The paper interprets the cubic nonlinear Schrödinger equation as a Hamiltonian system with infinite dimensional phase space. There exists 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, the nonlinear Schrödinger equation gives a Lax pair of coupled ordinary differential equations 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; the results are illustrated by numerical simulations. The paper contains quantitative estimates with well-controlled constants on the rate of convergence of the empirical distribution in Wasserstein metric.

## I. INTRODUCTION

*u*=

*P*+

*iQ*satisfies the nonlinear Schrödinger equation

*γ*= 2, we have the cubic nonlinear Schrödinger equation. The spatial variable is $x\u2208T$, and the functions are periodic, so that the system applies to fields parametrized by a circle. Throughout the paper, we write (

*M*,

*d*,

*μ*) for a complete and separable metric space with a Radon (inner regular) probability measure

*μ*on the

*σ*-algebra generated by the Borel subsets. The squared

*L*

^{2}norm

*H*

_{1}=

*∫*(

*P*

^{2}+

*Q*

^{2}) is formally invariant under the canonical equations of motion, so we can consider possible invariant measures on

*B*

_{K}for this micro-canonical ensemble is

*W*(

*du*) is Wiener loop measure, $IBK$ is the indicator function of

*B*

_{K}and

*Z*

_{K}(

*β*) is a normalizing constant. Lebowitz, Rose and Speer

^{26}proved existence of such an invariant measure, so that for all

*K*> 0 and $\beta \u2208R$ there exists

*Z*

_{K}(

*β*) > 0 such that

*μ*

_{K,β}is a Radon probability measure on $BK\u2282L2(T;R2)$. When

*β*= 0, we refer to the measure as free Wiener loop measure, indicating that the dynamics are free of potentials. For

*β*< 0, (1.3) is said to be focussing and the Hamiltonian is unbounded below, giving the source of the technical problem, which is addressed by restricting the measure to

*B*

_{K}.

Bourgain^{9} 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,^{30} that the sample paths are Hölder continuous, and a result from Theorem 1.2(iv) in Ref. 6 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 $x\u2208R$ captures soliton solutions, and Ref. 26 discuss the possible transition of the system between an ambient bounded random field and a soliton solution. For $x\u2208T$, the notion of a spatially localized solution is inapplicable, but some of the results are still relevant.^{26}

Bourgain^{10} p. 128 comments that invariant Gibbs measures for the periodic cubic Schrödinger equation can be constructed on other phase spaces, and one can consider Gibbs measures on *L*^{2} that have different normalizations than *μ*_{K,β}.

*H*and a

*k*-point density matrix. For

*μ*

_{0}a centered Gaussian measure on

*H*, we express a specific integral

*u*

^{⊗k}⟩⟨

*u*

^{⊗k}| is close to its mean value

*J*

^{(k)}on a set of large probability. This statement also holds when we replace

*μ*

_{0}by the Gibbs measure. In Sec. IV 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*↦

*u*(

*x*,

*t*) is

*γ*-Hölder continuous for 0 <

*γ*< 1/16, in the sense that $sup{\Vert u(x+h,t)\u2212u(x,t)\Vert Lx4/|h|\gamma );h\u22600}$ is almost surely finite, for fixed 0 <

*t*

_{0}<

*t*.

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 Sec. VII.

Hasimoto^{20} 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 $R3$. Cruzeiro and Malliavin^{14} developed stochastic differential geometry for frames, pursuing Cartan’s precedent.^{12} In Secs. V and VI we consider the evolution of the dynamical system corresponding to Hasimoto frames under the Gibbs measure. In Sec. VII 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). In the Appendix,^{20} Hasimoto expressed the change of variables in a polar decomposition $u=\rho exp(i\varphi )$ where *ρ* is a probability density and *ϕ* a phase, and derived Betchov’s intrinsic equation for vortex filaments from the nonlinear Schrödinger equation. We remark that Villani^{35} p. 691 carries out a similar a transformation to interpret the linear Schrödinger equation as a transport problem for densities *ρ* for a suitable action integral. The current paper is a further step at introducing transportation methods into PDE.

## II. TENSOR PRODUCTS AND *k*-POINT DENSITY MATRICES FOR GAUSSIAN MEASURE

*H*be a separable complex Hilbert space, with inner product ⟨·∣·⟩ which is linear in the second argument. The algebraic tensor product

*H*⊗

*H*is identified with the set of finite-rank operators on

*H*, and then we identify the injective tensor product $H\u2297\u030cH$ with the algebra $L(H)$ of bounded linear operators on

*H*and the projective tensor product $H\u2297\u0302H$ with the ideal $L1(H)$ of trace class operators on

*H*. By the theory of metric tensor products the dual space of $L1(H)$ is canonically $L(H)$. For

*H*=

*L*

^{2}, the identification is

*A*≤

*I*, and let

*μ*

_{0}be a Gaussian measure on

*H*of mean zero and covariance

*A*. By the spectral theorem, we can choose an orthonormal basis $(\phi j)j=1\u221e$ of

*H*such that

*Aφ*

_{j}=

*α*

_{j}

*φ*

_{j}where the spectrum of

*A*is the closure of {

*α*

_{j}:

*j*= 1, 2, …}. Then we introduce mutually independent Gaussian

*N*(0, 1) random variables $(\gamma j)j=1\u221e$ and the vector

*μ*

_{0}is the distribution of

*u*on

*H*, as one easily checks by computing the expectation

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

*k*-fold tensor product

*H*

^{⊗k}can be completed to give a Hilbert space, so that the space

*H*

^{s⊗k}of symmetric tensors gives a closed linear subspace. We consider

*k*-point density matrix, or equivalently a trace class operator $J(k)\u2208L1(Hs\u2297k)$. The following computation of Gaussian moments is known in Quantum field theory as Wick’s theorem.

*J*

^{(k)}which we have not been able to interpret, particularly line 8 of p. 79. Here we calculate

*J*

^{(2)}directly, before addressing the case of general

*k*. Evidently we have $E(\gamma j\gamma \u2113\gamma m\gamma n)=0$ if one of the indices

*j*,

*ℓ*,

*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

*H*⊗

_{s}

*H*, hence

*J*

^{(2)}is not a multiple of

*A*⊗

*A*.

_{k}be the set of all partitions of

*k*so that

*π*∈ Π

_{k}may be expressed as

*k*=

*k*

_{1}+

*k*

_{2}+ ⋯ +

*k*

_{n}where the row lengths $kj\u2208N$ have

*k*

_{1}≥

*k*

_{2}≥ ⋯ ≥

*k*

_{n}. Given such

*π*and a

*n*-element subset {

*j*

_{1}, …,

*j*

_{n}} of $N$, there is a symmetric tensor

*σ*of {

*j*

_{1}, …,

*j*

_{n}}. The set of all such tensors gives an orthonormal basis of the

*k*-fold symmetric tensor product

*H*

^{s⊗k}.

*u*as in (2.1) and consider the expansion

*H*

^{s⊗k}, and look for the terms that do not vanish after integration with respect to

*μ*

_{0}.

*π*∈ Π

_{k}consider a pair $(\lambda ,\rho )\u2208\Pi k2$ with rows

*λ*:

*k*=

*ℓ*

_{1}+

*ℓ*

_{2}+ ⋯ +

*ℓ*

_{n}where $\u2113j\u2208N\u222a{0}$ and

*ρ*:

*k*=

*r*

_{1}+

*r*

_{2}+ ⋯ +

*r*

_{n}where $rj\u2208N\u222a{0}$ and

*λ*and

*ρ*have equal numbers of odd rows; here rows may have zero lengths, and the rows are not necessarily in decreasing order. We refer to (

*λ*,

*ρ*) as an even decomposition of

*π*.

There are various alternative descriptions of even decompositions. We write *λ* ∼ *ρ* if *λ* and *ρ* are partitions that have equal numbers of boxes and equal numbers of odd rows; evidently ∼ is an equivalence relation on the set of partitions. By Ref. 24 Theorem 4 there is a bijection between symmetric matrices *A* that have entries in $N\u222a{0}$ with column sums *c*_{1}, …, *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 $N\u222a{0}$ 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*′ and *Q*′ have an equal number of odd rows, so *P*′ ∼ *Q*′.

