brownian loop in the style

$u(\theta)=\sum_{j\in{\mathbb{Z}}\setminus\\{0\\}}{\frac{z_{j}e^{ij\theta}}{|j|}},$
(3.1)

so that Wiener loop $W(du)$ is the distribution of $u\in H$. By orthogonality,
we have

$\int_{\mathbb{T}}|u(\theta)|^{2}{\frac{d\theta}{2\pi}}=\sum_{j\in{\mathbb{Z}}\setminus\\{0\\}}{\frac{|z_{j}|^{2}}{j^{2}}}$
(3.2)

so that $u\in B_{K/2\pi}$ if and only if $\sum_{j}\gamma_{j}^{2}/j^{2}\leq K.$
Chebyshev’s inequality and independence, we have

${\mathbb{P}}\Bigl{[}\sum_{j\in{\mathbb{Z}}\setminus\\{0\\}}{\frac{\gamma_{j}^{2}}{j^{2}}}\geq
K\Bigr{]}\leq
e^{-tK}{\frac{\pi\sqrt{2t}}{\sin(\pi\sqrt{2t})}}\qquad(0<t<1/2,K>0).$ (3.3)

The low Fourier modes are the predominant terms since one has the estimate

${\mathbb{P}}\Bigl{[}\sum_{j\in{\mathbb{Z}};|j|\geq
m}{\frac{\gamma_{j}^{2}}{j^{2}}}\geq
K\Bigr{]}\leq\exp\bigl{(}m-Km^{2}/4\bigr{)},\qquad(m\in{\mathbb{N}},K>0),$
(3.4)

which also follows from Chebyshev’s inequality and independence.

Let $\mu_{\lambda}(du)=\zeta(\lambda)^{-1}\exp(\lambda V(u))W(du)$ where

$\zeta(\lambda)=\int_{B_{K}}\exp(\lambda V(u))W(du)$

so that $\mu_{\lambda}$ is a probability measure; we can take
$V(u)=\int_{\mathbb{T}}u(\theta)^{4}d\theta/(2\pi)$ and $W$ to be Brownian
loop measure. Here $\mu_{\lambda}$ is Gibbs measure (1.5) with the inverse
temperature $\beta$, but we prefer to work with $\lambda=-\beta>0$ so that the
convexity statements are easier to interpret.

######  Theorem 3.1.

Under the family of Gibbs measures (1.5) associated with $NLS$ (1.3), the
random variable $u\mapsto\langle u^{\otimes k}\mid T\mid u^{\otimes k}\rangle$
with $u\in(B_{K},L^{2},\mu_{\lambda})$ and $T\in{\mathcal{L}}(H^{s\otimes k})$
satisfies a Gaussian concentration of measure, the mean is a Lipschitz
continuous function of $\beta$, and the mean for $\beta=0$ is a sum over
partitions of $2k$ over even decompositions.

The statements in this theorem will be proved in this section. They involve
the integral

$G_{\lambda}^{(k)}=\int_{B_{K}}\mid u^{\otimes k}\rangle\langle u^{\otimes
k}\mid\mu_{\lambda}(du)$ (3.5)

where $\mu_{\lambda}$ is the Gibbs measure for NLS. In the defocussing case,
the $k$-particle density matrix of an interacting quantum system with suitable
initial conditions converges to its classical analogue see [AGT] (2.16) for
the 1D case and [LNR2] for 2D and 3D.

We can write $u=P+iQ$ for real variables $(P,Q)$ and interpret $\langle
u^{\otimes k}\mid T\mid u^{\otimes k}\rangle$ as a homogeneous polynomial in
$(p,q)$ of total degree $2k$. The following result gives concentration of
measure for Lipschitz functions on $(B_{K},L^{2},\mu_{\lambda})$, and shows
that $k$-point matrices are concentrated near to their mean value.

######  Proposition 3.2.

For $T\in{\mathcal{L}}(H^{s\otimes k})$ with operator norm $\|T\|$, let
$g_{T}:B_{K}\rightarrow{\mathbb{C}}$ by $g_{T}(u)=\langle u^{\otimes k}\mid
T\mid u^{\otimes k}\rangle$. Then there exists $\alpha=\alpha(\beta,K)>0$ such
that

$\mu_{\lambda}\Bigl{(}\bigl{\\{}u\in
B_{K}:|g_{T}(u)-\mathop{\mathrm{trace}}(G_{\lambda}^{(k)}T)|>r\bigr{\\}}\Bigr{)}\leq
4\exp\Bigl{(}{\frac{-\alpha
r^{2}}{32k^{2}K^{2k-1}\|T\|^{2}}}\Bigr{)}\qquad(r>0).$ (3.6)

###### Proof.

Here $g_{T}$ has mean value

$\int_{B_{K}}g_{T}(u)\mu_{\lambda}(du)=\int_{B_{K}}\langle u^{\otimes k}\mid
T\mid u^{\otimes
k}\rangle\mu_{\lambda}(du)=\mathop{\mathrm{trace}}(G_{\lambda}^{(k)}T).$ (3.7)

Also $g_{T}$ is Lipschitz, with

$\displaystyle|g_{T}(u)-g_{T}(v)|$
$\displaystyle\leq\|T\|\sum_{j=0}^{2k-1}\|u\|^{j}\|v\|^{2k-j-1}\|u-v\|$
$\displaystyle\leq 2kK^{k-1/2}\|T\|\|u-v\|\qquad(u,v\in B_{K}).$ (3.8)

By the logarithmic Sobolev inequality Theorem 1.2(iv) of [BBD], there exists
$\alpha=\alpha(K,\beta)>0$ such that

$\int_{B_{K}}f(u)^{2}\log
f(u)^{2}\mu_{\lambda}(du)\leq\int_{B_{K}}f(u)^{2}\mu_{\lambda}(du)\log\Bigl{(}\int_{B_{K}}f(u)^{2}\mu_{\lambda}(du)\Bigr{)}+{\frac{2}{\alpha}}\int_{B_{K}}\|\nabla
f\|^{2}\mu_{\lambda}(du)$ (3.9)

for all continuously differentiable $f:B_{K}\rightarrow{\mathbb{R}}$. In
particular, we choose

$f(u)=\exp\bigl{(}\lambda\mathop{{\rm
Re}}\bigl{(}g_{T}(U)-\mathop{\mathrm{trace}}(G_{t}^{(k)}T)\bigr{)}/2\Bigr{)}$

and we deduce that the moment generating function

$\varphi(r)=\int_{B_{K}}\exp\Bigl{(}r\mathop{{\rm
Re}}\bigl{(}g_{T}(u)-\mathop{\mathrm{trace}}(G_{\lambda}^{(k)}T)\bigr{)}\Bigr{)}\mu_{\lambda}(du)$
(3.10)

satisfies $\varphi(0)=1$, $\varphi^{\prime}(0)=1$ and the differential
inequality

$r\varphi^{\prime}(r)\leq\varphi(r)\log\varphi(r)+{\frac{8k^{2}K^{2k-1}\|T\|^{2}r^{2}}{\alpha}}$
(3.11)

hence

$\varphi(r)\leq\exp\Bigl{(}{\frac{8k^{2}K^{2k-1}\|T\|^{2}r^{2}}{\alpha}}\Bigr{)}.$

One can conclude the proof by a standard application of Chebyshev’s inequality
to the integral for $\varphi$ in (3.10). ∎

To make full use of the previous result, one needs to know the mean
$\mathop{\mathrm{trace}}(G_{\lambda}^{(k)}T)$ as in (3.7), which depends upon
the measure in (3.5). The following shows how the mean can vary with the
inverse temperature $\beta=-\lambda$.

######  Proposition 3.3.

For $g:B_{K}\rightarrow{\mathbb{R}}$ an $L$-Lipschitz function, the mean
values of $g$ with respect to the measures $\mu_{\lambda}$ satisfy

$\Bigl{(}\int_{B_{K}}g(u)\bigl{(}\mu_{b}(du)-\mu_{a}(du)\bigr{)}\Bigr{)}^{2}\leq{\frac{L^{2}(b-a)^{2}}{2\alpha}}\int\\!\\!\\!\int_{B_{K}\times
B_{K}}(V(u)-V(w))^{2}\mu_{\lambda}(du)\mu_{\lambda}(dw)$ (3.12)

where $\alpha$ is the constant in (3.9) for $\mu_{a}$, and some
$\lambda\in(a,b)$.

###### Proof.

We observe that $\log\zeta(\lambda)$ is a convex function of $\lambda>0$ and
by the mean value theorem, there exists $a<\lambda<b$ such that

$\displaystyle\log\zeta(a)-\log\zeta(b)$
$\displaystyle=-(b-a){\frac{\zeta^{\prime}(b)}{\zeta(b)}}+{\frac{(b-a)^{2}}{2}}\Bigl{(}{\frac{\zeta^{\prime\prime}(\lambda)}{\zeta(\lambda)}}-{\frac{\zeta^{\prime}(\lambda)^{2}}{\zeta(\lambda)^{2}}}\Bigr{)}$
$\displaystyle=-(b-a)\int_{B_{K}}V(u)\mu_{b}(du)$
$\displaystyle\quad+{\frac{(b-a)^{2}}{4}}\int\\!\\!\\!\int_{B_{K}\times
B_{K}}(V(u)-V(w))^{2}\mu_{\lambda}(du)\mu_{\lambda}(dw).$ (3.13)

Let $W_{1}(\mu_{a},\mu_{b})$ be the Wasserstein transportation distance
between $\mu_{b}$ and $\mu_{a}$ for the cost function $\|u-v\|_{L^{2}}$, as in
page 34 of [V1]. Then by duality we have

$\Bigl{|}\int_{B_{K}}g(u)\mu_{b}(du)-\int_{B_{K}}g(u)\mu_{a}(du)\Bigr{|}\leq
LW_{1}(\mu_{b},\mu_{a}).$

By results of Otto and Villani discussed in [V1] pages 291-2, the logarithmic
Sobolev inequality of Theorem 1.2(iv) [BBD] implies a transportation cost
inequality

$W_{1}(\mu_{b},\mu_{a})\leq\Bigl{(}{\frac{2}{\alpha}}{\hbox{Ent}}(\mu_{b}\mid\mu_{a})\Bigr{)}^{1/2}$
(3.14)

in the style of Talagrand, where the relative entropy is

$\displaystyle{\hbox{Ent}}(\mu_{b}\mid\mu_{a})$
$\displaystyle=\int_{B_{K}}\log{\frac{d\mu_{b}}{d\mu_{a}}}\mu_{b}(da)$
$\displaystyle=(b-a)\int_{B_{K}}V(u)\mu_{b}(du)-(\log\zeta(b)-\zeta(a))$
$\displaystyle={\frac{(b-a)^{2}}{4}}\int\\!\\!\\!\int_{B_{K}\times
B_{K}}(V(u)-V(w))^{2}\mu_{\lambda}(du)\mu_{\lambda}(dw),$ (3.15)

where the final step follows from (3). The stated result follows on combining
these inequalities. ∎

######  Proposition 3.4.

The integral $G_{0}^{(k)}$ from (3.5) is the sum over the terms (2.6) that
arise from a $\pi\in\Pi_{k}$ with $n$ nonzero rows, a $n$-subset of
${\mathbb{N}}$, and an even decomposition of $\pi$ into a pair
$(\lambda,\rho)\in\Pi_{k}^{2}$ where $\lambda$ and $\rho$ have equal numbers
of odd rows.

###### Proof.

The measure $\mu_{0}$ is a Wiener loop measure restricted to $B_{k}$. For any
sequence $(\varepsilon_{n})_{n\in{\mathbb{Z}}}\in\\{\pm 1\\}^{\mathbb{Z}}$,
the sequence $(\gamma_{n})_{n\in{\mathbb{Z}}}$ with $\gamma_{n}$ mutually
independent $N(0,1)$ Gaussian random variables has the same distribution as
the sequence $(\varepsilon_{n}\gamma)_{n\in{\mathbb{Z}}}$. Also, the condition
$\sum_{n\in{\mathbb{Z}}\setminus\\{0\\}}\gamma_{n}^{2}/n^{2}\leq K$ does not
change under this transformation. Let
$u_{\varepsilon}(\theta)=\sum^{\prime}_{j}\varepsilon_{j}\zeta_{j}e^{ij\theta}/|j|$.
We therefore have

$G_{0}^{(k)}=\int_{B_{K}}\int_{\\{\pm 1\\}^{\mathbb{Z}}}\mid
u_{\varepsilon}^{\otimes k}\rangle\langle u_{\varepsilon}^{\otimes k}\mid
d\varepsilon\mu_{0}(du)$

where $d\varepsilon$ is the Haar probability measure on the Cantor group
$\\{\pm 1\\}^{\mathbb{Z}}$. We can therefore compute the inner integral in
this expression for $G_{0}^{(k)}$ by the same calculation that led to the
corresponding statement for $J^{(k)}$, since we only used the even
decomposition of partitions to derive (2.6).∎

We have

$\Bigl{[}\sum_{j\in{\mathbb{Z}}\setminus\\{0\\}}{\frac{\gamma_{j}^{2}}{j^{2}}}\leq
K\Big{]}\subseteq\bigcap_{j\in{\mathbb{Z}}\setminus\\{0\\}}\Bigl{[}\gamma_{j}^{2}\leq
Kj^{2}\Bigr{]}$ (3.16)

where the sets are independent under the Gaussian measure $d{\mathbb{P}}$, so
we have a substitute for (2.7)

$\int_{B_{K}}\gamma_{j_{1}}^{2k_{1}}\gamma_{j_{2}}^{2k_{2}}\dots\gamma_{j_{n}}^{2k_{n}}\mu_{0}(du)\leq{\mathbb{P}}(B_{K})^{-1}\prod_{\ell=1}^{n}\int_{[\gamma^{2}\leq
Kj_{\ell}^{2}]}\gamma^{2k_{\ell}}d{\mathbb{P}},$ (3.17)

where ${\mathbb{P}}(B_{K})$ satisfies (3.3) and there is an approximate
formula

$\int_{[\gamma^{2}\leq
Kj_{\ell}^{2}]}\gamma^{2k_{\ell}}d{\mathbb{P}}={\frac{(2k_{\ell})!}{2^{k_{\ell}}k_{\ell}!}}\exp\Bigl{(}-\Bigl{(}{\frac{j_{\ell}^{2}K}{2}}\Bigr{)}^{k_{\ell}-1/2}{\frac{e^{-_{\ell}^{2}K/2}}{\Gamma(k_{\ell}+1/2)}}\Bigr{)}.$
(3.18)

##  4\. Concentration for metric measure spaces

In a similar spirit, we give a concentration result for $k$-fold stochastic
integrals. This result is in the spirit of the integrability criteria of[CM]
which relates to a single variable. Let $H^{1}$ be the Sobolev space of $v\in
L^{2}({\mathbb{T}};{\mathbb{C}})$ that are absolutely continuous with
derivative $v^{\prime}\in L^{2}({\mathbb{T}};{\mathbb{C}})$. Let $h_{j}\in
H^{1}$ for $j=1,\dots,k$ be such that
$\sum_{j=1}^{k}\int(h_{j}^{\prime})^{2}dx\leq 1$, and consider
$\Phi:B_{K}^{k}\mapsto{\mathbb{R}}^{k}$ given by

$\Phi:(u_{j})_{j=1}^{k}\mapsto\Bigl{(}\int
u_{j}(x)h_{j}^{\prime}(x)dx\Bigr{)}_{j=1}^{k}=\Bigl{(}-\int
h_{j}(x)du_{j}\Bigr{)}_{j=1}^{k}.$ (4.1)

The following result describes the distribution of this
${\mathbb{C}}^{k}$-valued random variable.

