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

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, 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; 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.


Consider the Hamiltonian
on T = R/2πZ which gives the canonical equations of motion 0 1 2) so u = P + iQ satisfies the nonlinear Schrödinger equation (1.3) When γ = 2, we have the cubic nonlinear Schrödinger equation.The spatial variable is x ∈ T, 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.
Date: 13th May 2022.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 = u = P + iQ : P, Q ∈ L 2 (T; R) : (1.4) The Gibbs measure on B K for this micro-canonical ensemble is where W (du) is Wiener loop measure, and Z K (β) is a normalizing constant.Lebowitz, Rose and Speer [LRS] proved existence of such an invariant measure, so that for all K > 0 and β ∈ R there exists Z K (β) > 0 such that µ K,β is a Radon probability measure on ).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.
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 finitedimensional Hamiltonian systems.
The focusing case for spatial variable x ∈ 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 ∈ 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 µ 0 a centered Gaussian measure on H, we express a specific integral 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 ⊗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 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 → u(x, t) is γ-Hölder continuous [0, 2π] → L 4 for 0 < γ < 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 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).

Tensor products and k-point density matrices for Gaussian measure
Let H be a separable complex Hilbert space, with inner product • | • which is linear in the second argument.We identify the injective tensor product H ⊗H with the algebra L(H) of bounded linear operators on H and the projective tensor product H ⊗H with the ideal L 1 (H) of trace class operators on H.For H = L 2 , the identification is Let A ∈ L 1 (H) be self-adjoint such that 0 ≤ 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 (ϕ j ) ∞ j=1 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 (γ j ) ∞ j=1 and the vector so that µ 0 is the distribution of u on H, as one easily checks by computing the expectation Hence A is the mean of rank-one tensors with respect to Gaussian measure The 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 An element of H ⊗k ⊗H ⊗k determines a linear operator L(H ⊗k ), commonly referred to as a matrix, so J (k) ∈ (L 2 ) s⊗k ⊗(L 2 ) ⊗k gives a k-point density matrix, or equivalently a trace class operator J (k) ∈ L 1 (H s⊗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 E(γ j γ γ m γ n ) = 0 if one of the indices j, , 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 and we combine the second and fourth of these to obtain 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 For k ∈ N, let Π k be the set of all partitions of k so that π ∈ Π k may be expressed as π and a n-element subset {j 1 , . . ., j n } of N, there is a symmetric tensor where the sum is over all the permutations 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 .
We express u as in (2.1) and consider the expansion in terms of this orthonormal basis of H s⊗k , and look for the terms that do not vanish after integration with respect to µ 0 .
so that 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.
Remark 2.2.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 [K] Theorem 4 there is a bijection between symmetric matrices A that have entries in N ∪ {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 ∪ {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 .
For notational convenience, we also regard ϕ ⊗0 j as a factor which may be omitted in tensor products.Then given such a triple (π, λ, ρ) and an n-subset {j 1 , . . ., j n } of N, where gives a nonzero summand in J (k) .
Conversely, let (λ, ρ) ∈ Π 2 k 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 + j even for all j.
Then we introduce 2k j = j + r j and after a further reordering write k and we have π ∈ Π k as above.Given a n-subset {j 1 , . . ., j n } of N, we take 2k m copies of j m and split them as 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.
Proposition 2.3.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 (λ, ρ) ∈ Π 2 k where λ and ρ have equal numbers of odd rows.