*π*,

*λ*,

*ρ*) and an

*n*-subset {

*j*

_{1}, …,

*j*

_{n}} of $N$,

*J*

^{(k)}.

Conversely, let $(\lambda ,\rho )\u2208\Pi k2$ and suppose that *λ* and *ρ* have equal numbers of odd rows, so that after adding zero rows and reordering the rows we have *r*_{j} + *ℓ*_{j} even for all *j*. Then we introduce 2*k*_{j} = *ℓ*_{j} + *r*_{j} and after a further reordering write *k* = *k*_{1} + *k*_{2} + ⋯ + *k*_{n} where $kj\u2208N$ have *k*_{1} ≥ *k*_{2} ≥ ⋯ ≥ *k*_{n}, and we have *π* ∈ Π_{k} as above. Given a *n*-subset {*j*_{1}, …, *j*_{n}} of $N$, we take 2*k*_{m} copies of *j*_{m} and split them as *ℓ*_{m} on the bra side and *r*_{m} on the ket side of the tensor for *m* = 1, …, *n*, making a contribution as in (2.6). We summarize these results as follows.

*The integral* *J*^{(k)} *is the sum over the summands (2.6) that arise from a* *π* ∈ Π_{k} *with* *n* *nonzero rows, a* *n**-subset of* $N$*, and an even decomposition of* *π* *into a pair* $(\lambda ,\rho )\u2208\Pi k2$ *where* *λ* *and* *ρ* *have equal numbers of odd* *rows.*

## III. CONCENTRATION OF *k*-POINT MATRICES FOR GIBBS MEASURE

*N*(0, 1) random variables on some probability space (Ω,

*P*), where $zj=(\gamma j+i\gamma \u2212j)/2)$ and $z\u2212j=(\gamma j\u2212i\gamma \u2212j)/2$ for $j\u2208N$. Then we take Brownian loop to be the random Fourier series in the style

*W*(

*du*) is the distribution of

*u*∈

*H*, namely the probability measure induced by random variable

*u*via (Ω,

*P*) → (

*H*,

*W*). By orthogonality, we have

*u*∈

*B*

_{K/2π}if and only if $\u2211j\gamma j2/j2\u2264K$. Chernoff’s inequality, and independence we have

*μ*

_{λ}(

*du*) =

*ζ*(

*λ*)

^{−1}exp(

*λV*(

*u*))

*W*(

*du*) where

*μ*

_{λ}is a probability measure; we can take $V(u)=\u222bTu(\theta )4d\theta /(2\pi )$ and

*W*to be Brownian loop measure. Here

*μ*

_{λ}is Gibbs measure (1.5) with the inverse temperature

*β*, but we prefer to work with

*λ*= −

*β*> 0 so that the convexity statements are easier to interpret.

*Under the family of Gibbs measures (1.5) associated with the* *nonlinear Schrödinger equation (**NLS)* *(1.3), the random variable* *u* ↦ ⟨*u*^{⊗k}∣*T*∣*u*^{⊗k}⟩ *with* *u* ∈ (*B*_{K}, *L*^{2}, *μ*_{λ}) *and* $T\u2208L(Hs\u2297k)$ *satisfies a Gaussian concentration of measure (3.6), the mean is a Lipschitz continuous function of* *β**, and the mean for* *β* = 0 *is a sum over partitions of* 2*k* *over even* *decompositions.*

*μ*

_{λ}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 Ref. 1 (2.16) for the 1D case and Ref. 28 for 2D and 3D.

We can write *u* = *P* + *iQ* for real variables (*P*, *Q*) and interpret ⟨*u*^{⊗k}∣*T*∣*u*^{⊗k}⟩ as a homogeneous polynomial in (*p*, *q*) of total degree 2*k*. The following result gives concentration of measure for Lipschitz functions on (*B*_{K}, *L*^{2}, *μ*_{λ}), and shows that *k*-point matrices are concentrated near to their mean value.

*For*$T\u2208L(Hs\u2297k)$

*with operator norm*‖

*T*‖

*, let*$gT:BK\u2192C$

*by*

*g*

_{T}(

*u*) = ⟨

*u*

^{⊗k}∣

*T*∣

*u*

^{⊗k}⟩.

*Then there exists*

*α*=

*α*(

*β*,

*K*) > 0

*such that*

*g*

_{T}has mean value

*g*

_{T}is Lipschitz, with

*α*=

*α*(

*K*,

*β*) > 0 such that

*φ*(0) = 1,

*r*

^{−1}log

*φ*(

*r*) → 0 as

*r*→ 0+. The differential inequality

*φ*in (3.10).□

To make full use of the previous result, one needs to know the mean $trace(G\lambda (k)T)$ as in (3.7), which depends upon the measure in (3.5). The following shows how the mean can vary with the inverse temperature *β* = −*λ*.

*For*$g:BK\u2192R$

*an*

*L*

*-Lipschitz function, the mean values of*

*g*

*with respect to the measures*

*μ*

_{λ}

*satisfy*

*where*

*α*

*is the constant in (3.9) for*

*μ*

_{a}

*, and some*

*λ*∈ (

*a*,

*b*).

*ζ*(

*λ*) is a convex function of

*λ*> 0 and by the mean value theorem, there exists

*a*<

*λ*<

*b*such that

*W*

_{1}(

*μ*

_{a},

*μ*

_{b}) be the Wasserstein transportation distance between

*μ*

_{b}and

*μ*

_{a}for the cost function $\Vert u\u2212v\Vert L2$, as in p. 34 of Ref. 34. Then by duality we have

*μ*

_{0}is a Wiener loop measure restricted to

*B*

_{k}. For any sequence $(\epsilon n)n\u2208Z\u2208{\xb11}Z$, the sequence $(\gamma n)n\u2208Z$ with

*γ*

_{n}mutually independent

*N*(0, 1) Gaussian random variables has the same distribution as the sequence $(\epsilon n\gamma )n\u2208Z$. Also, the condition $\u2211n\u2208Z\{0}\gamma n2/n2\u2264K$ does not change under this transformation. Let $u\epsilon (\theta )=\u2211j\u2032\epsilon j\zeta jeij\theta /|j|$. We therefore have

*dɛ*is the Haar probability measure on the Cantor group ${\xb11}Z$. The measure on {±} is associated with tossing a fair coin, and Haar measure is the product of such probability measures. We can therefore compute the inner integral in this expression for $G0(k)$ 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).□

*B*

_{K},

## IV. CONCENTRATION FOR METRIC MEASURE SPACES

*k*-fold stochastic integrals. This result resemble the integrability criteria of Ref. 11 which relates to a single variable. Let $H01$ be the homogeneous Sobolev space of $v\u2208L2(T;C)$ that are absolutely continuous with derivative $v\u2032\u2208L2(T;C)$ with $\u222bTv(x)dx=0$. Let $hj\u2208H01$ for

*j*= 1, …,

*k*be such that $\u2211j=1k\u222b(hj\u2032)2dx\u22641$, and consider $\Phi :BKk\u21a6Rk$ given by

*Let*

*ν*

_{K}

*be the probability measure on*$Ck$

*that is induced from*$\mu K,\beta \u2297k$

*by*Φ

*. Then there exists*

*α*

_{K}> 0

*independent of*

*k*

*such that*

*for all*$G\u2208Cc1(Ck;R)$

*. The distribution*

*ν*

_{K}

*has mean*

*x*

_{0}

*and satisfies*

*B*

_{K},

*L*

^{2},

*μ*

_{K,β}) satisfies a logarithmic Sobolev inequality with constant

*α*

_{K}> 0 by Ref. 6, and the probability space $(BKk,\u21132(L2),\mu K,\beta \u2297k)$ is a direct product of the metric probability spaces (

*B*

_{K},

*L*

^{2},

*μ*

_{K,β}), hence also satisfies a logarithmic Sobolev inequality

*α*

_{K}independent of

*k*, by Sec. 22 of Ref. 35.