######  Proposition 4.1.

Let $\nu_{K}$ be the probability measure on ${\mathbb{C}}^{k}$ that is induced
from $\mu_{K,\beta}^{\otimes k}$ by $\Phi$. Then there exists $\alpha_{K}>0$
independent of $k$ such that

$\int_{{\mathbb{C}}^{k}}G(w)^{2}\log\Bigl{(}G(w)^{2}/\int
G^{2}d\nu_{K}\Bigr{)}\nu_{K}(dw)\leq{\frac{2}{\alpha_{K}}}\int_{{\mathbb{C}}^{k}}\|\nabla
G(w)\|^{2}\nu_{K}(dw)$ (4.2)

for all $G\in C^{1}_{c}({\mathbb{C}}^{k};{\mathbb{R}})$. The distribution
$\nu_{K}$ has mean $x_{0}$ and satisfies

$\int_{{\mathbb{C}}^{k}}e^{t^{2}\|x-x_{0}\|^{2}/2}\nu_{K}(dx)\leq\Bigl{(}1-{\frac{t^{2}}{\alpha}}\Bigr{)}^{-k}\qquad(t^{2}<\alpha).$
(4.3)

###### Proof.

We observe that for all $u=(u_{j})_{j=1}^{k}\in B_{K}^{k}$ and
$v=(v_{j})_{j=1}^{k}\in B_{K}^{k}$ we have

$\|\Phi(u)-\Phi(v)\|^{2}_{{\mathbb{R}}^{k}}\leq\sum_{j=1}^{k}\int(h_{j}^{\prime}(x))^{2}dx\sum_{j=1}^{k}\int|u_{j}(x)-v_{j}(x)|^{2}dx$
(4.4)

so $\Phi:(B_{K}^{k},\ell^{2}(L^{2}))\rightarrow({\mathbb{R}}^{k},\ell^{2})$ is
Lipschitz with constant one. Each metric probability space
$(B_{K},L^{2},\mu_{K,\beta})$ satisfies a logarithmic Sobolev inequality with
constant $\alpha_{K}>0$ by [BBD], and the probability space
$(B_{K}^{k},\ell^{2}(L^{2}),\mu_{K,\beta}^{\otimes k})$ is a direct product of
the metric probability spaces $(B_{K},L^{2},\mu_{K,\beta})$, hence also
satisfies a logarithmic Sobolev inequality

$\int_{B_{K}^{k}}G\circ\Phi(u)^{2}\log\Bigl{(}G\circ\Phi(u)^{2}/\int
G\circ\Phi^{2}d\mu_{K,\beta}^{\otimes k}\Bigr{)}\mu_{K,\beta}^{\otimes
k}(du)\leq{\frac{2}{\alpha_{K}}}\int_{B_{K}^{k}}\|\nabla
G\circ\Phi(u)\|^{2}\mu_{K,\beta}^{\otimes k}(du)$ (4.5)

with constant $\alpha_{K}$ independent of $k$, by section 22 of [V2].

We introduce $x_{0}=\int_{{\mathbb{C}}^{k}}x\nu_{K}(dx)$ and consider

$\varphi(t)=\int_{{\mathbb{C}}^{k}}e^{t\mathop{{\rm Re}}\langle
x-x_{0},y\rangle}\nu_{K}(dx)$ (4.6)

which satisfies $\varphi(0)=1$, $\varphi^{\prime}(0)=0$ and the differential
inequality

$t\varphi^{\prime}(t)\leq\varphi(t)\log\varphi(t)+{\frac{t^{2}\|y\|^{2}}{2\alpha}}$
(4.7)

follows from (4.2). This gives

$\int_{{\mathbb{C}}^{k}}e^{t\mathop{{\rm Re}}\langle
x-x_{0},y\rangle}\nu_{K}(dx)\leq e^{t^{2}\|y\|^{2}/(2\alpha)}$ (4.8)

which we integrate against $e^{-\|y\|^{2}/2}$, where
$y\in{\mathbb{C}}^{k}={\mathbb{R}}^{2k}$, to obtain

$\int_{{\mathbb{C}}^{k}}e^{t^{2}\|x-x_{0}\|^{2}/2}\nu_{K}(dx)\leq\Bigl{(}1-{\frac{t^{2}}{\alpha}}\Bigr{)}^{-k}\qquad(t^{2}<\alpha).$
(4.9)

∎

The probability space $(B_{K},L^{2},\mu_{K,\beta})$ has a tangent space
associated with infinitesimal translations. Let $H^{1}$ be the Sobolev space
of $v\in L^{2}({\mathbb{T}};{\mathbb{C}})$ that are absolutely continuous with
derivative $v^{\prime}\in L^{2}({\mathbb{T}};{\mathbb{C}})$; let
$H^{-1}=(H^{1})^{*}$ be the linear topological dual space for the pairing
$\langle v,w\rangle\mapsto\int_{\mathbb{T}}v(x)\overline{w(x)}dx/(2\pi)$ as
interpreted via Fourier series. Then there is a Radonifying triple of
continuous linear inclusions

$H^{1}\rightarrow L^{2}\rightarrow(H^{1})^{*}$ (4.10)

associated with the Gibbs measure $\mu_{K}$. The space $H^{1}$ has orthonormal
basis

$(e^{in\theta}/\sqrt{n^{2}+1})_{n=-\infty}^{\infty}$ and the covariance matrix
of Wiener loop is $R_{0}={\hbox{diag}}[1/(1+n^{2})]_{n=-\infty}^{\infty}$ with
respect to this basis. By simple estimates, one can deduce that there exists,
for each $\beta$ and $K>0$, a self-adjoint, nonnegative and trace class
operator $R$ such that

$\bigl{\langle}Rf,g\bigr{\rangle}_{H^{1}}=\int_{B_{K}}\,\int_{[0,2\pi]}f^{\prime}ud\theta\int_{[0,2\pi]}\bar{g}^{\prime}\bar{u}d\theta\,\mu_{K,\beta}(du),$
(4.11)

which gives the covariance matrix of the Gibbs measure on $H^{1}$. This is
essentially $G^{(1)}_{-\beta}$, up to the identification of Hilbert spaces in
(4.10).

Cameron and Martin computed the density with respect to the Wiener measure
that results from the linear translation $u\mapsto u+v$ for $v\in H^{1}$;
their results extends to Gibbs measure with some modifications.

We momentarily suppress the dependence of functions upon time, and consider
for $p,q\in H^{1}$, the linear transformation $P+iQ\mapsto P+p+i(Q+q)$.
Cameron and Martin proved that free Wiener measure $(\beta=0)$ is mapped to a
measure that absolutely continuous with respect to the free Wiener measure.
Likewise, Gibbs measure is mapped to a measure absolutely continuous with
respect to Gibbs measure, so we can regard $H^{1}$ as fibres of a tangent
space to $(B_{K},L^{2},\mu_{K,\beta})$. We make a polar decomposition
$P+iQ=\kappa e^{i\sigma}$ with $\kappa=\sqrt{P^{2}+Q^{2}}$ and consider
$\tau={\frac{\partial\sigma}{\partial x}}$.

######  Proposition 4.2.

For $p,q\in H^{1}$ the functional

$L(P,Q)=L(\kappa e^{i\sigma})=-\int_{\mathbb{T}}{\frac{\partial p}{\partial
x}}\kappa\cos\sigma dx-\int_{\mathbb{T}}{\frac{\partial q}{\partial
x}}\kappa\sin\sigma\,dx$ (4.12)

is a Lipschitz functional of $P+iQ=\kappa e^{i\sigma}$ such that

$\displaystyle\int_{B_{K}}\exp\Bigl{(}sL(P,Q)-$ $\displaystyle
s\int_{B_{K}}L(P,Q)\mu_{K,\beta}(dPdQ)\Bigr{)}\mu_{K,\beta}(dPdQ)$
$\displaystyle\leq\exp\Bigl{\\{}Cs^{2}\int_{\mathbb{T}}\Bigl{(}\Bigl{(}{\frac{\partial
p}{\partial x}}\Bigr{)}^{2}+\Bigl{(}{\frac{\partial q}{\partial
x}}\Bigr{)}^{2}\Bigr{)}dx\Bigr{\\}}\qquad(s\in{\mathbb{R}}).$ (4.13)

###### Proof.

Note that $P+iQ\mapsto\kappa$ is $1$-Lipschitz with

$\bigl{[}\sigma\kappa,\kappa\nabla\sigma\bigr{]}={\frac{1}{\sqrt{P^{2}+Q^{2}}}}\begin{bmatrix}P&-Q\\\
Q&P\end{bmatrix}\in SO(2).$

Also $\kappa^{2}\|{\hbox{Hess}}\,\sigma\|$ is bounded. We have

$L(P,Q)=\int p\Bigl{(}{\frac{\partial\kappa}{\partial
x}}\cos\sigma-\kappa\tau\sin\sigma\Bigr{)}dx+\int
q\Bigl{(}{\frac{\partial\kappa}{\partial
x}}\sin\sigma+\kappa\tau\cos\sigma\Bigr{)}dx$ (4.14)

which is bounded on $L^{2}$ with norm $\Lambda$ where

$\Lambda^{2}\leq\int_{\mathbb{T}}\Bigl{(}\Bigl{(}{\frac{\partial p}{\partial
x}}\Bigr{)}^{2}+\Bigl{(}{\frac{\partial q}{\partial
x}}\Bigr{)}^{2}\Bigr{)}dx.$ (4.15)

By the concentration of measure theorem for $\nu_{K}$, we deduce the stated
inequality. ∎

Definition We say that $(M,d,\mu)$ satisfies $T_{2}(\alpha)$ if

$W_{2}(\nu,\mu)^{2}\leq{\frac{2}{\alpha}}{\hbox{Ent}}(\nu\mid\mu)$ (4.16)

for all probability measures $\nu$ that are of finite relative entropy with
respect to $\mu$. The notation credits Talagrand, who developed the theory of
such transportation inequalities. Otto and Villani showed that $LSI(\alpha)$
implies $T_{2}(\alpha)$; see [V1] and [V2]

######  Theorem 4.3.

Let $(M,d,\mu)$ be a metric probability space that satisfies $T_{2}(\alpha)$;
let

$(M^{N},\ell^{2}(d),\mu^{\otimes N})$ be the direct product metric probability
space. Let $L_{N}^{\xi}=N^{-1}\sum_{j=1}^{N}\delta_{\xi_{j}}$ be the empirical
distribution for $\xi=(\xi_{j})_{j=1}^{N}\in M^{N}$ where $\xi_{j}$
distributed as $\mu$. Then the concentration inequality holds

$\mu^{\otimes N}\Bigl{(}\Bigl{\\{}\xi:\in
M^{N}:\bigl{|}W_{p}(L^{\xi}_{N},\mu)-{\mathbb{E}}W_{p}(L_{N},\mu)\bigr{|}>\varepsilon\,\Bigr{\\}}\Bigr{)}\leq
2e^{-N\alpha\varepsilon^{2}/2}\qquad(\varepsilon>0)$ (4.17)

for $p=1,2$.

###### Proof.

The map between metric spaces

$L_{N}:(M^{N},\ell^{2}(d))\rightarrow({\hbox{Prob}}(M),W_{2}):\quad(x_{j})_{j=1}^{N}\mapsto{\frac{1}{N}}\sum_{j=1}^{N}\delta_{x_{j}}$
(4.18)

associated with the empirical distribution is $1/\sqrt{N}$-Lipschitz. Let
$(x_{j})_{j=1}^{N},(y_{j})_{j=1}^{N}\in M^{N}$ and consider the probability
measure on $M\times M$ given by

$\pi={\frac{1}{N}}\sum_{j=1}^{N}\delta_{(x_{j},y_{j})}$ (4.19)

which has marginals $L_{N}^{x}=N^{-1}\sum_{j=1}^{N}\delta_{x_{j}}$ and
$L_{N}^{y}=N^{-1}\sum_{j=1}^{N}\delta_{y_{j}}$, hence gives a transport plan
with cost

$W_{2}(L_{N}^{x},L_{N}^{y})^{2}\leq\int\\!\\!\int_{M\times
M}d(x,y)^{2}\pi(dxdy)={\frac{1}{N}}\sum_{j=1}^{N}d(x_{j},y_{j})^{2}.$ (4.20)

Suppose that $(M,d,\mu)$ satisfies $T_{2}(\alpha)$. Then we take $N$
independent samples $\xi_{1},\dots,\xi_{N}$, each distributed as $\mu$ so they
have joint distribution $\mu^{\otimes N}$ on $(M^{N},\ell^{2}(d))$, where by
independence [BB] Theorem 1.2, $(M^{N},\ell^{2}(d),\mu^{\otimes N})$ also
satisfies $T_{2}(\alpha)$. By forming the empirical distribution, we obtain a
map $L_{N}:(M^{N},\ell^{2}(d))\rightarrow({\hbox{Prob}}M,W_{2})$. Then
$\varphi(\xi)=\sqrt{N}W_{p}(L^{\xi}_{N},\mu)$ is $1$-Lipschitz
$(M^{N},\ell^{2}(d))\rightarrow{\mathbb{R}}$, since by the triangle inequality
and (4.20),

$\displaystyle|\varphi(\xi)-\varphi(\eta)|$
$\displaystyle\leq\sqrt{N}W_{p}(L_{N}^{\xi},L_{N}^{\eta})$
$\displaystyle\leq\sqrt{N}W_{2}(L_{N}^{\xi},L_{N}^{\eta})$
$\displaystyle\leq\sum_{j=1}^{N}d(\xi_{j},\eta_{j})^{2};$ (4.21)

hence $\varphi$ satisfies the concentration inequality

$\int_{M^{N}}\exp\Bigl{(}t\varphi(\xi)-t\int\varphi d\mu^{\otimes
N}\Bigr{)}d\mu^{\otimes N}\leq\exp(t^{2}/(2\alpha))\qquad(t\in{\mathbb{R}}).$
(4.22)

Then the stated concentration inequality follows from Chebyshev’s inequality.