Concentration of k-point matrices for Gibbs measure
Let H = L 2 (T; R) and let (γ j ) j∈Z be mutually independent Gaussian N (0, 1) random variables, where z j = (γ j + iγ −j )/ √ 2) and z −j = (γ j − iγ −j )/ √ 2 for j ∈ N. Then we take Brownian loop in the style so that Wiener loop W (du) is the distribution of u ∈ H.By orthogonality, we have so that u ∈ B K/2π if and only if j γ 2 j /j 2 ≤ K. Chebyshev's inequality and independence, we have The low Fourier modes are the predominant terms since one has the estimate which also follows from Chebyshev's inequality and independence. Let so that µ λ is a probability measure; we can take V (u) = T u(θ) 4 dθ/(2π) 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.
Theorem 3.1.Under the family of Gibbs measures (1.5) associated with N LS (1.3), the random variable u → u ⊗k | T | u ⊗k with u ∈ (B K , L 2 , µ λ ) and T ∈ L(H s⊗k ) satisfies a Gaussian concentration of measure, the mean is a Lipschitz continuous function of β, and the mean for β = 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 where 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 [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 u ⊗k | T | u ⊗k 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 , µ λ ), and shows that k-point matrices are concentrated near to their mean value.
Proposition 3.2.For T ∈ L(H s⊗k ) with operator norm T , let g T : (3.6) Proof.Here g T has mean value By the logarithmic Sobolev inequality Theorem 1.2(iv) of [BBD], there exists α = α(K, β) > 0 such that for all continuously differentiable f : B K → R. In particular, we choose and we deduce that the moment generating function satisfies ϕ(0) = 1, ϕ (0) = 1 and the differential inequality One can conclude the proof by a standard application of Chebyshev's inequality to the integral for ϕ in (3.10).
To make full use of the previous result, one needs to know the mean trace(G λ 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.
Proposition 3.3.For g : B K → R 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).
Proof.We observe that log ζ(λ) is a convex function of λ > 0 and by the mean value theorem, there exists a < λ < b such that log Let W 1 (µ a , µ b ) be the Wasserstein transportation distance between µ b and µ a for the cost function u − v L 2 , as in page 34 of [V1].Then by duality we have 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 in the style of Talagrand, where the relative entropy is where the final step follows from (3.13).The stated result follows on combining these inequalities.
Proposition 3.4.The integral G (k) 0 from (3.5) is the sum over the terms (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 (λ, ρ) ∈ Π 2 k where λ and ρ have equal numbers of odd rows.
Proof.The measure µ 0 is a Wiener loop measure restricted to B k .For any sequence (ε n ) n∈Z ∈ {±1} Z , the sequence (γ n ) n∈Z with γ n mutually independent N (0, 1) Gaussian random variables has the same distribution as the sequence (ε n γ) n∈Z .Also, the con- where dε is the Haar probability measure on the Cantor group {±1} Z .We can therefore compute the inner integral in this expression for G (k) 0 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 where the sets are independent under the Gaussian measure dP, so we have a substitute for (2.7) γ 2k dP, (3.17) where P(B K ) satisfies (3.3) and there is an approximate formula