*φ*(0) = 1,

*φ*′(0) = 0 and the differential inequality

*B*

_{K},

*L*

^{2},

*μ*

_{K,β}) has a tangent space associated with infinitesimal translations. Let

*H*

^{1}be the Sobolev space of $v\u2208L2(T;C)$ that are absolutely continuous with derivative $v\u2032\u2208L2(T;C)$; let $H\u22121=(H1)*$ be the linear topological dual space for the pairing $\u27e8v,w\u27e9\u21a6\u222bTv(x)w(x)\u0304dx/(2\pi )$ as interpreted via Fourier series. Then there is a Radonifying triple of continuous linear inclusions

*μ*

_{K}. The space

*H*

^{1}has orthonormal basis $(hn)n=\u2212\u221e\u221e=(ein\theta /n2+1)n=\u2212\u221e\u221e$ and the covariance matrix of Wiener loop is $R0=diag[1/(1+n2)]n=\u2212\u221e\u221e$ with respect to this basis. By Cauchy–Schwarz, we have

*ɛ*> 0. By such simple estimates, one can deduce that there exists, for each

*β*and

*K*> 0, a self-adjoint, nonnegative and trace class operator

*R*such that

*H*

^{1}. This is essentially $G\u2212\beta (1)$, 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* ↦ *u* + *v* for *v* ∈ *H*^{1}; their results extends to Gibbs measure with some modifications.

*p*,

*q*∈

*H*

^{1}, the linear transformation

*P*+

*iQ*↦

*P*+

*p*+

*i*(

*Q*+

*q*). Cameron and Martin proved that free Wiener measure (

*β*= 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. The total space of the tangent bundle to the $K$ sphere in

*L*

^{2}is

*H*

^{1}. With this in mind, we make a polar decomposition

*P*+

*iQ*=

*κe*

^{iσ}with $\kappa =P2+Q2$ and consider $\tau =\u2202\sigma \u2202x$.

*For*

*p*,

*q*∈

*H*

^{1}

*the functional*

*is a Lipschitz functional of*

*P*+

*iQ*=

*κe*

^{iσ}

*such that*

*P*+

*iQ*↦

*κ*is 1-Lipschitz with

*κ*

^{2}‖Hess

*σ*‖ is bounded. We have

*L*

^{2}with norm Λ where

*ν*

_{K}, we deduce the stated inequality.□

*M*,

*d*,

*μ*) satisfies

*T*

_{2}(

*α*) if

*ν*that are of finite relative entropy with respect to

*μ*. The notation credits Talagrand, who developed the theory of such transportation inequalities. Otto and Villani showed that

*LSI*(

*α*) implies

*T*

_{2}(

*α*); see Refs. 34 and 35.

*Let*(

*M*,

*d*,

*μ*)

*be a metric probability space that satisfies*

*T*

_{2}(

*α*)

*; let*(

*M*

^{N},

*ℓ*

^{2}(

*d*),

*μ*

^{⊗N})

*be the direct product metric probability space. Let*$LN\xi =N\u22121\u2211j=1N\delta \xi j$

*be the empirical distribution for*$\xi =(\xi j)j=1N\u2208MN$

*where*

*ξ*

_{j}

*distributed as*

*μ*

*. Then the concentration inequality holds*

*for*

*p*= 1, 2.

*M*×

*M*given by

*M*,

*d*,

*μ*) satisfies

*T*

_{2}(

*α*). Then we take

*N*independent samples

*ξ*

_{1}, …,

*ξ*

_{N}, each distributed as

*μ*so they have joint distribution

*μ*

^{⊗N}on [

*M*

^{N},

*ℓ*

^{2}(

*d*)], where by independence

^{5}Theorem 1.2, [

*M*

^{N},

*ℓ*

^{2}(

*d*),

*μ*

^{⊗N}] also satisfies

*T*

_{2}(

*α*). By forming the empirical distribution, we obtain a map

*L*

_{N}: (

*M*

^{N},

*ℓ*

^{2}(

*d*)) → (Prob

*M*,

*W*

_{2}). Then $\phi (\xi )=NWp(LN\xi ,\mu )$ is 1-Lipschitz $(MN,\u21132(d))\u2192R$, since by the triangle inequality and (4.21),

*φ*satisfies the concentration inequality

Theorem IV.3 gives a metric version of Sanov’s theorem on the empirical distribution; see p. 70 of Ref. 15. There are related results in Bolley’s thesis.^{7} By Ref. 13, Theorem IV.3 applies to Haar probability measure on *SO*(3) and normalized area measure on $S2$, as is relevant in Sec. VII below. However, to ensure that $EW2(LN,\mu )\u21920$ as *N* → *∞*, it is convenient to reduce to one-dimensional distributions, where we use the following integral formula. For distributions *μ* and *ν* on $R$ with cumulative distribution functions *F* and *G*, we write *W*_{p}(*μ*, *ν*) = *W*_{p}(*F*, *G*).

*Let*

*ξ*

_{1}

*be a real random variable with finite fourth moment, and cumulative distribution function*

*F*

*. Let*

*ξ*

_{1}, …,

*ξ*

_{N}

*be mutually independent copies of*

*ξ*

_{1}

*giving an empirical measure*$LN\xi =N\u22121\u2211j=1N\delta \xi j$

*with cumulative distribution function*$FN\xi (t)$

*. Then*

*H*be Heaviside’s unit step function; then $FN\xi (t)=N\u22121\u2211j=1NH(t\u2212\xi j)$, so $N(FN(t)\u2212F(t))$ is a sum of mutually independent and bounded random variables with mean zero. Also, as in the weak law of large numbers, we have

*N*.□

*Suppose that*

*ξ*

_{1}

*has distribution*

*μ*

*on*$S2$

*where*

*μ*

*is absolutely continuous with respect to the normalized area measure*

*ν*

_{1}

*, and*

*dμ*=

*fdν*

_{1}

*where*

*f*

*is bounded with*‖

*f*‖

_{∞}≤

*M*

*. Let*

*ξ*

_{j}

*be mutually independent copies of*

*ξ*

_{1}

*, and let*$LN\xi $

*be the empirical measure from*

*N*

*samples. Then*

*g*has $\u222bS2g(x)\nu 1(dx)=0$; then

*g*is bounded with ‖

*g*‖

_{∞}≤

*π*. Given

*δ*> 0, by considering squares for coordinates in longitude and colatitude, we choose disjoint and connected subsets

*E*

_{ℓ}with diameter diam(

*E*

_{ℓ}) ≤

*δ*and

*ν*

_{1}(

*E*

_{ℓ}) ≤

*δ*

^{2}and

*μ*(

*E*

_{ℓ}) ≤

*Mν*

_{1}(

*E*) such that $\u222a\u2113E\u2113=S2$. We can arrange that there are

*S*

_{δ}such sets

*E*

_{ℓ}, where

*S*

_{δ}≤

*C*/

*δ*

^{2}. Let $F$ be the

*σ*-algebra that is generated by the

*E*

_{ℓ}, take conditional expectations in

*L*

^{2}(

*ν*

_{1}), and observe that

*N*

^{−1}

*μ*(

*E*

_{ℓ})(1 −

*μ*(

*E*

_{ℓ})), so

*δ*=

*N*

^{−1/4}we make both (4.26) and (4.29) small, which gives the stated result.□

Consider the discrete metric *δ* on [0, 1], and observe that $IA$ gives a 1-Lipschitz function on [0, 1] for all open *A* ⊆ [0, 1]. Then we have $\u222bIA(x)(d\mu (x)\u2212d\nu (x))=\mu (A)\u2212\nu (A)$, so by maximizing over *A* we obtain the total variation norm ‖*μ* − *ν*‖_{var}. With *μ* a continuous measure and *ν* a purely discrete measure, such as an empirical measure, we have ‖*μ* − *ν*‖_{var} = 1. The Propositions IV.4 and IV.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}, *μ*_{K,β}). To make this idea precise, we recall some definitions from Ref. 33.

(Convergence of metric measure spaces).