∎

Theorem 4.3 gives a metric version of Sanov’s theorem on the empirical
distribution; see page 70 of [DS]. There are related results in Bolley’s
thesis [Bl]. By [CEMS], Theorem 4.3 applies to Haar probability measure on
$SO(3)$ and normalized area measure on ${\mathbb{S}}^{2}$, as is relevant in
section 7 below. However, to ensure that
${\mathbb{E}}W_{2}(L_{N},\mu)\rightarrow 0$ as $N\rightarrow\infty$, it is
convenient to reduce to one-dimensional distributions, where we use the
following integral formula. For distributions $\mu$ and $\nu$ on
${\mathbb{R}}$ with cumulative distribution functions $F$ and $G$, we write
$W_{p}(\mu,\nu)=W_{p}(F,G)$.

######  Proposition 4.4.

Let $\xi_{1}$ be a real random variable with finite fourth moment, and
cumulative distribution function $F$. Let $\xi_{1},\dots,\xi_{N}$ be mutually
independent copies of $\xi_{1}$ giving an empirical measure
$L^{\xi}_{N}=N^{-1}\sum_{j=1}^{N}\delta_{\xi_{j}}$ with cumulative
distribution function $F^{\xi}_{N}(t).$

Then

$\int_{{\mathbb{R}}^{N}}W_{1}(F^{\xi}_{N},F)\mu^{\otimes
N}(d\xi)\leq{\frac{1}{\sqrt{N}}}\int_{-\infty}^{\infty}\sqrt{F(t)(1-F(t))}dt.$
(4.23)

###### Proof.

Let $H$ be Heaviside’s unit step function; then
$F^{\xi}_{N}(t)=N^{-1}\sum_{j=1}^{N}H(t-\xi_{j})$, so
$\sqrt{N}(F_{N}(t)-F(t))$ is a sum of mutually independent and bounded random
variables with mean zero. Also, as in the weak law of large numbers, we have

$\displaystyle{\mathbb{E}}W_{1}(F_{N},F)$
$\displaystyle=\int_{-\infty}^{\infty}{\mathbb{E}}|F_{N}(t)-F(t)|\,dt$
$\displaystyle\leq\int_{-\infty}^{\infty}\Bigl{(}{\mathbb{E}}\bigl{(}(F_{N}(t)-F(t))^{2}\bigr{)}\Bigr{)}^{1/2}\,dt$
$\displaystyle={\frac{1}{\sqrt{N}}}\int_{-\infty}^{\infty}\sqrt{F(t)(1-F(t))}\,dt,$
(4.24)

where the integral is finite by Chebyshev’s inequality since
${\mathbb{E}}\xi^{4}$ is finite. Compare [BGV].

∎

######  Proposition 4.5.

Suppose that $\xi_{1}$ has distribution $\mu$ on ${\mathbb{S}}^{2}$ where
$\mu$ is absolutely continuous with respect to the normalized area measure
$\nu_{1}$, and $d\mu=fd\nu_{1}$ where $f$ is bounded with $\|f\|_{\infty}\leq
M$. Let $\xi_{j}$ be mutually independent copies of $\xi_{1}$, and let
$L_{N}^{\xi}$ be the empirical measure from $N$ samples. Then

$\int_{({\mathbb{S}}^{2})^{N}}W_{1}(L_{N}^{\xi},\mu)d\mu^{\otimes
N}=O(N^{-1/4})\qquad(N\rightarrow\infty).$

###### Proof.

Let $g:{\mathbb{S}}^{2}\rightarrow{\mathbb{R}}$ be $1$-Lipschitz, and suppose
without loss of generality that $g$ has
$\int_{{\mathbb{S}}^{2}}g(x)\nu_{1}(dx)=0$; then $g$ is bounded with
$\|g\|_{\infty}\leq\pi$. Given $\delta>0$, by considering squares for
coordinates in longitude and colatitude, we choose disjoint and connected
subsets $E_{\ell}$ with diameter ${\hbox{diam}}(E_{\ell})\leq\delta$ and
$\nu_{1}(E_{\ell})\leq\delta^{2}$ and $\mu(E_{\ell})\leq M\nu_{1}(E)$ such
that $\cup_{\ell}E_{\ell}={\mathbb{S}}^{2}$. We can arrange that there are
$S_{\delta}$ such sets $E_{\ell}$, where $S_{\delta}\leq C/\delta^{2}$. Let
${\mathcal{F}}$ be the $\sigma$-algebra that is generated by the $E_{\ell}$,
take conditional expectations in $L^{2}(\nu_{1})$, and observe that

$\displaystyle\int_{{\mathbb{S}}^{2}}g(x)dL_{N}^{\xi}-\int_{{\mathbb{S}}^{2}}g(x)d\mu(x)$
$\displaystyle=\int_{{\mathbb{S}}^{2}}\bigl{(}g(x)-{\mathbb{E}}g\mid{\mathcal{F}})\bigr{)}dL_{N}^{\xi}+\int_{{\mathbb{S}}^{2}}\bigl{(}{\mathbb{E}}(g\mid{\mathcal{F}})-g(x)\bigr{)}d\mu(x)$
$\displaystyle\quad+\int_{{\mathbb{S}}^{2}}{\mathbb{E}}(g\mid{\mathcal{F}})(x)\bigl{(}dL_{N}^{\xi}(x)-d\mu(x)\bigr{)}$
(4.25)

where we have bounds

$\displaystyle\Bigl{|}\int_{{\mathbb{S}}^{2}}\bigl{(}g(x)-{\mathbb{E}}(g\mid{\mathcal{F}})\bigr{)}dL_{N}^{\xi}\Bigr{|}$
$\displaystyle\leq\sup_{\ell}{\hbox{Lip}}(g){\hbox{diam}}(E_{\ell})\leq\delta,$
(4.26)
$\displaystyle\Bigl{|}\int_{{\mathbb{S}}^{2}}\bigl{(}g(x)-{\mathbb{E}}(g\mid{\mathcal{F}})\bigr{)}d\mu|$
$\displaystyle\leq\sup_{\ell}{\hbox{Lip}}(g){\hbox{diam}}(E_{\ell})\leq\delta,$
(4.27)

and the identity

$\int_{{\mathbb{S}}^{2}}{\mathbb{E}}(g\mid{\mathcal{F}})(x)\bigl{(}dL_{N}^{\xi}(x)-d\mu(x)\bigr{)}=\sum_{\ell}{\frac{\int_{E_{\ell}}g(x)d\nu_{1}(x)}{\nu_{1}(E_{\ell})}}\int_{E_{\ell}}(dL_{N}^{\xi}-d\mu),$

so by Cauchy–Schwarz, we have

$\displaystyle\int_{({\mathbb{S}}^{2})^{N}}$
$\displaystyle\Bigl{(}\int_{{\mathbb{S}}^{2}}{\mathbb{E}}(g\mid{\mathcal{F}})(x)\bigl{(}dL_{N}^{\xi}(x)-d\mu(x)\bigr{)}\Bigr{)}^{2}d\mu^{\otimes
N}$
$\displaystyle\leq\sum_{\ell}\nu_{1}(E_{\ell})\|g\|_{\infty}^{2}\sum_{\ell}{\frac{1}{\nu_{1}(E_{\ell})}}\int_{({\mathbb{S}}^{2})^{N}}(L_{N}^{\xi}(E_{\ell})-\mu(E_{\ell}))^{2}d\mu^{\otimes
N}$ (4.28)

where
$L_{N}^{\xi}(E_{\ell})-\mu(E_{\ell})=N^{-1}\sum_{j=1}^{N}({\mathbb{I}}_{E_{\ell}}(\xi_{j})-\mu(E_{\ell}))$
is a sum of independent random variables with mean zero and variance
$N^{-1}\mu(E_{\ell})(1-\mu(E_{\ell}))$, so

$\displaystyle\int_{({\mathbb{S}}^{2})^{N}}\Bigl{(}\int_{{\mathbb{S}}^{2}}{\mathbb{E}}(g\mid{\mathcal{F}})(x)\bigl{(}dL_{N}^{\xi}(x)-d\mu(x)\bigr{)}\Bigr{)}^{2}d\mu^{\otimes
N}\leq{\frac{1}{N}}\|g\|_{\infty}^{2}\sum_{\ell}M\leq{\frac{\pi^{2}CM}{\delta^{2}N}}.$
(4.29)

Choosing $\delta=N^{-1/4}$ we make both (4.26) and (4.29) small, which gives
the stated result.

∎

Remark Consider the discrete metric $\delta$ on $[0,1]$, and observe that
${\mathbb{I}}_{A}$ gives a $1$-Lipschitz function on $[0,1]$ for all open
$A\subseteq[0,1]$. Then we have
$\int{\mathbb{I}}_{A}(x)(d\mu(x)-d\nu(x))=\mu(A)-\nu(A)$, so by maximizing
over $A$ we obtain the total variation norm $\|\mu-\nu\|_{var}$. With $\mu$ a
continuous measure and $\nu$ a purely discrete measure, such as an empirical
measure, we have $\|\mu-\nu\|_{var}=1$. The Propositions 4.4 and 4.5 depend
upon the choice of cost function as well as the measures.

The Gibbs measure (1.5) was defined using random Fourier series. This
construction gives us a sequence of finite-dimensional probability spaces
which approximate the space $(B_{K},L^{2},\mu_{K,\beta})$. To make this idea
precise, we recall some definitions from [S].

######  Definition 4.6.

(Convergence of metric measure spaces) (i) For $M$ a nonempty set, a
pseudometric is a function $\delta:M\rightarrow[0,\infty]$ such that

$\delta(x,y)=\delta(y,x),\quad\delta(x,x)=0,\quad\delta(x,z)\leq\delta(x,y)+\delta(y,z)\qquad(x,y,z\in
M);$ (4.30)

then $(M,\delta)$ is a pseudometric space.

(ii) Given pseudo metric spaces $(M_{1},\delta_{1})$ and $(M,\delta_{2})$, a
coupling is a pseudo metric $\delta:M\rightarrow[0,\infty]$ where
$M=M_{1}\sqcup M_{2}$ such that $\delta\mid M_{1}\times M_{1}=\delta_{1}$ and
$\delta\mid M_{2}\times M_{2}=\delta_{2}$.

(iii) Suppose that $\hat{M}_{1}=(M_{1},\delta_{1},\mu_{1})$ and
$\hat{M}_{2}=(M_{2},\delta_{2},\mu_{2})$ are complete separable metric spaces
endowed with probability measures. Consider a coupling $(M,\delta)$ and a
probability measure $\pi$ on $M_{1}\times M_{2}$ with marginals
$\pi_{1}=\mu_{1}$ and $\pi_{2}=\mu_{2}$. Then the $L^{2}$ distance between
$\hat{M}_{1}$ and $\hat{M}_{2}$ is

${\mathcal{D}}_{L^{2}}(\hat{M}_{1},\hat{M}_{2})=\inf_{\delta,\pi}\Bigl{(}\int_{M\times
M}\delta(x,y)^{2}\pi(dxdy)\Bigr{)}^{1/2}$ (4.31)

Let $D_{n}$ be the Dirichlet projection taking
$\sum_{k=-\infty}^{\infty}(a_{k}+ib_{k})e^{ikx}$ to
$\sum_{k=-n}^{n}(a_{k}+ib_{k})e^{ikx}$. Following [B02], we truncate the
random Fourier series of
$u=P+iQ=\sum_{k=-\infty}^{\infty}(a_{k}+ib_{k})e^{ikx}$ to
$u_{n}=P_{n}+iQ_{n}=\sum_{k=-n}^{n}(a_{k}+ib_{k})e^{ikx}$ and correspondingly
modify the Hamiltonian to

$H_{3}^{(n)}((a_{k}),(b_{k}))={\frac{1}{2}}\sum_{k=-n}^{n}k^{2}(a_{k}^{2}+b_{k}^{2})+{\frac{\beta}{4}}\int\Bigl{|}\sum_{k=-n}^{n}(a_{k}+ib_{k})e^{ik\theta}\Bigr{|}^{4}{\frac{d\theta}{2\pi}}$
(4.32)

for the real canonical variables $((a_{k},b_{k}))_{k=-n}^{n}$. Then the
canonical equations become a coupled system of ordinary differential
differential equations in the Fourier coefficients. We introduce the polar
decomposition $P_{n}+iQ_{n}=\kappa_{n}e^{i\sigma_{n}}$, and observe that in
terms of these noncanonical variables, the Hamiltonians
$H_{1}^{(n)}=\int_{\mathbb{T}}\kappa_{n}^{2}d\theta$ and

$H_{3}^{(n)}={\frac{1}{2}}\int_{\mathbb{T}}\Bigl{(}\Bigl{(}{\frac{\partial\kappa_{n}}{\partial\theta}}\Bigr{)}^{2}+\kappa_{n}^{2}\Bigl{(}{\frac{\partial\sigma_{n}}{\partial\theta}}\Bigr{)}^{2}\Bigr{)}{\frac{d\theta}{2\pi}}+{\frac{\beta}{4}}\int_{\mathbb{T}}\kappa_{n}^{4}{\frac{d\theta}{2\pi}}$
(4.33)

are invariants under the flow. The corresponding Gibbs measure is

$d\mu_{K,\beta}^{(n)}=Z(K,\beta,n)^{-1}{\bf
I}_{B_{K}}(u)\exp\Bigl{(}{\frac{-\beta}{4}}\int_{\mathbb{T}}|u(\theta)|^{4}{\frac{d\theta}{2\pi}}\Bigr{)}\prod_{\theta\in[0,2\pi]}du(\theta)$
(4.34)

in which

$\prod_{\theta\in[0,2\pi]}du(\theta)=\prod_{j=-n;j\neq
0}^{n}\exp\Bigl{(}-{\frac{j^{2}}{2}}(a_{j}^{2}+b_{j}^{2})\Bigr{)}{\frac{j^{2}da_{j}db_{j}}{2\pi}}.$
(4.35)

Consider the map $u(x,t)\mapsto u(x+h,t)$ of translation in the space
variable. This commutes with $D_{n}$, and the Gibbs measures
$\mu_{K,\beta}^{(n)}$ are all invariant under this translation. In terms of
Fourier components, we have $M_{\infty}=B_{K}$ and

$M_{n}=\Bigl{\\{}(a_{j},b_{j})_{j=-n}^{n}:a_{j},b_{j}\in{\mathbb{R}}:\sum_{j=-n}^{n}(a_{j}^{2}+b_{j}^{2})\leq
K\Bigr{\\}}$ (4.36)

with the canonical inclusions of metric spaces
$(M_{1},\ell^{2})\subset(M_{2},\ell^{2})\subset\dots\subset(M_{\infty},\ell^{2})$
defined by adding zeros at the start and end of the sequences, which gives a
sequence of isometric embeddings for the $\ell^{2}$ metric on sequences. When
we identify $(a_{j},b_{j})_{j=-n}^{n}$ with
$\sum_{j=-n}^{n}(a_{j}+ib_{j})e^{ijx}$, then we have a corresponding embedding
for the $L^{2}$ metric.

Here $(M_{n},L^{2},\mu_{K,\beta}^{(n)})$ is a finite-dimensional manifold and
a metric probability space. We now show that these spaces converge to
$(M_{\infty},L^{2},\mu_{K,\beta})$ as $n\rightarrow\infty$.

######  Lemma 4.7.

(i) Suppose that $0<-\beta K<3/(14\pi^{2})$. Then
$\hat{M}_{n}=(M_{n},{L^{2}},\mu_{K,\beta}^{(n)})$ has

${\mathcal{D}}_{L^{2}}(\hat{M}_{n},\hat{M}_{\infty})\rightarrow
0\qquad(n\rightarrow\infty).$ (4.37)

(ii) The measures $\mu_{K,\beta}^{(n)}$ converge in total variation norm to
$\mu_{K,\beta}$ as $n\rightarrow\infty$.

###### Proof.

(i) This is proved in Theorem 3.2 of [B2]; see also Example 3.8 of [S]. Let
$W_{2}(D_{n}\sharp\mu,\mu)$ be the Wasserstein transportation distance between
$D_{n}\sharp\mu$ and free Brownian loop measure $\mu$ for the cost function
$\|u-v\|^{2}_{L^{2}}$, where $D_{n}$ is the Dirichlet projection. The key
point is

$\displaystyle W_{2}(D_{n}\sharp\mu,\mu)^{2}$
$\displaystyle\leq\int\|D_{n}u-u\|^{2}_{L^{2}}\mu(du)$
$\displaystyle={\mathbb{E}}\sum_{k:|k|>n}{\frac{|\gamma_{k}|^{2}}{k^{2}}}=O\Bigl{(}{\frac{1}{n}}\Bigr{)}\qquad(n\rightarrow\infty).$
(4.38)

(ii) The measures $\mu_{K,\beta}^{(n)}$ converge in total variation norm to
$\mu_{K}$, by an observation of McKean in step 7 of [M]. By M. Riesz’s
theorem, there exists $c_{4}>0$ such that
$\int_{\mathbb{T}}|D_{n}u|^{4}d\theta\leq
c_{4}\int_{\mathbb{T}}|u|^{4}d\theta$, and by [LRS] the integral

$\int_{B_{K}}\exp\Bigl{(}\lambda
c_{4}\int_{\mathbb{T}}|u(\theta)|^{4}d\theta\Bigr{)}W(du)$ (4.39)

is finite, so we can use the integrand as a dominating function to show

$\int_{B_{K}}\Bigl{|}\exp\Bigl{(}\lambda\int_{\mathbb{T}}|D_{n}u(\theta)|^{4}d\theta\Bigr{)}-\exp\Bigl{(}\lambda\int_{\mathbb{T}}|u(\theta)|^{4}d\theta\Bigr{)}\Bigr{|}W(du)\rightarrow
0\qquad(n\rightarrow\infty).$ (4.40)

∎

######  Proposition 4.8.

Let $(M_{n}\sqcup M_{\infty},\delta_{n})$ be a coupling of $(M_{n},L^{2})$ and
$M_{\infty},L^{2})$, and let $\varphi:(M_{n}\sqcup
M_{\infty},\delta_{n})\rightarrow{\mathbb{R}}$ be a Lipschitz function. Then

$\int_{M_{n}}\varphi(u_{n})\mu_{K,\beta}^{(n)}(du_{n})\rightarrow\int_{M_{\infty}}\varphi(u)\mu_{K,\beta}(du)\qquad(n\rightarrow\infty).$
(4.41)

###### Proof.

We can introduce a pseudo metric $\delta_{n}$ on $M_{n}\cup M_{\infty}$ that
restricts to the $L^{2}$ metric on $M_{n}$ and $M_{\infty}$, and apply (4) to
Lipschitz functions $\varphi:(M_{n}\sqcup
M_{\infty},\delta_{n})\rightarrow{\mathbb{R}}$. We can regard $M_{n}\times
M_{\infty}$ as a subset of $M\times M=(M_{n}\sqcup
M_{\infty})\times(M_{n}\sqcup M_{\infty})$. Note that for a Lipschitz function
$\varphi:M\rightarrow{\mathbb{R}}$ such that
$|\varphi(x)-\varphi(y)|\leq\delta(x,y)$ for all $x,y\in M$, we have

$\displaystyle\int_{M_{n}}\varphi(u_{n})\mu_{K,\beta}^{(n)}(du_{n})-\int_{M_{\infty}}\varphi(u)\mu_{K,\beta}(du)$
$\displaystyle=\int\\!\\!\\!\int_{M_{n}\times
M_{\infty}}(\varphi(u_{n})-\varphi(u))\pi(dudu_{n})$
$\displaystyle\leq\int\\!\\!\\!\int_{M_{n}\times
M_{\infty}}\delta(u_{n},u)\pi(du_{n}du)$
$\displaystyle\leq\Bigl{(}\int\\!\\!\\!\int_{M_{n}\times
M_{\infty}}\delta(u_{n},u)^{2}\pi(du_{n}du)\Bigr{)}^{1/2}$
$\displaystyle={\mathcal{D}}_{L^{2}}(\hat{M}_{n},\hat{M}_{\infty}).$ (4.42)

∎

For example, with $u=\sum_{n=-\infty}^{\infty}(a_{k}+ib_{k})e^{ik\theta}$ we
introduce $D_{n}u=\sum_{k=-n}^{n}(a_{k}+ib_{k})e^{ik\theta}$; then
$\varphi(u)=\|D_{n}u\|_{L^{2}}$ and $\psi(u)=\|u-D_{n}u\|_{L^{2}}$ give
Lipschitz functions $\varphi,\psi:(B_{K},L^{2})\rightarrow{\mathbb{R}}$.

######  Proposition 4.9.

For $0<\gamma<1/16$ and fixed $0<t<t_{0}$, the map $x\mapsto u(x,t)\in L^{4}$
is $\gamma$\- Hölder continuous.

###### Proof.

We prove that for $0<t<t_{0}$, we have $C=C(t_{0})$ such that

$\int_{B_{K}}\bigl{\|}u(x+h,t)-u(x,t)\bigr{\|}^{16}_{L^{4}_{x}}\mu_{K,\beta}(du)\leq
Ch^{2}$ (4.43)

so $x\mapsto u(x,t)\in L^{4}_{x}$ is $\gamma$\- Hölder continuous for
$0<\gamma<1/16$ by the Censov–Kolmogorov’s theorem. To obtain (4.43), let
$J_{3/8}(x)=\sum^{\prime}e^{ikx}/|k|^{3/8}$ so that $J_{3/8}(x)|x|^{5/8}$ is
bounded on $(-\pi,\pi)$ and $J_{3/8}\in L^{4/3}(-\pi,\pi)$. Then by Young’s
inequality for convolutions, with $|D|:e^{inx}\mapsto|n|e^{inx}$ we have

$\bigl{\|}u(x+h,t)-u(x,t)\bigr{\|}_{L^{4}_{x}}\leq\bigl{\|}J_{3/8}\|_{L^{4/3}}\bigl{\|}|D|^{3/8}u(x+h,t)-|D|^{3/8}u(x,t)\bigr{\|}_{L^{2}_{x}}$
(4.44)

Then by Bourgain’s estimate on the solutions of NLS [B], there exists
$C(t_{0})$ such that

$\bigl{\|}u(x+h,t)-u(x,t)\bigr{\|}_{H^{3/8}_{x}}\leq
C(t_{0})\bigl{\|}u(x+h,t)-u(x,t)\bigr{\|}_{H^{3/8}_{x}}$ (4.45)

where

$\displaystyle\int_{B_{K}}$
$\displaystyle\bigl{\|}u(x+h,t)-u(x,t)\bigr{\|}^{16}_{H^{3/8}_{x}}\mu_{K,\beta}(du)$
$\displaystyle\leq\Bigl{(}\int_{B_{K}}\bigl{\|}u(x+h,t)-u(x,t)\bigr{\|}^{32}_{H^{3/8}_{x}}W(du)\Bigr{)}^{1/2}\Bigl{(}\int_{B_{K}}\Bigl{(}{\frac{d\mu_{K,\beta}}{dW}}\Bigr{)}^{2}W(du)\Bigr{)}^{1/2}.$
(4.46)

By basic results about Gaussian series, the first factor on the right-hand
side is bounded by the eighth power of

$\sum_{k=1}^{\infty}{\frac{k^{3/4}(1-\cos hk)}{k^{2}}}\leq Ch^{1/4},$ (4.47)

so we obtain (4.43). Also, by rotation invariance of the Gibbs measure, we
have

$\int_{B_{K}}\bigl{|}u(\theta+h,t)-u(\theta,t)\bigr{|}^{4}\mu_{K,\beta}(du)=\int_{B_{K}}\bigl{\|}u(x+h,t)-u(x,t)\bigr{\|}^{4}_{L^{4}_{x}}\mu_{K,\beta}(du),$

which is

$\displaystyle\leq\Bigl{(}\int_{B_{K}}\bigl{\|}u(x+h,0)-u(x,0)\bigr{\|}_{L^{4}_{x}}^{8}W(du)\Bigr{)}^{1/2}\Bigl{(}\int_{B_{K}}\Bigl{(}{\frac{d\mu}{dW}}\Bigr{)}^{2}W(du)\Bigr{)}^{1/2}$
$\displaystyle\leq C\Bigl{(}\sum_{k=1}^{\infty}{\frac{1-\cos
hk}{k^{2}}}\Bigr{)}^{2}\leq Ch^{2},$

so $x\mapsto u(x,t)$ is $1/4$-Hölder continuous along solutions in the support
of the Gibbs measure. ∎

##  5\. Hasimoto transform

We recall the Hasimoto [H] transform, which associates with a solution $u\in
C^{2}$ of (1.3) a space curve in ${\mathbb{R}}^{3}$ with moving frame
$\\{T,N,B\\}$; Hasimoto considered the case $\beta=-1/2$. In the present
context, $u$ is associated with the space derivative of a tangent vector $T$
to a unit speed space curve, so the curvature is $\kappa=\|{\frac{\partial
T}{\partial x}}\|$. We have a polar decomposition $u=\kappa e^{i\sigma}$ where
$\sigma(x,t)=\int_{0}^{x}\tau(y,t)dy$ and $\tau$ is the torsion. Then the
Serret–Frenet formula is

${\frac{\partial}{\partial x}}\begin{bmatrix}T\\\ N\\\
B\end{bmatrix}=\begin{bmatrix}0&\kappa&0\\\ -\kappa&0&\tau\\\
0&-\tau&0\end{bmatrix}\begin{bmatrix}T\\\ N\\\ B\end{bmatrix},$ (5.1)

so the frame develops along the space curve. Let $X=[T;N;B]\in SO(3)$, and
$\Omega_{1}(x,t)$ the matrix in (5.1). The space

${\mathcal{L}}(SO(3))=\\{g:[0,2\pi]\rightarrow
SO(3);g\quad{\hbox{continuous}},\quad g(0)=g(2\pi)\\}$

with pointwise multiplication is a loop group, and its Lie algebra may be
regarded as

$\displaystyle H_{0}^{1}(so(3))=\Bigl{\\{}h;[0,2\pi]\rightarrow so(3);$
$\displaystyle h\quad{\hbox{absolutely continuous}},\quad h(0)=h(2\pi)=0,$
$\displaystyle\quad\int_{0}^{2\pi}\|h^{\prime}(x)\|^{2}_{so(3)}dx<\infty\Bigr{\\}};$

see [DL]. When $\Omega_{1}(\cdot,t)\in C({\mathbb{T}};so(3))$, the solution
$X(\cdot,t)\in C([0,2\pi];SO(3))$ to (5.1) is $2\pi$ periodic up to a
multiplicative monodromy factor $U(t)\in SO(3)$ such that
$X(x+2\pi,t)=X(x,t)U(t)$.

The frame also evolves with respect to time, so that with
$\mu=-{\frac{\partial\sigma}{\partial t}}-\beta\kappa^{2}$, we have

${\frac{\partial}{\partial t}}\begin{bmatrix}T\\\ N\\\
B\end{bmatrix}=\begin{bmatrix}0&-\tau\kappa&{\frac{\partial\kappa}{\partial
x}}\\\ \tau\kappa&0&-\mu\\\ -{\frac{\partial\kappa}{\partial
x}}&\mu&0\end{bmatrix}\begin{bmatrix}T\\\ N\\\ B\end{bmatrix}.$ (5.2)

######  Lemma 5.1.

(Hasimoto) If $u$ is a $C^{2}$ function that satisfies the nonlinear
Schrödinger equation, then the coupled pair of differential equations is
consistent, giving a Lax pair.