- For
*M*a nonempty set, a pseudometric is a function*δ*:*M*→ [0,*∞*] such thatthen ((4.30)$\delta (x,y)=\delta (y,x),\delta (x,x)=0,\delta (x,z)\u2264\delta (x,y)+\delta (y,z)(x,y,z\u2208M);$*M*,*δ*) is a pseudometric space. Given pseudo metric spaces (

*M*_{1},*δ*_{1}) and (*M*,*δ*_{2}), a coupling is a pseudo metric*δ*:*M*→ [0,*∞*] where*M*=*M*_{1}⊔*M*_{2}such that*δ*∣*M*_{1}×*M*_{1}=*δ*_{1}and*δ*∣*M*_{2}×*M*_{2}=*δ*_{2}.- Suppose that $M\u03021=(M1,\delta 1,\mu 1)$ and $M\u03022=(M2,\delta 2,\mu 2)$ are complete separable metric spaces endowed with probability measures. Consider a coupling (
*M*,*δ*) and a probability measure*π*on*M*_{1}×*M*_{2}with marginals*π*_{1}=*μ*_{1}and*π*_{2}=*μ*_{2}. Then the*L*^{2}distance between $M\u03021$ and $M\u03022$ is(4.31)$DL2(M\u03021,M\u03022)=inf\delta ,\pi \u222bM\xd7M\delta (x,y)2\pi (dxdy)1/2$

*D*

_{n}be the Dirichlet projection taking $\u2211k=\u2212\u221e\u221e(ak+ibk)eik\theta $ to $\u2211k=\u2212nn(ak+ibk)eik\theta $. Following,

^{10}we truncate the random Fourier series of $u=P+iQ=\u2211k=\u2212\u221e\u221e(ak+ibk)eik\theta $ to $un=Pn+iQn=\u2211k=\u2212nn(ak+ibk)eik\theta $ and correspondingly modify the Hamiltonian to

*W*(

*du*

_{n}) is the finite dimensional projection of Wiener loop measure and is defined in terms of the Fourier modes as

*u*(

*x*,

*t*) ↦

*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*

_{∞}=

*B*

_{K}and

*M*

_{1},

*ℓ*

^{2}) ⊂ (

*M*

_{2},

*ℓ*

^{2}) ⊂ ⋯ ⊂ (

*M*

_{∞},

*ℓ*

^{2}) defined by adding zeros at the start and end of the sequences, which gives a sequence of isometric embeddings for the

*ℓ*

^{2}metric on sequences. When we identify $(aj,bj)j=\u2212nn$ with $\u2211j=\u2212nn(aj+ibj)eij\theta $, then we have a corresponding embedding for the

*L*

^{2}metric.

Here $(Mn,L2,\mu K,\beta (n))$ is a finite-dimensional manifold and a metric probability space. We now show that these spaces converge to (*M*_{∞}, *L*^{2}, *μ*_{K,β}) as *n* → *∞*.

*Suppose that*0 < −*βK*< 3/(14*π*^{2})*. Then*$M\u0302n=(Mn,L2,\mu K,\beta (n))$*has*(4.37)$DL2(M\u0302n,M\u0302\u221e)\u21920(n\u2192\u221e).$*The measures*$\mu K,\beta (n)$*converge in total variation norm to**μ*_{K,β}*as**n*→*∞**.*

This is proved in Theorem 3.2 of Ref. 4; see also Example 3.8 of Ref. 33. Let

*W*_{2}(*μ*^{(n)},*μ*) be the Wasserstein transportation distance between free Brownian loop measure*μ*and the pushforward of*μ*under the Dirichlet projection,*μ*^{(n)}=*D*_{n}*♯μ*, for the cost function $\Vert u\u2212v\Vert L22$.The key point is(4.38)$W2(\mu (n),\mu )2\u2264\u222b\Vert Dnu\u2212u\Vert L22\mu (du)=E\u2211k:|k|>n|\gamma k|2k2=O1n(n\u2192\u221e).$- The measures $\mu K,\beta (n)$ converge in total variation norm to
*μ*_{K}, by an observation of McKean^{30}in his step 7. By Riesz’s theorem, there exists*c*_{4}> 0 such that $\u222bT|Dnu|4d\theta \u2264c4\u222bT|u|4d\theta $, and by Ref. 26 the integralis finite, so we can use the integrand as a dominating function to show(4.39)$\u222bBK\u2061exp\lambda c4\u222bT|u(\theta )|4d\theta W(du)$□(4.40)$\u222bBKexp\lambda \u222bT|Dnu(\theta )|4d\theta \u2212exp\lambda \u222bT|u(\theta )|4d\theta W(du)\u21920(n\u2192\u221e).$

*Let*(

*M*

_{n}⊔

*M*

_{∞},

*δ*

_{n})

*be a coupling of*(

*M*

_{n},

*L*

^{2})

*and*

*M*

_{∞},

*L*

^{2})

*, and let*$\phi :(Mn\u2294M\u221e,\delta n)\u2192R$

*be a Lipschitz function. Then*

*δ*

_{n}on

*M*

_{n}∪

*M*

_{∞}that restricts to the

*L*

^{2}metric on

*M*

_{n}and

*M*

_{∞}, and apply (4.42) to Lipschitz functions $\phi :(Mn\u2294M\u221e,\delta n)\u2192R$. We can regard

*M*

_{n}×

*M*

_{∞}as a subset of

*M*×

*M*= (

*M*

_{n}⊔

*M*

_{∞}) × (

*M*

_{n}⊔

*M*

_{∞}). Note that for a Lipschitz function $\phi :M\u2192R$ such that |

*φ*(

*x*) −

*φ*(

*y*)| ≤

*δ*(

*x*,

*y*) for all

*x*,

*y*∈

*M*, we have

For example, with $u=\u2211n=\u2212\u221e\u221e(ak+ibk)eik\theta $ we introduce $Dnu=\u2211k=\u2212nn(ak+ibk)eik\theta $; then $\phi (u)=\Vert Dnu\Vert L2$ and $\psi (u)=\Vert u\u2212Dnu\Vert L2$ give Lipschitz functions $\phi ,\psi :(BK,L2)\u2192R$.

*For* 0 < *γ* < 1/16 *and fixed* 0 < *t* < *t*_{0}*, the map* *x* ↦ *u*(*x*, *t*) ∈ *L*^{4} *is* *γ**-Hölder continuous, so that* $sup{\Vert u(x+h,t)\u2212u(x,t)\Vert Lx4/|h|\gamma );h\u22600}$ *is almost surely* *finite.*

*t*<

*t*

_{0}, we have

*C*=

*C*(

*t*

_{0}) such that

*γ*-Hölder continuous for 0 <

*γ*< 1/16 by the Kolmogorov–$C\u030c$entsov theorem, as Ref. 23. To obtain (4.43), let

*J*

_{3/8}(

*x*) =

*∑*′

*e*

^{ikx}/|

*k*|

^{3/8}so that

*J*

_{3/8}(

*x*)|

*x*|

^{5/8}is bounded on (−

*π*,

*π*) and

*J*

_{3/8}∈

*L*

^{4/3}(−

*π*,

*π*). Then by Young’s inequality for convolutions, with |

*D*|:

*e*

^{inx}↦ |

*n*|

*e*

^{inx}we have

*C*(

*t*

_{0}) such that

*x*↦

*u*(

*x*,

*t*) is 1/4-Hölder continuous along solutions in the support of the Gibbs measure.□

## V. HASIMOTO TRANSFORM

^{20}transform, which associates with a solution

*u*∈

*C*

^{2}of (1.3) a space curve in $R3$ with moving frame {

*T*,

*N*,

*B*}; Hasimoto considered the case

*β*= −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 =\Vert \u2202T\u2202x\Vert $. We have a polar decomposition

*u*=

*κe*

^{iσ}where $\sigma (x,t)=\u222b0x\tau (y,t)dy$ and

*τ*is the torsion. Then the Serret–Frenet formula is

*X*= [

*T*;

*N*;

*B*] ∈

*SO*(3), and Ω

_{1}(

*x*,

*t*) the matrix in (5.1). When $\Omega 1(\u22c5,t)\u2208C(T;so(3))$, the solution

*X*(·,

*t*) ∈

*C*([0, 2

*π*];

*SO*(3)) to (5.1) is 2

*π*periodic up to a multiplicative monodromy factor

*U*(

*t*) ∈

*SO*(3) such that

*X*(

*x*+ 2

*π*,

*t*) =

*X*(

*x*,

*t*)

*U*(

*t*).

_{2}denote the matrix in Eq. (5.2). For a pair of coupled ordinary differential equations (ODE)

*dX*/

*dx*− Ω

_{1}

*X*= 0 and

*dX*/

*dt*− Ω

_{2}

*X*= 0, the corresponding Lax pair is

*(Hasimoto). If* *u* *is a* *C*^{2} *function that satisfies the nonlinear Schrödinger equation, then the coupled pair of differential equations is consistent in the sense that there exists a local solution of the pair of ODE, and there exists a local solution of Lax* *pair.*

*X*∈

*SO*(3) evolves along the solution

*P*+

*iQ*∈

*B*

_{K}of NLS, and we can regard

*d*/

*dx*− Ω

_{1}and

*d*/

*dt*− Ω

_{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

*Let*

*−H*_{2}*is convergent almost surely and is invariant under the flow associated with**NLS**,*- −
*H*_{2}*represents the area that is enclosed by the contour*{*u*(*x*):*x*∈ [0, 2*π*]}*in the complex plane, and*(5.7)$H2=12\u222bT\kappa 2\tau dx;$ $H22\u22644\u22121H1H3$

*for**β*> 0,*where**H*_{1}*is given in Eq. (5.5) and**H*_{3}*is given in Eq. (1.1).*

- The invariance of
*H*_{2}was noted in Ref. 31 and can be proved by differentiating through the integral sign and using the canonical equations. We have a serieswhich converges almost surely. This follows since$\u222bTu\u0304(\theta ,t)\u2202u\u2202\theta (\theta ,t)d\theta 2\pi =limN\u2192\u221e\u2211j=\u2212NNu\u0302(j)\u0304iju\u0302(j)$where the final integral involves the series(5.8)$\u222bBKsupN\u2211j=\u2212NNu\u0302(j)\u0304iju\u0302(j)p\mu K,\beta (du)\u2264\u222bBKd\mu K,\beta dW2dW1/2\u222bBKsupN\u2211j=\u2212NNu\u0302(j)\u0304iju\u0302(j)2pW(du)1/2,$which is a martingale; by Fatou’s Lemma, we have(5.9)$limN\u2192\u221e\u2211j=\u2212NNu\u0302(j)\u0304iju\u0302(j)=\u2211j=1\u221e|zj|2\u2212|z\u2212j|2j$so the series in (5.9) is marginally exponentially integrable. Hence the integrals in (5.8) converge by the(5.10)$\u222bL2\u2061exp\lambda \u2211j=1\u221e|zj|2\u2212|z\u2212j|2jdW=\u220fj=1\u221e\u222bL2\u2061exp\lambda (|zj|2\u2212|z\u2212j|2)jdW=2\pi \lambda sin2\pi \lambda 1/2(\u22121/2<\lambda <1/2),$*L*^{p}martingale maximal theorem for all 1 <*p*<*∞*. - One can write
*H*_{2}in terms of*P*+*iQ*=*κe*^{iσ}, and make a change of variables to obtainand$\kappa =\u2202(P,Q)\u2202(\kappa ,\sigma )$To interpret this as an area, we write$H2=12\u222bTP\u2032Q\u2212PQ\u2032dx=12\u222bT\kappa 2\tau dx.$*θ*∈ [0, 2*π*] for the space variable and extend functions on [0, 2*π*] 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 $D$ under the map to*P*+*iQ*, as inThis is similar to Lévy’s stochastic area, as discussed in Example 5.1 of Ref. 21.(5.11)$\u2212H2=\u222b\u222bD\u2202(P,Q)\u2202(x,y)dxdy.$ - We then havewhich is bounded in terms of other invariants, with $H22\u22644\u22121H1H3$.□$\u222bT\kappa 2\tau dx2\u2264\u222bT\kappa 2dx\u222bT\kappa 2\tau 2dx$

Bourgain

^{10}interprets*H*_{2}in terms of momentum (5.70).With

*M*_{n}as in (4.36), the space $C\u221e(Mn;R)$ is a Poisson algebra for the bracket ${f,g}=\u2211j=\u2212nn\u2202(f,g)\u2202(aj,bj)$, and the canonical equations arise with Hamiltonian $H3(n)$ on*M*_{n}. Let*Q*be the ring of quaternions, and extend the Poisson bracket to*C*^{∞}(*M*_{n};*Q*) via {*f*⊗*X*,*g*⊗*Y*} = {*f*,*g*} ⊗*XY*. Then $(R3,\xd7)$ may be realised as $Q/RI$ and $(so(3),[\u22c5,\u22c5])\u2245(R3,\xd7)$; see Example 2.3 of Ref. 37. This Lie algebra is also the Lie algebra of*SU*(2), and there is a 2 − 1 group homomorphism*SU*(2) →*SO*(3). Hence some of the following results may be expressed in terms of*SU*(2), which is the form in which a Lax pair for the nonlinear Schrödinger equation was presented, see Refs. 36 and 38 (Subsection 8.3.2).- Suppose that $T\u2208C2([0,a]\xd7[0,b];S2)$, so that
*T*(*x*,*t*) represents the spin of the particle at (*x*,*t*) and letwhich corresponds to our (5.5). One can consider infinitesimal variations(5.12)$E(T)=\u222b0a\u2202T\u2202x(x,t)2dx,$*T*↦*T*+*T*×*V*and thereby compute $\u2202E\u2202T$. In the focusing case*β*= −1, Ding^{16}introduces a symplectic structure on the space of such maps such that the Hamiltonian flow iswhich 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 ferro-magnet. 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(5.13)$\u2202T\u2202t=T\xd7\u22022T\u2202x2$(5.14)$\u22022T\u2202x22=\u2202\kappa \u2202x2+\kappa 2\tau 2+\kappa 4=\u2202u\u2202x2+|u|4.$ - As in Ref. 17, the spacewith pointwise multiplication is a loop group, and its Lie algebra may be regarded as$L(SO(3))={g:[0,2\pi ]\u2192SO(3);gcontinuous,g(0)=g(2\pi )}$$H01(so(3))=h;[0,2\pi ]\u2192so(3);habsolutely\u2009continuous,h(0)=h(2\pi )=0,\u222b02\pi \Vert h\u2032(x)\Vert so(3)2dx<\u221e.$

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.

## VI. GIBBS MEASURE TRANSPORTED TO THE FRAMES

The compact Lie group *SO*(3) of real orthogonal matrices with determinant one is a subset of $M3\xd73(R)$, which has the scalar product ⟨*X*, *Y*⟩ = trace(*XY*^{⊤}) and associated metric *d*(*X*, *Y*) = ⟨*X* − *Y*, *X* − *Y*⟩^{1/2} such that ⟨*XU*, *YU*⟩ = ⟨*X*, *Y*⟩ and *d*(*XU*, *YU*) = *d*(*X*, *Y*) for all *U* ∈ *SO*(3) and $X,Y\u2208M3\xd73(R)$. 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* ∈ *SO*(3) consists of {Ω*X*: Ω ∈ *so*(3)}, where *so*(3) is a Lie algebra for [*x*, *y*] = *xy* − *yx*, *x*, *y* ∈ *so*(3), and the exponential map is surjective *so*(3) → *SO*(3).

*t*∈ [0, 1] is the evolving time, and

*X*∈

*SO*(3). We consider a column vector $x\u2208R3$, satisfying $dxdt=\Omega x$ which gives a velocity, and ‖

*x*‖ = 1 because Ω ∈

*so*(3). Following Otto’s interpretation

^{34}of optimal transport in the setting of partial differential equations, one constructs a weakly continuous family of probability measures, $\nu \u0303t$ on $S2$ for

*t*∈ [0, 1], which satisfy the weak continuity equation,

*ν*

_{t}on

*SO*(3). If the integral

*X*is locally bounded, then Ω

*X*is locally Lipschitz and

*ν*

_{t}is the unique solution to the weak continuity equation by Theorem 5.34 of Ref. 34. Recall that for the operator norm on $M3\xd73(R)$, $\Vert A\Vert =sup{\Vert Ay\Vert :y\u2208R3}$, where ‖

*X*‖ = 1 for all

*X*∈

*SO*(3) so ‖Ω

*X*‖ ≤ ‖Ω‖.

*X*

_{0}↦

*X*

_{t}(

*X*

_{0}) gives the dependence of the solution of (6.1) on the initial condition. The velocity field Ω

*X*is associated with a transportation plan taking $\nu t1$ to $\nu t2$ which is possibly not optimal, but does give an upper bound on the Wasserstein distance for the cost

*d*(

*X*,

*Y*)

^{2}on

*SO*(3) of

*ν*

_{t}) of probability measures is absolutely continuous, so there exists

*ℓ*∈

*L*

^{1}[0, 1] such that $W2(\nu t2,\nu t1)\u2264\u222bt1t2\u2113(t)dt$ and 1/2-Hölder continuous, so there exists

*C*> 0 such that $W2(\nu t2,\nu t1)\u2264C|t2\u2212t1|1/2$.