Thus the frame $X\in SO(3)$ evolves along the solution $P+iQ\in B_{K}$ of NLS,
and we can regard $d/dx-\Omega_{1}$ and $d/dt-\Omega_{2}$ as connections for
this evolution. Both of the coefficient matrices are real and skew symmetric.
One can check that a solution of the integral equation

$X(x,t)=X_{0}(x)+t\Omega_{2}(0,0)X_{0}(0)+\int_{0}^{x}\int_{0}^{t}\Bigl{(}{\frac{\partial\Omega_{1}(y,s)}{\partial
t}}+\Omega_{1}(y,s)\Omega_{2}(y,s)\Bigr{)}X(y,s)dsdy$ (5.3)

satisfies

$X(x,0)=X_{0}(x),\qquad{\frac{\partial X(x,0)}{\partial
t}}=\Omega_{2}(x,0)X_{0}(x),$ ${\frac{\partial^{2}X(x,t)}{\partial x\partial
t}}=\Bigl{(}{\frac{\partial\Omega_{1}(x,t)}{\partial
t}}+\Omega_{1}(x,t)\Omega_{2}(x,t)\Bigr{)}X(x,t),$

so smooth solutions are given in terms of an integral equation.

From the Serret–Frenet formulas the components of the acceleration along the
space curve satisfy

$\displaystyle\Bigl{\|}T\times{\frac{\partial^{2}T}{\partial
x^{2}}}\Bigr{\|}^{2}$ $\displaystyle=\Bigl{(}{\frac{\partial\kappa}{\partial
x}}\Bigr{)}^{2}+\kappa^{2}\tau^{2}=\Bigl{(}{\frac{\partial Q}{\partial
x}}\Bigr{)}^{2}+\Bigl{(}{\frac{\partial P}{\partial x}}\Bigr{)}^{2},$
$\displaystyle\Bigl{(}T\cdot{\frac{\partial^{2}T}{\partial
x^{2}}}\Bigr{)}^{2}$
$\displaystyle=\kappa^{4}=\bigl{(}P^{2}+Q^{2}\bigr{)}^{2}.$ (5.4)

The total curvature of the space curve is

$\int_{\mathbb{T}}\kappa(x)^{2}dx=\int_{\mathbb{T}}(P^{2}+Q^{2})dx=H_{1}(P,Q),$
(5.5)

which is an invariant under the flow associated with the NLS.

######  Proposition 5.2.

Let

$H_{2}(P,Q)=-\int_{\mathbb{T}}P(x)Q^{\prime}(x)dx.$ (5.6)

(i) Then $-H_{2}$ is convergent almost surely and is invariant under the flow
associated with $NLS$,

(ii) $-H_{2}$ represents the area that is enclosed by the contour
$\\{u(x):x\in[0,2\pi]\\}$ in the complex plane, and

$\qquad H_{2}={\frac{1}{2}}\int_{\mathbb{T}}\kappa^{2}\tau dx;$ (5.7)

(iii) $H_{2}^{2}\leq 4^{-1}H_{1}H_{3}$ for $\beta>0.$

###### Proof.

(i) The invariance of $H_{2}$ was noted in [MV1] and can be proved by
differentiating through the integral sign and using the canonical equations.
We have a series

$\int_{\mathbb{T}}\bar{u}(\theta,t){\frac{\partial
u}{\partial\theta}}(\theta,t){\frac{d\theta}{2\pi}}=\lim_{N\rightarrow\infty}\sum_{j=-N}^{N}\overline{\hat{u}(j)}ij\hat{u}(j)$

which converges almost surely. This follows since

$\displaystyle\int_{B_{K}}\sup_{N}\Bigl{|}\sum_{j=-N}^{N}$
$\displaystyle\overline{\hat{u}(j)}ij\hat{u}(j)\Bigr{|}^{p}\mu_{K,\beta}(du)$
$\displaystyle\leq\Bigl{(}\int_{B_{K}}\Bigl{(}{\frac{d\mu_{K,\beta}}{dW}}\Bigr{)}^{2}dW\Bigr{)}^{1/2}\Bigl{(}\int_{B_{K}}\sup_{N}\Bigl{|}\sum_{j=-N}^{N}\overline{\hat{u}(j)}ij\hat{u}(j)\Bigr{|}^{2p}W(du)\Bigr{)}^{1/2},$
(5.8)

where the final integral involves the series

$\displaystyle\lim_{N\rightarrow\infty}\sum_{j=-N}^{N}\overline{\hat{u}(j)}ij\hat{u}(j)=\sum_{j=1}^{\infty}{\frac{|z_{j}|^{2}-|z_{-j}|^{2}}{j}}$
(5.9)

which is a martingale; by Fatou’s Lemma, we have

$\displaystyle\int_{L^{2}}\exp\Bigl{(}\lambda\sum_{j=1}^{\infty}{\frac{|z_{j}|^{2}-|z_{-j}|^{2}}{j}}\Bigr{)}dW$
$\displaystyle=\prod_{j=1}^{\infty}\int_{L^{2}}\exp\Bigl{(}{\frac{\lambda(|z_{j}|^{2}-|z_{-j}|^{2})}{j}}\Bigr{)}dW$
$\displaystyle=\Bigl{(}{\frac{2\pi\lambda}{\sin
2\pi\lambda}}\Bigr{)}^{1/2}\qquad(-1/2<\lambda<1/2),$ (5.10)

so the series in (5.9) is marginally exponentially integrable. Hence the
integrals in (5) converge by the $L^{p}$ martingale maximal theorem for all
$1<p<\infty$.

(ii) One can write $H_{2}$ in terms of $P+iQ=\kappa e^{i\sigma}$, and make a
change of variables to obtain

$\kappa={\frac{\partial(P,Q)}{\partial(\kappa,\sigma)}}$

and

$\displaystyle H_{2}$
$\displaystyle={\frac{1}{2}}\int_{\mathbb{T}}\bigl{(}P^{\prime}Q-PQ^{\prime})dx={\frac{1}{2}}\int_{\mathbb{T}}\kappa^{2}\tau
dx.$

To interpret this as an area, We write $\theta\in[0,2\pi]$ for the space
variable and extend functions on $[0,2\pi]$ to harmonic functions on the unit
disc via the Poisson kernel. Then by Green’s theorem, we can express this
invariant in terms of the area of the image of ${\mathbb{D}}$ under the map to
$P+iQ$, as in

$\displaystyle-
H_{2}=\int\\!\\!\\!\int_{\mathbb{D}}{\frac{\partial(P,Q)}{\partial(x,y)}}dxdy.$
(5.11)

This is similar to Lévy’s stochastic area, as discussed in Example 5.1 of
[IW].

(iii) We then have

$\Bigl{(}\int_{\mathbb{T}}\kappa^{2}\tau
dx\Bigr{)}^{2}\leq\int_{\mathbb{T}}\kappa^{2}dx\int_{\mathbb{T}}\kappa^{2}\tau^{2}dx$

which is bounded in terms of other invariants, with $H_{2}^{2}\leq
4^{-1}H_{1}H_{3}$.

∎

(i) Bourgain interprets $H_{2}$ in terms of momentum (5.70) [B02].

(ii) Suppose that $T\in C^{2}([0,a]\times[0,b];{\mathbb{S}}^{2})$, so that
$T(x,t)$ represents the spin of the particle at $(x,t)$ and let

$E(T)=\int_{0}^{a}\Bigl{\|}{\frac{\partial T}{\partial
x}}(x,t)\Bigr{\|}^{2}dx,$ (5.12)

which corresponds to our 5.5. One can consider infinitesimal variations
$T\mapsto T+T\times V$ and thereby compute ${\frac{\partial E}{\partial T}}$.
In the focusing case $\beta=-1$, Ding [D] introduces a symplectic structure on
the space of such maps such that the Hamiltonian flow is

${\frac{\partial T}{\partial t}}=T\times{\frac{\partial^{2}T}{\partial
x^{2}}}$ (5.13)

which corresponds to Heisenberg’s equation for the one-dimensional ferro-
magnet, and gives the top entry of (5.2). There is a a gauge equivalence
between the focussing NLS and Heisenberg’s ferromagnet. There is also a gauge
equivalence between the defocussing NLS and a hyperbolic version of the
ferromagnet in which the standard cross product is modified. We have

$\displaystyle\Bigl{|}{\frac{\partial^{2}T}{\partial x^{2}}}\Bigr{|}^{2}$
$\displaystyle=\Bigl{(}{\frac{\partial\kappa}{\partial
x}}\Bigr{)}^{2}+\kappa^{2}\tau^{2}+\kappa^{4}$ (5.14)
$\displaystyle=\bigl{|}{\frac{\partial u}{\partial x}}\bigr{|}^{2}+|u|^{4}.$
(5.15)

The aim of the next section is to interpret the Lax pair suitably for
solutions which are typically not differentiable and for which we have a pair
of stochastic differential equations with random matrix coefficients.

##  6\. Gibbs measure transported to the frames

The compact Lie group $SO(3)$ of real orthogonal matrices with determinant one
is a subset of $M_{3\times 3}({\mathbb{R}})$, which has the scalar product
$\langle X,Y\rangle=\mathop{\mathrm{trace}}(XY^{\dagger})$ and associated
metric $d(X,Y)=\langle X-Y,X-Y\rangle^{1/2}$ such that $\langle
XU,YU\rangle=\langle X,Y\rangle$ and $d(XU,YU)=d(X,Y)$ for all $U\in SO(3)$
and $X,Y\in M_{3\times 3}({\mathbb{R}})$. The Lie group $SO(3)$ has tangent
space at the identity element give by the skew symmetric matrices $so(3)$, so
the tangent space $T_{X}SO(3)$ at $X\in SO(3)$ consists of $\\{\Omega
X:\Omega\in so(3)\\}$, where $so(3)$ is a Lie algebra for $[x,y]=xy-yx$,
$x,y\in so(3)$.

With $M_{n}$ as in (4.36), the space $C^{\infty}(M_{n};{\mathbb{R}})$ is a
Poisson algebra for the bracket
$\\{f,g\\}=\sum_{j=-n}^{n}{\frac{\partial(f,g)}{\partial(a_{j},b_{j})}}$, and
the canonical equations arise with Hamiltonian $H_{3}^{(n)}$ on $M_{n}$. Let
$Q$ be the ring of quaternions, and extend the Poisson bracket to
$C^{\infty}(M_{n};Q)$ via $\\{f\otimes X,g\otimes Y\\}=\\{f,g\\}\otimes XY$.
Then $({\mathbb{R}}^{3},\times)$ may be realised as $Q/{\mathbb{R}}I$ and
$(so(3),[\cdot,\cdot])\cong({\mathbb{R}}^{3},\times)$; see Example 2.3 of [X].

We consider the differential equation

${\frac{dX}{dt}}=\Omega(t)X;\quad X(0)=X_{0}$ (6.1)

where $t\in[0,1]$ is the evolving time. Let $\nu_{t}$ for $t\in[0,1]$ be a
weakly continuous family of probability measures on $SO(3)$, such that

$\int_{0}^{1}\int_{SO(3)}\|\Omega\|^{2}_{so(3)}\nu_{t}(dX)dt<\infty$ (6.2)

converges and $\Omega X$ is locally bounded so $X\mapsto\Omega X$ is locally
Lipschitz, and such that the family $(\nu_{t})$ satisfies the weak continuity
equation

${\frac{\partial\nu_{t}}{\partial t}}+\nabla\cdot\Bigl{(}\Omega
X\nu_{t}\Bigr{)}=0.$ (6.3)

The weak continuity equation is equivalent to

$\int_{SO(3)}f(X)\nu_{t}(dX)=\int_{SO(3)}f(X_{t}(X_{0}))\nu_{0}(dX_{0})$ (6.4)

for all $f\in C(SO(3);{\mathbb{R}}),$ where $X_{0}\mapsto X_{t}(X_{0})$ gives
the dependence of the solution of (6.1) on the initial condition. It is not
asserted that the velocity field $\Omega X$ gives an optimal transportation
plan, but it does an upper bound on the Wasserstein transportation cost for
the cost $d(X,Y)^{2}$ on $SO(3)$ of

${\frac{W_{2}(\nu_{t_{2}},\nu_{t_{1}})^{2}}{t_{t}-t_{1}}}\leq\int_{t_{1}}^{t_{2}}\int_{so(3)}\|\Omega\|^{2}_{so(3)}\nu_{t}(dX)dt\qquad(0<t_{1}<t_{2}<1).$
(6.5)

Then by Theorem 23.9 of [V2], the path $(\nu_{t})$ of probability measures is
absolutely continuous, so there exists $\ell\in L^{1}[0,1]$ such that
$W_{2}(\nu_{t_{2}},\nu_{t_{1}})\leq\int_{t_{1}}^{t_{2}}\ell(t)dt$ and
$1/2$-Hölder continuous, so there exists $C>0$ such that
$W_{2}(\nu_{t_{2}},\nu_{t_{1}})\leq C|t_{2}-t_{1}|^{1/2}$.

######  Example 6.1.

(i) If $\Omega_{t}\in M_{3\times 3}({\mathbb{R}})$ is skew, and $X_{t},Y_{t}$
give solutions of the differential equation

${\frac{dX}{dt}}=\Omega_{t}X,X(0)=X_{0};\quad{\frac{dY}{dt}}=\Omega_{t}Y,Y(0)=Y_{0}$
(6.6)

then $d(X_{t},Y_{t})=d(X_{0},Y_{0})$. We deduce that if $X_{0}$ is distributed
according to Haar measure on $SO(3)$, then $X_{t}$ is also distributed
according to Haar measure since the measure, the metric and solutions are all
preserved via $X\mapsto XU$.

(ii) As an alternative, we can consider $X_{0}$ to have first column $[0;0;1]$
and observe the evolution of the first column $T$ of $X$ under the (6.1) where
$T$ evolves on ${\mathbb{S}}^{2}$.

We now consider the case in which $\Omega$ is a $so(3)$-valued random variable
over $(M_{\infty},\mu_{K,\beta},L^{2})$.

######  Proposition 6.2.

Suppose that $\Omega=\Omega(u(\cdot,t))$ where $u(x,t)$ is a solution of NLS
and that

$\int_{B_{K}}\|\Omega(u(\cdot,0))\|^{2}_{so(3)}\mu_{K,\beta}(du)$ (6.7)

converges. Then for almost all $u$ with respect to $\mu_{K,\beta}$, there
exists a flow $(\nu_{t}(dX;u))$ of probability measures on $SO(3)$.

###### Proof.

Each solution $u$ of NLS determines $\Omega$ so that the associated ODE (6.1)
transports the initial distribution of $X_{0}\in SO(3)$ to a probability
measure on $SO(3)$; then we average over the $u$ with respect to
$\mu_{K}(du)$. This Gibbs measure is invariant under the NLS flow, so by
Fubini’s theorem

$\int_{B_{K}}\int_{0}^{1}\int_{SO(3)}\|\Omega(u(\cdot,t))\|^{2}_{so(3)}\nu_{t}(dX)dt\mu_{K}(du)$
(6.8)

converges. Hence the condition (6.2) is satisfied, for almost all $u$, and we
can invoke Theorem 23.9 of [V2].∎

For the finite-dimensional $M_{n}$ of (4.36) and solutions
$u_{n}=\kappa_{n}e^{i\sigma_{n}}$, the modified Hasimoto differential
equations are

${\frac{\partial}{\partial x}}X^{(n)}(x,t)=\begin{bmatrix}0&\kappa_{n}&0\\\
-\kappa_{n}&0&\tau_{n}\\\ 0&-\tau_{n}&0\end{bmatrix}X^{(n)}(x,t),$ (6.9)

and

${\frac{\partial}{\partial
t}}X^{(n)}(x,t)=\begin{bmatrix}0&-\tau_{n}\kappa_{n}&{\frac{\partial\kappa_{n}}{\partial
x}}\\\ \tau_{n}\kappa_{n}&0&{\frac{\partial\sigma_{n}}{\partial
t}}-\beta\kappa_{n}^{2}\\\ -{\frac{\partial\kappa_{n}}{\partial
x}}&-{\frac{\partial\sigma_{n}}{\partial
t}}-\beta\kappa_{n}^{2}&0\end{bmatrix}X^{(n)}(x,t)$ (6.10)

involves $\tau_{n}={\frac{\partial\sigma_{n}}{\partial x}}$ and
$({\frac{\partial\kappa_{n}}{\partial
x}})^{2}+\tau_{n}^{2}\kappa_{n}^{2}=({\frac{\partial P_{n}}{\partial
x}})^{2}+({\frac{\partial Q_{n}}{\partial x}})^{2}$ which is continuous, so
there exists a solution $X^{(n)}(x,t)\in SO(3)$. We can interpret the
solutions as elements of a fibre bundle over $(M_{n},\mu_{K}^{(n)},L^{2})$
with fibres that are isomorphic to $SO(3)$.

Let $P+iQ=\kappa e^{i\sigma}$ be a solution of NLS and let

$\Omega_{1}=\begin{bmatrix}0&\kappa&0\\\ -\kappa&0&\tau\\\
0&-\tau&0\end{bmatrix}.$ (6.11)

######  Proposition 6.3.

(i) Let $P+iQ=\kappa e^{i\sigma}$ be a solution of NLS with initial data in
$P(x,0)+iQ(x,0)\in B_{K}\cap H^{1}$. Then $\Omega_{1}$ in (6.11) gives an
$so(3)$-valued vector field in $L^{2}(\kappa^{2}(x,t)dx)$.

(ii) Let $P+iQ=\kappa e^{i\sigma}$ be a solution of NLS with initial data
$P(x,0)+iQ(x,0)\in H^{1}\cap B_{K}$, and let
$P_{n}+iQ_{n}=\kappa_{n}e^{i\sigma_{n}}$ be the corresponding solution of the
NLS truncated in Fourier space, giving matrix $\Omega_{1}^{(n)}$. Let
$X^{(n)}_{t}(x)$ be a solution of (6.9) and suppose that $X^{(n)}$ converges
weakly in $L^{2}$ to $X_{t}(x)$. Then $X_{t}$ gives a weak solution of (5.1).

###### Proof.

(i) With $\omega=\sqrt{\kappa^{2}+\tau^{2}}$, we have

$\exp(h\Omega_{1})=I+{\frac{\sin h\omega}{\omega}}\Omega_{1}+{\frac{1-\cos
h\omega}{\omega^{2}}}\Omega_{1}^{2}$

where the entries of $\Omega_{1}^{2}$ are bounded by $\kappa^{2}+\tau^{2}$,
hence

$\int_{\mathbb{T}}\|\Omega_{1}(x,t)\|^{2}\kappa(x,t)^{2}dx<\infty$ (6.12)

for $u\in H^{1}$; however, there is no reason to suppose that $\tau$ itself is
integrable with respect to $dx$.

(ii) By (5) and (5.5), we have $\kappa\Omega_{1}\in L^{2}_{x}$ for all $u\in
H^{1}$. Moreover, Bourgain [Bo] has shown that for initial data
$P(x,0)+iQ(x,0)=\kappa(x,0)e^{i\sigma(x,0)}$ in $H^{1}\cap B_{K}$, the map

$\kappa(x,0)e^{i\sigma(x,0)}\mapsto\kappa(x,t)\Omega_{1}(x,t)\in L^{2}$ (6.13)

is Lipschitz continuous for $0\leq t\leq t_{0}$ with Lipschitz constant
depending upon $t_{0},K>0$. We have

$\displaystyle{\frac{\|\kappa(x+h,t)X(x+h,t)-\kappa(x,t)X(x,t)\|^{2}}{h^{2}}}$
$\displaystyle\leq
2\Bigl{(}{\frac{1}{h}}\int_{x}^{x+h}\bigl{|}{\frac{\partial\kappa}{\partial
y}}(y,t)\bigr{|}dy\Bigr{)}^{2}$
$\displaystyle\quad+2\Bigl{(}{\frac{1}{h}}\int_{x}^{x+h}\kappa(y,t)\|\Omega_{1}(y,t)\|dy\Bigr{)}^{2}$
(6.14)

where the right-hand side is integrable with respect to $x$ by the
Hardy–Littlewood maximal inequality and (6.12). Suppose that $X^{(n)}$ is a
solution of (5.1). We take $\tau_{n}$ to be locally bounded. Then by applying
Cauchy–Schwarz inequality to the integral

$X^{(n)}(x+h,t)-X^{(n)}(x,t)=\int_{0}^{h}\Omega_{1}^{(n)}(x+s,t)X^{(n)}(x+s,t)ds,$

we deduce that

$\displaystyle\int_{[0,2\pi]}$
$\displaystyle\|X^{(n)}(x+s,t)-X^{(n)}(x,t)\|_{M_{3\times
3}({\mathbb{R}})}^{2}\kappa_{n}(x,t)^{2}dx$ $\displaystyle\leq
h\int_{0}^{h}\int_{[0,2\pi]}\|\Omega_{1}^{(n)}(x+s,t)\|^{2}_{M_{3\times
3}({\mathbb{R}})}\kappa_{n}(x,t)^{2}dxds$ (6.15)

where the integral is finite by (6.12).

Also

$\displaystyle\sum_{j=1}^{N}{\frac{\|X^{(n)}(x_{j},t)-X^{(n)}(x_{j-1},t)\|^{2}_{M_{3\times
3}({\mathbb{R}})}}{x_{j}-x_{j-1}}}\leq\int_{x_{0}}^{x_{N}}\|\Omega_{1}^{(n)}(x,t)\|^{2}dx$

for $0<x_{1}<x_{2}<\dots<x_{N}<2\pi$.

We have

${\frac{\partial}{\partial
x}}\bigl{(}\kappa_{n}X^{(n)}\bigr{)}={\frac{\partial\kappa_{n}}{\partial
x}}X^{(n)}+\kappa^{(n)}\Omega_{1}^{(n)}X^{(n)}$ (6.16)

so for $Z\in C^{\infty}([0,2\pi];M_{3\times 3}({\mathbb{R}}))$ and the inner
product on $M_{3\times 3}({\mathbb{R}})$, we have

$\displaystyle\langle$
$\displaystyle\kappa_{n}(2\pi)X^{(n)}(2\pi),Z(2\pi)\rangle-\langle\kappa_{n}(0)X^{(n)}(0),Z(0)\rangle-\int_{0}^{2\pi}\kappa_{n}(x)\langle
X^{(n)}(x),Z(x)\rangle\,dx$
$\displaystyle=\int_{0}^{2\pi}{\frac{\partial\kappa_{n}}{\partial x}}\langle
X^{(n)}(x),Z(x)\rangle\,dx+\int_{0}^{2\pi}\langle
X^{(n)},\kappa_{n}(x)\Omega_{1}^{(n)}(x)^{T}Z(x)\rangle dx$ (6.17)

where $\kappa_{n}\rightarrow\kappa$ in $H^{1}$, so with norm convergence, we
have ${\frac{\partial\kappa_{n}}{\partial
x}}\rightarrow{\frac{\partial\kappa}{\partial x}}$ in $L^{2}$, and
$\kappa_{n}\Omega^{(n)}\rightarrow\kappa\Omega_{1}$ as $n\rightarrow\infty$,
and with weak convergence in $L^{2}$, we have $X^{(n)}\rightarrow X$, so

$\displaystyle\langle\kappa(2\pi)X(2\pi),Z(2\pi)\rangle$
$\displaystyle-\langle\kappa(0)X(0),Z(0)\rangle-\int_{0}^{2\pi}\kappa(x)\langle
X(x),Z(x)\rangle\,dx$
$\displaystyle=\int_{0}^{2\pi}{\frac{\partial\kappa}{\partial x}}\langle
X(x),Z(x)\rangle\,dx+\int_{0}^{2\pi}\langle
X,\kappa(x)\Omega_{1}(x)^{T}Z(x)\rangle dx.$ (6.18)

∎

The simulation of this differential equation computes
$X_{x}\in{\mathbb{S}}^{2}$ starting with $X_{0}=[0;0;1]$ and produces a frame
$\\{X_{x},\Omega_{x}X_{x},X_{x}\times\Omega_{x}X_{x}\\}$ of orthogonal
vectors. Geodesics on ${\mathbb{S}}^{2}$ are the curves such that the
principal normal is parallel to the position vector, namely the great circles.
For a geodesic, $X_{x}\times\Omega_{x}X_{x}$ is perpendicular to the plane
that contains the great circle.

Let $P+iQ=\kappa e^{i\sigma}$ be a solution of NLS and let

$\Omega_{2}=\begin{bmatrix}0&-\kappa\tau&{\frac{\partial\kappa}{\partial
x}}\\\ \kappa\tau&0&0\\\ -{\frac{\partial\kappa}{\partial
x}}&0&0\end{bmatrix}.$ (6.19)

######  Proposition 6.4.

(i) Let $P+iQ=\kappa e^{i\sigma}$ be a solution of NLS with initial data
$P(x,0)+iQ(x,0)\in B_{K}$. Then $x\mapsto\int_{0}^{x}\Omega_{2}(y,t)dy$ gives
a $so(3)$-valued stochastic of finite quadratic variation on $[0,2\pi]$ almost
surely with respect to $\mu_{K}(dPdQ)$.

(ii) Let $P+iQ=\kappa e^{i\sigma}$ be a solution of NLS with initial data
$P(x,0)+iQ(x,0)\in H^{1}\cap B_{K}$, and let
$P_{n}+iQ_{n}=\kappa_{n}e^{i\sigma_{n}}$ be the corresponding solution of the
NLS truncated in Fourier space, giving matrix $\Omega_{2}^{(n)}$. Let
$X^{(n)}_{t}$ be a solution of 6.10). Then $X_{t}^{(n)}$ converges in
$L^{2}_{x}$ norm to $X_{t}$ as $n\rightarrow\infty$ where $X_{t}$ gives a weak
solution of (5.2).

###### Proof.

(i) The essential estimate is

$\displaystyle\int_{B_{K}}\sum_{j}$
$\displaystyle|\kappa(x_{j+1},t)-\kappa(x_{j},t)|^{2}\mu_{K}(du)$
$\displaystyle\leq\sum_{j}\Bigl{(}\int_{B_{K}}|u(x_{j+1},t)-u(x_{j},t)|^{2}\mu_{K}(du)\Bigr{)}$
$\displaystyle\leq\sum_{j}\Bigl{(}\int_{B_{K}}|u(x_{j+1},t)-u(x_{j},t)|^{4}W_{K}(du)\Bigr{)}^{1/2}\Bigl{(}\int_{B_{K}}\Bigl{(}{\frac{d\mu_{K}}{dW}}\Bigr{)}^{2}dW\Bigr{)}^{1/2}$
$\displaystyle\leq
C\sum_{j}\Bigl{(}\int_{B_{K}}|u(x_{j+1},t)-u(x_{j},t)|^{2}W(du)\Bigr{)}^{1/2}$
$\displaystyle\leq C\sum_{j}(x_{j+1}-x_{j})\leq 2\pi C.$ (6.20)

Also, we can control the $\kappa\tau$ term via

$\int_{0}^{x}\kappa d\sigma-2^{-1}\kappa^{2}d\sigma\wedge
d\sigma=\int_{0}^{x}\kappa\nabla\sigma\cdot\begin{bmatrix}dP\\\
dQ\end{bmatrix}=\int_{0}^{x}{\frac{-QdP+PdQ}{\sqrt{P^{2}+Q^{2}}}}$

which is a bounded martingale transform of Wiener loop.

(ii) By (5) and (5.5), we have $\Omega_{2}\in L^{2}_{x}$ for all $u\in H^{1}$.
Bourgain [Bo] has shown that for initial data
$P(x,0)+iQ(x,0)=\kappa(x,0)e^{i\sigma(x,0)}$ in $H^{1}\cap B_{K}$, the map

$\kappa(x,0)e^{i\sigma(x,0)}\mapsto\Omega_{2}(x,t)\in L_{x}^{2}$ (6.21)

is Lipschitz continuous for $0\leq t\leq t_{0}$ with Lipschitz constant
depending upon $t_{0},K>0$. We have

$\int_{0}^{2\pi}\|\Omega_{2}(x)\|^{2}dx\leq
2\int_{0}^{2\pi}\Bigl{(}\Bigl{(}{\frac{\partial\kappa}{\partial
x}}\Bigr{)}^{2}+\kappa(x)^{2}\tau(x)^{2}+\kappa(x)^{4}\Bigr{)}dx,$

where the final integral is part of the Hamiltonian. With $Z\in
C^{\infty}({\mathbb{T}};M_{3\times 3}({\mathbb{R}}))$, we have the integral
equation for the pairing $\langle\cdot,\cdot\rangle$ on
$L^{2}([0,2\pi],M_{3\times 3}({\mathbb{R}}))$

$\langle X^{(n)}_{t},Z\rangle=\langle X_{0}^{(n)},Z\rangle+\int_{0}^{t}\langle
X^{(n)}_{s},(\Omega^{(n)}_{s})^{T}Z\rangle\,ds.$ (6.22)

Consider the variational differential equation in $L^{2}([0,2\pi],M_{3\times
3}({\mathbb{R}}))$

$\displaystyle{\frac{d}{dt}}(X^{(m)}(x,t)-X^{(n)}(x,t))$
$\displaystyle=\Omega_{2}^{(n)}(x,t)(X^{(m)}(x,t)-X^{(n)}(x,t))$
$\displaystyle\quad+(\Omega_{2}^{(m)}(x,t)-\Omega_{2}^{(n)}(x,t))X^{(m)}(x,t)$
(6.23)

where $\Omega_{2}^{(n)}(x,t)$ and
$\Omega_{2}^{(m)}(x,t)-\Omega_{2}^{(n)}(x,t)$ are skew.