- If $\Omega t\u2208M3\xd73(R)$ is skew, and
*X*_{t},*Y*_{t}give solutions of the differential equationthen(6.6)$dXdt=\Omega tX,X(0)=X0;dYdt=\Omega tY,Y(0)=Y0$*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*↦*XU*. 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 $S2$.

We now consider the case in which Ω as in (5.2) is a *so*(3)-valued random variable over (*M*_{∞}, *μ*_{K,β}, *L*^{2}).

*Suppose that*Ω = Ω(

*u*(·,

*t*))

*where*

*u*(

*x*,

*t*)

*is a solution of NLS and that*

*converges. Then for almost all*

*u*

*with respect to*

*μ*

_{K,β}

*, there exists a flow*(

*ν*

_{t}(

*dX*;

*u*))

*of probability measures on*

*SO*(3).

*u*of NLS determines Ω so that the associated ODE (6.1) transports the initial distribution of

*X*

_{0}∈

*SO*(3) to a probability measure on

*SO*(3); then we average over the

*u*with respect to

*μ*

_{K}(

*du*). This Gibbs measure is invariant under the NLS flow, so by Fubini’s theorem

*u*, and we can invoke Theorem 23.9 of Ref. 35.□

*M*

_{n}of (4.36) and solutions $un=\kappa nei\sigma n$, the modified Hasimoto differential equations are

*X*

^{(n)}(

*x*,

*t*) ∈

*SO*(3). We can interpret the solutions as elements of a fibre bundle over $(Mn,\mu K(n),L2)$ with fibres that are isomorphic to

*SO*(3).

*P*+

*iQ*=

*κe*

^{iσ}be a solution of NLS and let

*Let**P*+*iQ*=*κe*^{iσ}*be a solution of NLS with initial data in**P*(*x*, 0) +*iQ*(*x*, 0) ∈*B*_{K}∩*H*^{1}*. Then*Ω_{1}*in (6.11) gives an**so*(3)-valued vector field in*L*^{2}(*κ*^{2}(*x*,*t*)*dx*).*Let**P*+*iQ*=*κe*^{iσ}*be a solution of NLS with initial data**P*(*x*, 0) +*iQ*(*x*, 0) ∈*H*^{1}∩*B*_{K}*, and let*$Pn+iQn=\kappa nei\sigma n$*be the corresponding solution of the NLS truncated in Fourier space, giving matrix*$\Omega 1(n)$*. Let*$Xt(n)(x)$*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).*

- With $\omega =\kappa 2+\tau 2$, we havewhere the entries of $\Omega 12$ are bounded by$exp(h\Omega 1)=I+sinh\omega \omega \Omega 1+1\u2212cosh\omega \omega 2\Omega 12$
*κ*^{2}+*τ*^{2}, hencefor(6.12)$\u222bT\Vert \Omega 1(x,t)\Vert M3\xd73(R)2\kappa (x,t)2dx<\u221e$*u*∈*H*^{1}; however, there is no reason to suppose that*τ*itself is integrable with respect to*dx*. - By (5.4) and (5.5), we have $\kappa \Omega 1\u2208Lx2$ for all
*u*∈*H*^{1}. Moreover, Bourgain^{9}has shown that for initial data*P*(*x*, 0) +*iQ*(*x*, 0) =*κ*(*x*, 0)*e*^{iσ(x,0)}in*H*^{1}∩*B*_{K}, the mapis Lipschitz continuous for 0 ≤(6.13)$\kappa (x,0)ei\sigma (x,0)\u21a6\kappa (x,t)\Omega 1(x,t)\u2208L2$*t*≤*t*_{0}with Lipschitz constant depending upon*t*_{0},*K*> 0. We havewhere the right-hand side is integrable with respect to(6.14)$\Vert \kappa (x+h,t)X(x+h,t)\u2212\kappa (x,t)X(x,t)\Vert 2h2\u226421h\u222bxx+h\u2202\kappa \u2202y(y,t)dy2+21h\u222bxx+h\kappa (y,t)\Vert \Omega 1(y,t)\Vert dy2$*x*by the Hardy–Littlewood maximal inequality and (6.12). Suppose that*X*^{(n)}is a solution of (5.1). We take*τ*_{n}to be locally bounded. Then by applying Cauchy–Schwarz inequality to the integralwe deduce that$X(n)(x+h,t)\u2212X(n)(x,t)=\u222b0h\Omega 1(n)(x+s,t)X(n)(x+s,t)ds,$where the integral is finite by (6.12). Also(6.15)$\u222b[0,2\pi ]\Vert X(n)(x+s,t)\u2212X(n)(x,t)\Vert M3\xd73(R)2\kappa n(x,t)2dx\u2264h\u222b0h\u222b[0,2\pi ]\Vert \Omega 1(n)(x+s,t)\Vert M3\xd73(R)2\kappa n(x,t)2dxds$for 0 <$\u2211j=1N\Vert X(n)(xj,t)\u2212X(n)(xj\u22121,t)\Vert M3\xd73(R)2xj\u2212xj\u22121\u2264\u222bx0xN\Vert \Omega 1(n)(x,t)\Vert 2dx$*x*_{1}<*x*_{2}<^{…}<*x*_{N}< 2*π*. We haveso for $Z\u2208C\u221e([0,2\pi ];M3\xd73(R))$ and the inner product on $M3\xd73(R)$, we have(6.16)$\u2202\u2202x\kappa nX(n)=\u2202\kappa n\u2202xX(n)+\kappa (n)\Omega 1(n)X(n)$where(6.17)$\u27e8\kappa n(2\pi )X(n)(2\pi ),Z(2\pi )\u27e9\u2212\u27e8\kappa n(0)X(n)(0),Z(0)\u27e9\u2212\u222b02\pi \kappa n(x)\u27e8X(n)(x),Z(x)\u27e9dx=\u222b02\pi \u2202\kappa n\u2202x\u27e8X(n)(x),Z(x)\u27e9dx+\u222b02\pi \u27e8X(n),\kappa n(x)\Omega 1(n)(x)\u22a4Z(x)\u27e9dx$*κ*_{n}→*κ*in*H*^{1}, so with norm convergence, we have $\u2202\kappa n\u2202x\u2192\u2202\kappa \u2202x$ in*L*^{2}, and*κ*_{n}Ω^{(n)}→*κ*Ω_{1}as*n*→*∞*, and with weak convergence in*L*^{2}, we have*X*^{(n)}→*X*, so□(6.18)$\u27e8\kappa (2\pi )X(2\pi ),Z(2\pi )\u27e9\u2212\u27e8\kappa (0)X(0),Z(0)\u27e9\u2212\u222b02\pi \kappa (x)\u27e8X(x),Z(x)\u27e9dx=\u222b02\pi \u2202\kappa \u2202x\u27e8X(x),Z(x)\u27e9dx+\u222b02\pi \u27e8X,\kappa (x)\Omega 1(x)\u22a4Z(x)\u27e9dx.$

The simulation of this differential equation computes $Xx\u2208S2$ starting with *X*_{0} = [0; 0; 1] and produces a frame {*X*_{x}, Ω_{x}*X*_{x}, *X*_{x} × Ω_{x}*X*_{x}} of orthogonal vectors. Geodesics on $S2$ are the curves such that the principal normal is parallel to the position vector, namely the great circles. For a geodesic, *X*_{x} × Ω_{x}*X*_{x} is perpendicular to the plane that contains the great circle.