We introduce a family of matrices $U^{(n)}(x;t,s)$ such that
$U^{(n)}(x;t,r)U^{(n)}(x;r,s)=U^{(n)}(x;t,s)$ for $t>r>s$ and
$U^{(n)}(x;t,t)=I$ such that

${\frac{\partial}{\partial
t}}U^{(n)}(x;t,s)=\Omega_{2}^{(n)}(x;t)U^{(n)}(x;t,s).$ (6.24)

Then the variational equation has solution

$\displaystyle X^{(m)}(x,t)-X^{(n)}(x,t)$
$\displaystyle=U^{(n)}(x;t,0)(X^{(m)}(x,0)-X^{(n)}(x,0))$
$\displaystyle\quad+\int_{0}^{t}U^{(n)}(x;t,r)(\Omega_{2}^{(m)}(x;r)-\Omega_{2}^{(n)}(x;r))X^{(m)}(x,r)dr.$

Then

$\displaystyle{\frac{d}{dt}}$
$\displaystyle\bigl{\langle}X^{(m)}(t)-X^{(n)}(t),X^{(m)}(t)-X^{(n)}(t)\rangle_{L^{2}_{x}}$
$\displaystyle=2\mathop{{\rm
Re}}\bigl{\langle}(\Omega_{2}^{(m)}(t)-\Omega_{2}^{(n)}(t))X^{(m)}(t),X^{(m)}(t)-X^{(n)}(t)\bigr{\rangle}_{L^{2}_{x}}$
$\displaystyle\leq\|\Omega_{2}^{(m)}(t)-\Omega_{2}^{(n)}(t)\|^{2}_{L^{2}_{x}}\|X^{(m)}(t)\|^{2}_{L^{2}_{x}}+\|X^{(m)}(t)-X^{(n)}(t)\|^{2}_{L^{2}_{x}}$
(6.25)

so from this differential inequality we have

$\|X^{(m)}(t)-X^{(n)}(t)\|_{L^{2}_{x}}^{2}\leq
e^{t}\|X^{(m)}(0)-X^{(n)}(0)\|_{L^{2}_{x}}^{2}+\int_{0}^{t}e^{t-s}\|\Omega_{2}^{(m)}(s)-\Omega_{2}^{(n)}(s)\|_{L^{2}_{x}}^{2}ds.$
(6.26)

Now $X^{(m)}(0)-X^{(n)}(0)\rightarrow 0$ and
$\Omega_{2}^{(m)}(s)-\Omega_{2}^{(n)}(s)\rightarrow 0$ in $L_{x}^{2}$ norm as
$n,m\rightarrow\infty$, so there exists $X(x,t)\in L_{x}^{2}$ such that
$X(x,t)-X^{(n)}(x,t)\rightarrow 0$ in $L^{2}_{x}$ norm as
$n\rightarrow\infty$.

We deduce that

$\langle X(t),Z\rangle_{L^{2}_{x}}=\langle
X_{0},Z\rangle_{L^{2}_{x}}+\int_{0}^{t}\langle
X_{u},(\Omega_{2}(u))^{T}Z\rangle_{L^{2}_{x}}\,du,$ (6.27)

so we have a weak solution of the ODE. ∎

Let $\Omega^{(n,u_{n})}_{2}(x,t)$ be the Fourier truncated matrix that
corresponds to a solution $u_{n}$ of the Fourier truncated equation $NLS_{n}$,
then let $X^{n,u_{n}}(x,t)$ be the solution of the ODE (6.10). By Proposition
6.2, the map $u_{n}\mapsto X^{n,u_{n}}(\cdot,t)$ pushes forward the modified
Gibbs measure $\mu_{K}^{(n)}$ to a measure on $(C(M_{n};SO(3)),L^{2})$ that
satisfies a Gaussian concentration of measure inequality with constant
$\alpha(\beta,K)/n^{2}$; compare (3.9).

######  Corollary 6.5.

For each $Z\in L^{2}([0,2\pi];M_{3\times 3}({\mathbb{R}}))$, introduce the
${\mathbb{R}}$-valued random variable on $(M_{n},L^{2},\mu_{K}^{(n)})$ by

$Z_{n}(u_{n})=\int_{[0,2\pi]}\langle X^{(n,u_{n})}(x,t),Z(x)\rangle dx.$
(6.28)

(i) Then the distribution $\nu^{(n)}$ of $Z_{n}$ satisfies the Gaussian
concentration inequality

$\int_{M_{n}}\exp\Bigl{(}tZ_{n}-t\int_{M_{n}}Z_{n}d\mu_{K}^{(n)}\Bigr{)}\mu_{K}^{(n)}(du_{n})\leq\exp\bigl{(}n^{2}t^{2}/\alpha(\beta,K)\bigr{)}\qquad(t\in{\mathbb{R}}).$
(6.29)

(ii) Let $\nu_{N}^{(n)}=N^{-1}\sum_{j=1}^{N}\delta_{Z_{n}^{(j)}}$ be the
empirical distribution of $N$ independent copies of $Z_{n}$. Then
$W_{1}(\nu_{N}^{(n)},\nu^{(n)})\rightarrow 0$ almost surely as
$N\rightarrow\infty$.

###### Proof.

(i) As with $u_{n}$, we introduce the corresponding data for another solution
$v_{n}$. As in (6.26), we have

$\displaystyle\|X^{(n,u_{n})}(x,t)-X^{(n,v_{n})}(x,t)\|_{{\mathbb{R}}^{3}}^{2}$
$\displaystyle\leq
e^{t}\|X^{(n,u_{n})}(0)-X^{(n,v_{n})}(0)\|_{{\mathbb{R}}^{3}}^{2}$
$\displaystyle\quad+\int_{0}^{t}e^{t-s}\|\Omega_{2}^{(n,u_{n})}(x,s)-\Omega_{2}^{(n,v_{n})}(x,s)\|_{so(3)}^{2}ds.$
(6.30)

For given initial condition $X^{n,v_{n})}(0)=X^{(n,u_{n})}(0)$, and $T>0$, we
can take the supremum over $t$, then integrate this with respect to $x$ and
obtain

$\displaystyle\int_{0}^{2\pi}\sup_{0<t<T}\|X^{(n,u_{n})}(x,t)$
$\displaystyle-X^{(n,v_{n})}(x,t)\|_{{\mathbb{R}}^{3}}^{2}dx$
$\displaystyle\leq
e^{T}\int_{0}^{T}\|\Omega_{2}^{(n,u_{n})}(x,s)-\Omega_{2}^{(n,v_{n})}(x,s)\|_{L^{2}_{x}}^{2}ds$
(6.31)

so $\Omega^{(u)}\mapsto X^{u}$ is a Lipschitz function
$L^{2}([0,2\pi]\times[0,T],so(3))\rightarrow
L^{2}([0,2\pi];L^{\infty}([0,T],{\mathbb{R}}^{3}))$. By Bourgain’s results,
there exists $C>0$ such that

$\displaystyle\|\Omega_{2}^{(n,u_{n})}(x,s)-\Omega_{2}^{(n,v_{n})}(x,s)\|_{L^{2}_{x}}$
$\displaystyle\leq C\|u_{n}(x,s)-v_{n}(x,s)\|_{H_{x}^{1}}$ $\displaystyle\leq
Cn\|u_{n}(x,0)-v_{n}(x,0)\|_{L^{2}_{x}},$ (6.32)

so $u_{n}\mapsto X^{(n,u_{n})}$ is a Lipschitz function on $L^{2}_{x}$, albeit
with a constant growing with $n$. Thus we can push forward the modified Gibbs
measure $(M_{n},L^{2},\mu_{K,\beta}^{(n)})\rightarrow
L^{2}([0,2\pi];M_{3\times 3}({\mathbb{R}}))$ so that the image measure
satisfies a Gaussian concentration inequality with constant
$\alpha(\beta,K)/n^{2}$ dependent upon $n$. For each $Z\in
L^{2}([0,2\pi];M_{3\times 3}({\mathbb{R}}))$, we introduce $Z_{n}$, so that
where $u_{n}\mapsto Z_{n}$ is $Cn$-Lipschitz function from
$(M_{n},L^{2},\mu_{K,\beta}^{(n)})$ to ${\mathbb{R}}$. The random variable
$Z_{n}$ therefore satisfies the Gaussian concentration inequality (6.29).

(ii) By Theorem 4.3, we can use the Borel–Cantelli Lemma to show that

${\mathbb{P}}\Bigl{[}\bigl{|}W_{1}(\nu_{N}^{(n)},\nu^{(n)})-{\mathbb{E}}W_{1}(\nu_{N}^{(n)},\nu^{(n)})\bigr{|}>\varepsilon\quad{\hbox{for
infinitely many}}\quad N\Bigr{]}=0\qquad(\varepsilon>0),$

where by Proposition 4.4,
${\mathbb{E}}W_{1}(\nu_{N}^{(n)},\nu^{(n)})\rightarrow 0$ as
$N\rightarrow\infty$. ∎

Consider a coupling of $(M_{n},L^{2},\mu^{(n)}_{K,\beta})$ and
$(M_{\infty},L^{2},\mu_{K,\beta})$ involving measure $\pi_{n}$. For any
bounded continuous $\varphi:{\mathbb{C}}\rightarrow{\mathbb{R}}$ we can
consider

$\displaystyle\int_{M_{n}}\varphi(Z_{n}(u_{n}))\mu^{(n)}_{K,\beta}(du_{n})-$
$\displaystyle\int_{M_{\infty}}\varphi(Z(u))\mu_{K,\beta}(du)$
$\displaystyle=\int\\!\\!\\!\int_{M_{n}\times
M_{\infty}}\bigl{(}\varphi(Z_{n}(u_{n}))-\varphi(Z(u))\bigr{)}\pi_{n}(du_{n}du)$
(6.33)

where

$\displaystyle{\mathcal{D}}_{L^{2}}(\hat{M}_{n},\hat{M}_{\infty})^{2}=\int\\!\\!\\!\int_{M_{n}\times
M_{\infty}}\delta(u_{n},u)^{2}\pi_{n}(du_{n}du)\rightarrow
0\qquad(n\rightarrow\infty).$

######  Proposition 6.6.

Let $(\varphi_{j})_{j=1}^{\infty}$ be a dense sequence in
$Ball(C_{c}({\mathbb{C}};{\mathbb{R}}))$ and $(Y_{\ell})_{\ell=1}^{\infty}$ a
dense sequence in $Ball(L^{2})$. Then there exists a subsequence $(n_{k})$
such that

$\int_{M_{n_{k}}}\varphi_{j}\bigl{(}\langle
X^{(n_{k},u_{n_{k}})},Y_{\ell}\rangle\bigr{)}\mu_{K,\beta}^{(n_{k})}(du_{n_{k}})$
(6.34)

converges as $n_{k}\rightarrow\infty$ for all $j,\ell\in{\mathbb{N}}$.

###### Proof.

We can introduce a metric so that $M=\prod_{n=1}^{\infty}M_{n}\sqcup
M_{\infty}$ becomes a complete and separable metric space, and we can
transport $\mu_{n}$ onto $M$. Then
$\omega=2^{-1}\mu_{\infty}+\sum_{n=1}^{\infty}2^{-n-1}\mu_{n}$ is a
probability measure on $M$, and $\mu_{n}$ is absolutely continuous with
respect to $\omega$, so $d\mu_{n}=f_{n}d\omega$ for some probability density
function $f_{n}\in L^{1}(\omega)$. By convergence in total variation from
Lemma 4.7 (ii), here exists $f_{\infty}\in L^{1}(\omega)$ such that
$f_{n}\rightarrow f_{\infty}$ in $L^{1}$ as $n\rightarrow\infty$. Given a
bounded sequence $(g_{n})_{n=1}^{\infty}$ in $L^{\infty}(\omega)$, there
exists $g_{\infty}\in L^{\infty}(\omega)$ and a subsequence $(n_{k})$ such
that

$\int g_{n_{k}}d\mu_{n_{k}}=\int g_{n_{k}}f_{n_{k}}d\omega\rightarrow\int
g_{\infty}f_{\infty}d\omega=\int g_{\infty}d\mu_{\infty}.$ (6.35)

∎

######  Remark 6.7.

For $u\in M_{\infty},$ we have $u_{n}=D_{n}u\in M_{n}$ so that
$u_{n}\rightarrow u$ in $L^{2}$ norm as $n\rightarrow\infty$. It is plausible
that (6) tends to $0$ as $n\rightarrow\infty$, but we do not have a proof.
Unfortunately, the constants are not sharp enough to allow us to use (4.8) to
deduce $W_{2}$ convergence for the distributions on $SO(3)$.

##  7\. Experimental Results

Our objective in this section is to obtain a (random) numerical approximation
to the solution of (6.9). We consider the case where the parameter $\beta$ in
(1.3) is equal to $0$. Note that in this case, the Gibbs measure reduces to
Wiener loop measure and stochastic processes with the Wiener loop measure as
their law are by definition Brownian loop. Equation (6.9) is a PDE with
respect to the space variable $x$, while the parameter of a stochastic process
in an SDE is colloquially referred to as time. To avoid confusion, in this
section we refer to $x$ as $s$; whereas the time variable $t$ is suppressed.