*P*+

*iQ*=

*κe*

^{iσ}be a solution of NLS and let

*Let**P*+*iQ*=*κe*^{iσ}*be a solution of NLS with initial data**P*(*x*, 0) +*iQ*(*x*, 0) ∈*B*_{K}*. Then*$x\u21a6\u222b0x\Omega 2(y,t)dy$*gives a**so*(3)*-valued stochastic of finite quadratic variation on*[0, 2*π*]*almost surely with respect to**μ*_{K}(*dPdQ*)*.**Let**P*+*iQ*=*κe*^{iσ}*be a solution of NLS with initial data**P*(*x*, 0) +*iQ*(*x*, 0) ∈*H*^{1}∩*B*_{K}*, and let*$Pn+iQn=\kappa nei\sigma n$*be the corresponding solution of the NLS truncated in Fourier space, giving matrix*$\Omega 2(n)$*. Let*$Xt(n)$*be a solution of (6.10). Then*$Xt(n)$*converges in*$Lx2$*norm to**X*_{t}*as**n*→*∞**where**X*_{t}*gives a weak solution of (5.2).*

- The essential estimate is(6.20)$\u222bBK\u2211j|\kappa (xj+1,t)\u2212\kappa (xj,t)|2\mu K(du)\u2264\u2211j\u222bBK|u(xj+1,t)\u2212u(xj,t)|2\mu K(du)\u2264\u2211j\u222bBK|u(xj+1,t)\u2212u(xj,t)|4WK(du)1/2\u222bBKd\mu KdW2dW1/2\u2264C\u2211j\u222bBK|u(xj+1,t)\u2212u(xj,t)|2W(du)1/2\u2264C\u2211j(xj+1\u2212xj)\u22642\pi C.$The function
*σ*is a progressively measurable stochastic process adapted with respect to a suitable filtration, and with differential satisfying an Ito integral equation.^{18}Therefore, we can control the*κτ*term viawhich is a bounded martingale transform of Wiener loop. This formula is reminiscent of Levy’s stochastic area as in Example 5.1 of Ref. 21.(6.21)$\u222b0x(\kappa d\sigma \u22122\u22121\kappa 2\u27e8d\sigma ,d\sigma \u27e9)=\u222b0x\kappa \u2207\sigma \u22c5dPdQ=\u222b0x\u2212QdP+PdQP2+Q2$ - By (5.4) and (5.5), we have $\Omega 2\u2208Lx2$ for all
*u*∈*H*^{1}. Bourgain^{9}has shown that for initial data*P*(*x*, 0) +*iQ*(*x*, 0) =*κ*(*x*, 0)*e*^{iσ(x,0)}in*H*^{1}∩*B*_{K}, the mapis Lipschitz continuous for 0 ≤(6.22)$\kappa (x,0)ei\sigma (x,0)\u21a6\Omega 2(x,t)\u2208Lx2$*t*≤*t*_{0}with Lipschitz constant depending upon*t*_{0},*K*> 0. We havewhere the final integral is part of the Hamiltonian. With $Z\u2208C\u221e(T;M3\xd73(R))$, we have the integral equation for the pairing ⟨·, ·⟩ on $L2([0,2\pi ],M3\xd73(R))$$\u222b02\pi \Vert \Omega 2(x)\Vert 2dx\u22642\u222b02\pi \u2202\kappa \u2202x2+\kappa (x)2\tau (x)2+\kappa (x)4dx,$(6.23)$\u27e8Xt(n),Z\u27e9=\u27e8X0(n),Z\u27e9+\u222b0t\u27e8Xs(n),(\Omega s(n))\u22a4Z\u27e9ds.$Consider the variational differential equation in $L2([0,2\pi ],M3\xd73(R))$where $\Omega 2(n)(x,t)$ and $\Omega 2(m)(x,t)\u2212\Omega 2(n)(x,t)$ are skew.(6.24)$ddt(X(m)(x,t)\u2212X(n)(x,t))=\Omega 2(n)(x,t)(X(m)(x,t)\u2212X(n)(x,t))+(\Omega 2(m)(x,t)\u2212\Omega 2(n)(x,t))X(m)(x,t)$

*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

*X*

^{(m)}(0) −

*X*

^{(n)}(0) → 0 and $\Omega 2(m)(s)\u2212\Omega 2(n)(s)\u21920$ in $Lx2$ norm as

*n*,

*m*→

*∞*, so there exists $X(x,t)\u2208Lx2$ such that

*X*(

*x*,

*t*) −

*X*

^{(n)}(

*x*,

*t*) → 0 in $Lx2$ norm as

*n*→

*∞*.

Let $\Omega 2(n,un)(x,t)$ be the Fourier truncated matrix that corresponds to a solution *u*_{n} of the Fourier truncated equation *NLS*_{n}, then let $Xn,un(x,t)$ be the solution of the ODE (6.10). By Proposition VI.2, the map $un\u21a6Xn,un(\u22c5,t)$ 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 *α*(*β*, *K*)/*n*^{2}; compare (3.9).

*For each*$Z\u2208L2([0,2\pi ];M3\xd73(R))$

*, introduce the*$R$

*-valued random variable on*$(Mn,L2,\mu K(n))$

*by*

*Then the distribution**ν*^{(n)}*of**Z*_{n}*satisfies the Gaussian concentration inequality*(6.30)$\u222bMn\u2061exptZn\u2212t\u222bMnZnd\mu K(n)\mu K(n)(dun)\u2264expn2t2/\alpha (\beta ,K)(t\u2208R).$*Let*$\nu N(n)=N\u22121\u2211j=1N\delta Zn(j)$*be the empirical distribution of**N**independent copies of**Z*_{n}*. Then*$W1(\nu N(n),\nu (n))\u21920$*almost surely as**N*→*∞*.

- As with
*u*_{n}, we introduce the corresponding data for another solution*v*_{n}. As in (6.27), we haveFor given initial condition $Xn,vn)(0)=X(n,un)(0)$, and(6.31)$\Vert X(n,un)(x,t)\u2212X(n,vn)(x,t)\Vert R32\u2264et\Vert X(n,un)(0)\u2212X(n,vn)(0)\Vert R32+\u222b0tet\u2212s\Vert \Omega 2(n,un)(x,s)\u2212\Omega 2(n,vn)(x,s)\Vert so(3)2ds.$*T*> 0, we can take the supremum over*t*, then integrate this with respect to*x*and obtainso Ω(6.32)$\u222b02\pi sup0<t<T\Vert X(n,un)(x,t)\u2212X(n,vn)(x,t)\Vert R32dx\u2264eT\u222b0T\Vert \Omega 2(n,un)(x,s)\u2212\Omega 2(n,vn)(x,s)\Vert Lx22ds$^{(u)}↦*X*^{u}is a Lipschitz function $L2([0,2\pi ]\xd7[0,T],so(3))\u2192L2([0,2\pi ];L\u221e([0,T],R3))$. By Bourgain’s results, there exists*C*> 0 such thatso $un\u21a6X(n,un)$ is a Lipschitz function on $Lx2$, albeit with a constant growing with(6.33)$\Vert \Omega 2(n,un)(x,s)\u2212\Omega 2(n,vn)(x,s)\Vert Lx2\u2264C\Vert un(x,s)\u2212vn(x,s)\Vert Hx1\u2264Cn\Vert un(x,0)\u2212vn(x,0)\Vert Lx2,$*n*. Thus we can push forward the modified Gibbs measure $(Mn,L2,\mu K,\beta (n))\u2192L2([0,2\pi ];M3\xd73(R))$ so that the image measure satisfies a Gaussian concentration inequality with constant*α*(*β*,*K*)/*n*^{2}dependent upon*n*. For each $Z\u2208L2([0,2\pi ];M3\xd73(R))$, we introduce*Z*_{n}, so that where*u*_{n}↦*Z*_{n}is*Cn*-Lipschitz function from $(Mn,L2,\mu K,\beta (n))$ to $R$. The random variable*Z*_{n}therefore satisfies the Gaussian concentration inequality (6.30). - By Theorem IV.3, we can use the Borel–Cantelli Lemma to show thatwhere by Proposition IV.4, $EW1(\nu N(n),\nu (n))\u21920$ as$PW1(\nu N(n),\nu (n))\u2212EW1(\nu N(n),\nu (n))>\epsilon for\u2009infinitely\u2009manyN=0(\epsilon >0),$
*N*→*∞*.□

*M*

_{∞},

*L*

^{2},

*μ*

_{K,β}) involving measure

*π*

_{n}. For any bounded continuous $\phi :C\u2192R$ we can consider

*Let*$(\phi j)j=1\u221e$

*be a dense sequence in*$Ball(Cc(C;R))$

*and*$(Y\u2113)\u2113=1\u221e$

*a dense sequence in*

*Ball*(

*L*

^{2})

*. Then there exists a subsequence*(

*n*

_{k})

*such that*

*converges as*

*n*

_{k}→

*∞*

*for all*$j,\u2113\u2208N$.