Recall the polar decomposition $P+iQ=\kappa e^{i\sigma}$ where,
$\kappa=\sqrt{P^{2}+Q^{2}}$ and $\sigma$ is such that
$\tau={\frac{\partial\sigma}{\partial s}}$. Define
$\sigma_{\epsilon}(P,Q):=\tan^{-1}(\frac{PQ}{P^{2}+\epsilon^{2}})$ as the
regularised Ito integral of $\tau$. The Ito differential $d\sigma_{\epsilon}$
can be written as $d\sigma_{\epsilon}=f_{1}(P,Q)dP+f_{2}(P,Q)dQ+f_{3}(P,Q)ds$,
where

$\displaystyle
f_{1}(P,Q):=\frac{(\epsilon^{2}-P^{2})Q}{(\epsilon^{2}+P^{2})^{2}+P^{2}Q^{2}},$
$\displaystyle
f_{2}(P,Q):=\frac{P(\epsilon^{2}+P^{2})}{(\epsilon^{2}+P^{2})^{2}+P^{2}Q^{2}},$
$\displaystyle
f_{3}(P,Q):=-\frac{2P^{3}Q(\epsilon^{2}+P^{2})}{((\epsilon^{2}+P^{2})^{2}+P^{2}Q^{2})^{2}}-\frac{2PQ\left((\epsilon^{2}+P^{2})^{2}+P^{2}Q^{2}\right)}{\left((\epsilon^{2}+P^{2})^{2}+P^{2}Q^{2}\right)^{2}}$
$\displaystyle\qquad\qquad\qquad-\frac{(\epsilon^{2}-P^{2})Q\left(2PQ^{2}+4P(\epsilon^{2}+P^{2})\right)}{\left((\epsilon^{2}+P^{2})^{2}+P^{2}Q^{2}\right)^{2}}.$

We can write (6.9) in the form of a SDE, including a correction to convert
from a Stranovich SDE into an Ito SDE as follows

$\displaystyle dX_{s}=$
$\displaystyle\mathbf{A}X_{s}ds+\mathbf{B}X_{s}dP+\mathbf{C}X_{s}dQ$ (7.1)

where,

$\displaystyle\mathbf{A}=\begin{bmatrix}0&\sqrt{P^{2}+Q^{2}}&0\\\
-\sqrt{P^{2}+Q^{2}}&\frac{1}{2}f_{1}^{2}(P,Q)+\frac{1}{2}f_{2}^{2}(P,Q)&f_{3}(P,Q)\\\
0&-f_{3}(P,Q)&\frac{1}{2}f_{1}^{2}(P,Q)+\frac{1}{2}f_{2}^{2}(P,Q)\end{bmatrix},$
$\displaystyle\mathbf{B}=\begin{bmatrix}0&0&0\\\ 0&0&f_{1}(P,Q)\\\
0&-f_{1}(P,Q)&0\end{bmatrix},\quad\mathbf{C}=\begin{bmatrix}0&0&0\\\
0&0&f_{2}(P,Q)\\\ 0&-f_{2}(P,Q)&0\end{bmatrix}.$ (7.2)

As justified above $P$ and $Q$ are each a Brownian bridge with period
$T=2\pi$, thus they can be expressed in terms of Brownian motions $W_{1}$ and
$W_{2}$; that is, $P(s)=W_{1}(s)-sW_{1}(2\pi)/2\pi$ and likewise for $Q$.
Equation (7.1) is now written as a standard Ito SDE,

$\displaystyle dX_{s}=$
$\displaystyle\Bigl{(}\mathbf{A}+\frac{W_{1}(2\pi)}{2\pi}\mathbf{B}+\frac{W_{2}(2\pi)}{2\pi}\mathbf{C}\Bigr{)}X_{s}ds+\mathbf{B}X_{s}dW_{1}+\mathbf{C}X_{s}dW_{2}$
(7.3)

where $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ are defined as in Equation
(7.2). The resulting stochastic process $X_{s}\in SO(3)$ is then used to
rotate the unit vector $y_{0}=[0,0,1]^{T}$ on $\mathbb{S}^{2}$ by
$y_{s}=X_{s}y_{0}$. The sample paths of this process can be described by
construction of a frame $\\{y_{s},y^{\prime}_{s},y_{s}\times
y_{s}^{\prime}\\}$. In order to simulate this SDE, we make use of a numerical
scheme for matrix SDEs in $SO(3)$ developed by Marjanovic and Solo [MS]. This
involves a single step geometric Euler-Maruyama method, called g-EM, in the
associated Lie algebra. Figure 1 demonstrates a sample-path of $y_{s}$
generated via this method.

Figure 1. One sample path of the stochastic process $X_{s}$, over $[0,10]$
with step size of $h=10^{-5}$.

The sample paths start off on the great circle perpendicular to the y-axis,
and so have constant binormal $y_{s}\times y_{s}^{\prime}$. As a sample path
extends past the great circle the binormal vector at each point deviates
slowly, thus a sample path can be thought of as a precessing orbit.

The Ito process $y_{s}$ is derived from the solution to Equation (7.3) and
takes values in $\mathbb{R}^{3}$. Let $\hat{y}_{s,h}$ denote the numerical
approximation to $y_{s}$ on $[0,T]$ with step size $h$, which is calculated
using the g-EM method. The approximation error converges to zero in the
$L^{2}$ space of Ito processes as the step size $h\rightarrow 0$,

$\mathbb{E}\left[\sup_{0\leq
s<T}\|y_{s}-\hat{y}_{s,h}\|_{\mathbb{R}^{3}}^{2}\right]=\mathcal{O}(h^{1-\varepsilon}),$
(7.4)

for some $\varepsilon>0$ [PS]. A value of $\varepsilon=1/4$ allows us to
maintain control of the implied constants on the interval $[0,1]$, and $h$ is
taken to be $10^{-5}$. We apply g-EM to Equation (7.3) on the interval
$[0,10]$, upon which a smaller value of $h$ would be welcomed. However, we are
attempting to calculate a distribution, so we need a large number of sample-
paths.

The computational complexity of simulating a single sample-path is
$\mathcal{O}(T/h)$ where $T$ denotes the length of the interval simulated.
Therefore, for a total of $N$ samples, the computational complexity of our
simulation algorithm is $\mathcal{O}(NT/h)$. We run our simulations using a
machine equipped with an 8-core Intel Xeon Gold 6248R CPU with a clock speed
of $2993$ Mhz; we take advantage of integrated parallelisation in MATLAB. With
$h=10^{-5}$ and $N=2\times 10^{6}$ the algorithm takes around $1$ week to run
on our system.

Since the sample paths are constrained to $\mathbb{S}^{2}$ the points $y_{s}$
can be specified in spherical coordinates of longitude
$\theta_{s}\in[-\pi,\pi)$ and colatitude $\phi_{s}\in[0,\pi]$. Figure 2
demonstrates the empirical joint distribution of $\theta_{s}$ and $\phi_{s}$
for two different values of $s$.

Figure 2. Left: A histogram of $(\theta_{s},\phi_{s})$ over one hemisphere at
$s=1$. Right: A histogram of $(\theta_{s},\phi_{s})$ over the sphere at
$s=10$.

As can be observed, the distribution of $(\theta_{s},\phi_{s})$ varies with
$s$. We hypothesise that the angles $\theta_{s}$ and $\phi_{s}$ evolve to
become statistically independent, and that $y_{s}$ will eventually be
uniformly distributed on the sphere. In the remainder of the section, we test
this hypothesis statistically.

### Wasserstein distance between measures on $\mathbb{S}^{2}$.

We start by calculating the Wasserstein distance $W_{1}(\nu_{1},\nu_{2})$
between probability measures $\nu_{1}$ and $\nu_{2}$ on ${\mathbb{S}}^{2}$,
which are absolutely continuous with respect to area and have disintegrations

$\displaystyle d\nu_{j}$
$\displaystyle=f_{j}(\theta)g_{j}(\phi\mid\theta)\sin\phi\,d\phi
d\theta\qquad(\theta\in[-\pi,\pi],\phi\in[0,\pi],j=1,2)$

where $f_{j}$ $(j=1,2)$ are probability density functions on $[-\pi,\pi]$ that
give the marginal distributions of $\nu_{j}$ in the longitude $\theta$
variable, and $g_{j}$ in the colatitude variable. Let $F_{j}$ be the
cumulative distribution function of $f_{j}(\theta)d\theta$ and $G_{j}$ be the
cumulative distribution function of $g_{j}(\phi)\sin\phi\,d\phi$. We measure
$W_{1}(\nu_{1},\nu_{2})$ in terms of one-dimensional distributions. Given
distributions on ${\mathbb{R}}$ with cumulative distribution functions $F_{1}$
and $F_{2}$, we write $W_{1}(F_{1},F_{2})$ for the Wasserstein distance
between the distributions for cost function $|x-y|$. Let
$\psi:[-\pi,\pi]\rightarrow[-\pi,\pi]$ be an increasing function that induces
$f_{2}(\theta)d\theta$ from $f_{1}(\theta)d\theta$; then

$W_{1}(\nu_{1},\nu_{2})\leq
W_{1}\bigl{(}F_{1},F_{2})+\int_{-\pi}^{\pi}W_{1}\bigl{(}G_{2}(\cdot\mid\psi(\theta)),G_{1}(\cdot\mid\theta)\bigr{)}f_{1}(\theta)d\theta.$

In particular, for $f_{1}(\theta)=1/(2\pi)$ and $g_{1}(\phi)=1/2$, we have a
product measure $\nu_{1}(d\theta d\phi)=(4\pi)^{-1}\sin\phi d\phi d\theta$
giving normalized surface area on the sphere. Then
$F_{1}(\theta)=(\theta+\pi)/(2\pi)$ and
$F_{2}(\psi(\theta))=(\theta+\pi)/(2\pi)$, so $\psi(2\pi(\tau-1/2))$ for
$\tau\in[0,1]$ gives the inverse function of $F_{2}$. We deduce that

$W_{1}(F_{1},F_{2})=\int_{-\pi}^{\pi}\Bigl{|}{\frac{\theta+\pi}{2\pi}}-F_{2}(\theta)\Bigr{|}d\theta$
(7.5)

and

$W_{1}\bigl{(}G_{2}(\cdot\mid\psi(\theta)),G_{1}(\cdot\mid\theta)\bigr{)}=\int_{0}^{\pi}\Bigl{|}\int_{0}^{\phi}\bigl{(}g_{2}(\phi^{\prime}\mid\psi(\theta))-(1/2)\bigr{)}\sin\phi^{\prime}d\phi^{\prime}\Bigr{|}d\phi$
(7.6)

Hence the Wasserstein distance can be bounded in terms of the cumulative
distribution functions by

$\displaystyle W_{1}(\nu_{1},\nu_{2})$ $\displaystyle\leq
W_{1}(F_{2},F_{1})+W_{1}(G_{2},G_{1})+\int_{-\pi}^{\pi}W_{1}(G_{2}(\cdot\mid\theta),G_{2})dF_{1}(\theta)$
$\displaystyle=\int_{-\pi}^{\pi}\Bigl{|}{\frac{\theta+\pi}{2\pi}}-F_{2}(\theta)\Bigr{|}d\theta+\int_{0}^{\pi}\Bigl{|}G_{2}(\phi)-{\frac{1-\cos\phi}{2}}\Bigr{|}d\phi$
$\displaystyle\quad+\int_{-\pi}^{\pi}\int_{0}^{\pi}\bigl{|}G_{2}(\phi\mid\theta)-G_{2}(\phi)\bigr{|}dF_{1}(\theta)d\phi$
(7.7)

where we have used the triangle inequality to obtain a more symmetrical
expression involving the Wasserstein distances for the marginal distributions
and the $G$ conditional distributions, namely the dependence of the colatitude
distribution on longitude.

Figure 3. Wasserstein distance between $F_{1}(\theta)=(\theta+\pi)/(2\pi)$ and
$F_{N}^{\theta_{s}}$ (left) and Wasserstein distance between $G_{1}(\phi)$
which has $g_{1}(\phi)=1/2$ as pdf, and $G_{N}^{\phi_{s}}$ (right) with
$N=10^{5}$ and $s$ ranging over $0.3,0.6,0.9,\dots,6.0$.

For each $s\in[0,10]$, let $F^{\theta_{s}}$ and $G^{\phi_{s}}$ be the marginal
CDFs of $\theta_{s}$ and $\phi_{s}$ respectively. For $N\in\mathbb{N}$, denote
by $F_{N}^{\theta_{s}}$ and $G_{N}^{\phi_{s}}$ the empirical CDFs of
$\theta_{s}$ and $\phi_{s}$. We generate empirical CDFs $F_{N}^{\theta_{s}}$
and $G_{N}^{\phi_{s}}$ with $s=0.3,0.6,0.9,\dots,6.0$, and $N=10^{5}$. Figure
3 demonstrates that $W_{1}(F_{1},F_{N}^{\theta_{s}})$ and
$W_{1}(G_{1},G_{N}^{\phi_{s}})$, each decreases with increasing $s$. As a
consequence of Theorem 4.3 and Proposition 4.4 for $N=10^{5}$ with probability
at least $0.99$ it holds that $W_{1}(F_{N}^{\theta_{s}},F^{\theta_{s}})\leq
0.025$ and $W_{1}(G_{N}^{\phi_{s}},G^{\phi_{s}})\leq 0.018$. Thus, we observe
that $F^{\theta_{s}}$ and $G^{\phi_{s}}$ converge to $F_{1}$ and $G_{1}$
respectively.

### Hypothesis tests for independence and goodness-of-fit.

We run a total of $22$ hypothesis tests to examine the evolution of the joint
distribution of the angles $\theta_{s}$ and $\phi_{s}$. In order to account
for multiple testing, we set the significance level of each test to $0.00045$,
leading to an overall level of $0.01$. First, we generate sample paths to
obtain $N=10^{5}$ realisations of $(\theta_{s},\phi_{s})$ for each value of
$s=0.3,0.6,0.9,\dots,6.0$. For each $s$ we test the null hypothesis $H_{0,s}$
that the angles $\theta_{s}$ and $\phi_{s}$ are statistically independent,
against the alternative hypothesis $H_{1,s}$ that they are dependent. To this
end, we rely on a widely used nonparametric independence test, which is based
on the Hilbert-Schmidt Independence Criterion (HSIC) dependence measure [GG,
ALMM]; the implementation is due to [JZG]. It is observed that while the null
hypothesis is rejected for $s=0.3,\dots,2.1$, the test is unable to reject
$H_{0,s}$ from $s=2.4,\dots,6.0$ at (an overall) significance level $0.01$.
For $s=10$, we run two Kolmogorov-Smirnov goodness-of-tests as follows. The
first one tests the null hypothesis $H_{0}^{\theta_{s}}$ that
$\theta_{s},~{}s=10$ is distributed according to $F_{1}$ against the
alternative that it is not; the second one tests the null hypothesis
$H_{0}^{\phi_{s}}$ that $\phi_{s},~{}s=10$ is distributed according to $G_{1}$
against the alternative that it is not. At significance level $0.01$, the test
is unable to reject the null hypotheses $H_{0}^{\theta_{s}}$ and
$H_{0}^{\phi_{s}}$ for $s=10$.

##  8\. Acknowledgements

The authors thank Nadia Mazza for helpful remarks on combinatorics. MKS is
funded by a Faculty of Science and Technology studentship, Lancaster
University.

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