*μ*

_{n}onto

*M*. Then $\omega =2\u22121\mu \u221e+\u2211n=1\u221e2\u2212n\u22121\mu n$ is a probability measure on

*M*, and

*μ*

_{n}is absolutely continuous with respect to

*ω*, so

*dμ*

_{n}=

*f*

_{n}

*dω*for some probability density function

*f*

_{n}∈

*L*

^{1}(

*ω*). By convergence in total variation from Lemma IV.7(ii), here exists

*f*

_{∞}∈

*L*

^{1}(

*ω*) such that

*f*

_{n}→

*f*

_{∞}in

*L*

^{1}as

*n*→

*∞*. Given a bounded sequence $(gn)n=1\u221e$ in

*L*

^{∞}(

*ω*), there exists

*g*

_{∞}∈

*L*

^{∞}(

*ω*) and a subsequence (

*n*

_{k}) such that

For *u* ∈ *M*_{∞}, we have *u*_{n} = *D*_{n}*u* ∈ *M*_{n} so that *u*_{n} → *u* in *L*^{2} norm as *n* → *∞*. It is plausible that (6.34) tends to 0 as *n* → *∞*, but we do not have a proof. Unfortunately, the constants are not sharp enough to allow us to use Proposition IV.8 to deduce *W*_{2} convergence for the distributions on *SO*(3).

## VII. 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 *β* 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 partial differential equation (PDE) with respect to the space variable *x*, while the parameter of a stochastic process in a stochastic differential equation (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.

*P*+

*iQ*=

*κe*

^{iσ}where, $\kappa =P2+Q2$ and

*σ*is such that $\tau =\u2202\sigma \u2202s$. Define $\sigma \epsilon (P,Q)\u2254tan\u22121(PQP2+\epsilon 2)$ as the regularised Itô integral of

*τ*. The Itô differential can be written as

*P*and

*Q*are each a Brownian bridge with period

*T*= 2

*π*, 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

*π*)/2

*π*and likewise for

*Q*. Equation (7.1) is now written as a standard Itô SDE,

**A**,

**B**and

**C**are defined as in Eq. (7.2). The resulting stochastic process

*X*

_{s}∈

*SO*(3) is then used to rotate the unit vector

*y*

_{0}= [0, 0, 1]

^{⊤}on $S2$ to

*y*

_{s}=

*X*

_{s}

*y*

_{0}, the third column of

*X*

_{s}. The sample paths of this process can be described by construction of a frame ${ys,ys\u2032,ys\xd7ys\u2032}$. In order to simulate this SDE, we make use of a numerical scheme for matrix SDEs in

*SO*(3) developed by Marjanovic and Solo.

^{29}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, and the code used to simulate a sample path is available.

^{25}The sample paths start off on the great circle perpendicular to the

*y*-axis, and so have constant binormal $ys\xd7ys\u2032$. 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.

*y*

_{s}is derived from the solution to Eq. (7.3) and takes values in $R3$. Let $y\u0302s,h$ 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 Itô processes as the step size

*h*→ 0,

*ɛ*> 0 (See Piggott and Solo

^{32}). A value of

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

Since the sample paths are constrained to $S2$ the points *y*_{s} can be specified in spherical coordinates of longitude *θ*_{s} ∈ [−*π*, *π*) and colatitude *ϕ*_{s} ∈ [0, *π*]. Figure 2 demonstrates the empirical joint distribution of *θ*_{s} and *ϕ*_{s} for two different values of *s*. As can be observed, the distribution of (*θ*_{s}, *ϕ*_{s}) varies with *s*. We hypothesise that the angles *θ*_{s} and *ϕ*_{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.

### A. Wasserstein distance between measures on $S2$

*W*

_{1}(

*ν*

_{1},

*ν*

_{2}) between probability measures

*ν*

_{1}and

*ν*

_{2}on $S2$, which are absolutely continuous with respect to area and have disintegrations

*f*

_{j}(

*j*= 1, 2) are probability density functions on [−

*π*,

*π*] that give the marginal distributions of

*ν*

_{j}in the longitude

*θ*variable, and

*g*

_{j}in the colatitude variable. Let

*F*

_{j}be the cumulative distribution function of

*f*

_{j}(

*θ*)

*dθ*and

*G*

_{j}be the cumulative distribution function of

*g*

_{j}(

*ϕ*)sin

*ϕdϕ*. We measure

*W*

_{1}(

*ν*

_{1},

*ν*

_{2}) in terms of one-dimensional distributions. Given distributions on $R$ 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

*ψ*: [−

*π*,

*π*] → [−

*π*,

*π*] be an increasing function that induces

*f*

_{2}(

*θ*)

*dθ*from

*f*

_{1}(

*θ*)

*dθ*; then

*f*

_{1}(

*θ*) = 1/(2

*π*) and

*g*

_{1}(

*ϕ*) = 1/2, we have a product measure

*ν*

_{1}(

*dθdϕ*) = (4

*π*)

^{−1}sin

*ϕdϕdθ*giving normalized surface area on the sphere. Then

*F*

_{1}(

*θ*) = (

*θ*+

*π*)/(2

*π*) and

*F*

_{2}(

*ψ*(

*θ*)) = (

*θ*+

*π*)/(2

*π*), so

*ψ*(2

*π*(

*τ*− 1/2)) for

*τ*∈ [0, 1] gives the inverse function of

*F*

_{2}. We deduce that

*G*conditional distributions, namely the dependence of the colatitude distribution on longitude.

For each *s* ∈ [0, 10], let $F\theta s$ and $G\varphi s$ be the marginal cumulative distribution functions (CDFs) of *θ*_{s} and *ϕ*_{s} respectively. For $N\u2208N$, denote by $FN\theta s$ and $GN\varphi s$ the empirical CDFs of *θ*_{s} and *ϕ*_{s}. We generate empirical CDFs $FN\theta s$ and $GN\varphi s$ with *s* = 0.3, 0.6, 0.9, …, 6.0, and *N* = 10^{5}. Figure 3 demonstrates that $W1(F1,FN\theta s)$ and $W1(G1,GN\varphi s)$, each decreases with increasing *s*. As a consequence of Theorem IV.3 and Proposition IV.4 for *N* = 10^{5} with probability at least 0.99 it holds that $W1(FN\theta s,F\theta s)\u22640.025$ and $W1(GN\varphi s,G\varphi s)\u22640.018$. Thus, we observe that $F\theta s$ converges to *F*_{1} and $G\varphi s$ converges to *G*_{1}.

### B. 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 *θ*_{s} and *ϕ*_{s}. In order to account for multiple testing, we set the significance level of each test to 0.000 45, leading to an overall level of 0.01. First, we generate sample paths to obtain *N* = 10^{5} realisations of (*θ*_{s}, *ϕ*_{s}) for each value of *s* = 0.3, 0.6, 0.9, …, 6.0. For each *s*, we test the null hypothesis *H*_{0,s} that the angles *θ*_{s} and *ϕ*_{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;^{2,19} the implementation is due to Jitkrittum *et al*.^{22} It is observed that while the null hypothesis is rejected for *s* = 0.3, …, 2.1, the test is unable to reject *H*_{0,s} from *s* = 2.4, …, 6.0 at (an overall) significance level 0.01.

We run two Kolmogorov–Smirnov goodness-of-fit tests for *s* = 10 as follows. The first tests the null hypothesis $H0\theta s$ that *θ*_{s} is distributed according to *F*_{1} against the alternative that it is not; the second tests the null hypothesis $H0\varphi s$ that *ϕ*_{s} is distributed according to *G*_{1} against the alternative that it is not. At significance level 0.01, the tests are unable to reject the null hypotheses $H0\theta s$ and $H0\varphi s$.

## ACKNOWLEDGMENTS

The authors are grateful to the referee for insightful suggestions which improved the paper. Also, the authors thank Nadia Mazza for her helpful remarks on combinatorics. M.K.S. is funded by a Faculty of Science and Technology studentship, Lancaster University.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Gordon Blower**: Investigation (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). **Azadeh Khaleghi**: Investigation (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). **Moe Kuchemann-Scales**: Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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