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 ∈ L 2 (T; C) that are absolutely continuous with deriv- The following result describes the distribution of this C k -valued random variable.
Proposition 4.1.Let ν K be the probability measure on C k that is induced from µ ⊗k K,β by Φ.Then there exists α K > 0 independent of k such that ) is Lipschitz with constant one.Each metric probability space (B K , L 2 , µ K,β ) satisfies a logarithmic Sobolev inequality with constant α K > 0 by [BBD], and the probability space ) is a direct product of the metric probability spaces (B K , L 2 , µ K,β ), hence also satisfies a logarithmic Sobolev inequality with constant α K independent of k, by section 22 of [V2].
We introduce which satisfies ϕ(0) = 1, ϕ (0) = 0 and the differential inequality follows from (4.2).This gives which we integrate against e − y 2 /2 , where y ∈ C k = R 2k , to obtain (4.9) The probability space (B K , L 2 , µ K,β ) has a tangent space associated with infinitesimal translations.Let H 1 be the Sobolev space of v ∈ L 2 (T; C) that are absolutely continuous ) * be the linear topological dual space for the pairing v, w → T v(x)w(x)dx/(2π) as interpreted via Fourier series.Then there is a Radonifying triple of continuous linear inclusions associated with the Gibbs measure µ K .The space H 1 has orthonormal basis n=−∞ with respect to this basis.By 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 which gives the covariance matrix of the Gibbs measure on H 1 .This is essentially G (1) −β , 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.
We momentarily suppress the dependence of functions upon time, and consider for 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, so we can regard H 1 as fibres of a tangent space to (B K , L 2 , µ K,β ).We make a polar decomposition P +iQ = κe iσ with κ = P 2 + Q 2 and consider τ = ∂σ ∂x .
Proposition 4.2.For p, q ∈ H 1 the functional is a Lipschitz functional of P + iQ = κe iσ such that Proof.Note that P + iQ → κ is 1-Lipschitz with σκ, κ∇σ = 1 Also κ 2 Hess σ is bounded.We have which is bounded on L 2 with norm Λ where By the concentration of measure theorem for ν K , we deduce the stated inequality.
for all probability measures 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 [V1] and [V2] Theorem 4.3.Let (M, d, µ) be a metric probability space that satisfies T 2 (α); let (M N , 2 (d), µ ⊗N ) be the direct product metric probability space.Let L ξ N = N −1 N j=1 δ ξ j be the empirical distribution for ξ = (ξ j ) N j=1 ∈ M N where ξ j distributed as µ.Then the concentration inequality holds Proof.The map between metric spaces associated with the empirical distribution is 1/ √ N -Lipschitz.Let (x j ) N j=1 , (y j ) N j=1 ∈ M N and consider the probability measure on M × M given by Suppose that (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 , 2 (d)), where by independence [BB] Theorem 1.2, (M N , 2 (d), µ ⊗N ) also satisfies T 2 (α).By forming the empirical distribution, we obtain a map → R, since by the triangle inequality and (4.20), hence ϕ satisfies the concentration inequality 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 S 2 , as is relevant in section 7 below.However, to ensure that EW 2 (L N , µ) → 0 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 Proposition 4.4.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 Proof.Let H be Heaviside's unit step function; then ) is a sum of mutually independent and bounded random variables with mean zero.Also, as in the weak law of large numbers, we have where the integral is finite by Chebyshev's inequality since Eξ 4 is finite.Compare [BGV].
Proposition 4.5.Suppose that ξ 1 has distribution µ on S 2 where µ is absolutely continuous with respect to the normalized area measure ν 1 , and dµ = f dν 1 where f is bounded with f ∞ ≤ M .Let ξ j be mutually independent copies of ξ 1 , and let L ξ N be the empirical measure from N samples.Then Proof.Let g : S 2 → R be 1-Lipschitz, and suppose without loss of generality that g has S 2 g(x)ν 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 with diameter diam(E ) ≤ δ and ν 1 (E ) ≤ δ 2 and µ(E ) ≤ M ν 1 (E) such that ∪ E = S 2 .We can arrange that there are S δ such sets E , where S δ ≤ C/δ 2 .Let F be the σ-algebra that is generated by the E , take conditional expectations in L 2 (ν 1 ), and observe that where we have bounds and the identity so by Cauchy-Schwarz, we have where ) is a sum of independent random variables with mean zero and variance Choosing δ = N −1/4 we make both (4.26) and (4.29) small, which gives the stated result.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 [S]. (ii) Given pseudo metric spaces (M 1 , δ 1 ) and (M, δ 2 ), a coupling is a pseudo metric

Remark
(iii) Suppose that M1 = (M 1 , δ 1 , µ 1 ) and M2 = (M 2 , δ 2 , µ 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 M1 and M2 is Following [B02], we truncate the random Fourier series of and correspondingly modify the Hamiltonian to for the real canonical variables ((a k , b k )) n k=−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 = κ n e iσn , and observe that in terms of these noncanonical variables, the Hamiltonians H are invariants under the flow.The corresponding Gibbs measure is Consider the map u(x, t) → u(x+h, t) of translation in the space variable.This commutes with D n , and the Gibbs measures µ K,β are all invariant under this translation.In terms of Fourier components, we have M ∞ = B K and with the canonical inclusions of metric spaces defined by adding zeros at the start and end of the sequences, which gives a sequence of isometric embeddings for the 2 metric on sequences.When we identify (a j , b j ) n j=−n with n j=−n (a j + ib j )e ijx , then we have a corresponding embedding for the L 2 metric.
Here (M n , L 2 , µ ) 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 → ∞.
Lemma 4.7.(i) Suppose that 0 < −βK < 3/(14π 2 ).Then Mn = (M n , L 2 , µ K,β converge in total variation norm to µ K,β as n → ∞. Proof.(i) This is proved in Theorem 3.2 of [B2]; see also Example 3.8 of [S].Let The key point is K,β converge in total variation norm to µ K , by an observation of McKean in step 7 of [M].By M. Riesz's theorem, there exists c 4 > 0 such that T |D n u| 4 dθ ≤ c 4 T |u| 4 dθ, and by [LRS] the integral is finite, so we can use the integrand as a dominating function to show Proof.We can introduce a pseudo metric 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 ϕ : Proposition 4.9.For 0 < γ < 1/16 and fixed 0 < t < t 0 , the map x → u(x, t) ∈ L 4 is γ-Hölder continuous.
Proof.We prove that for 0 < t < t 0 , we have C = C(t 0 ) such that x is γ-Hölder continuous for 0 < γ < 1/16 by the Censov-Kolmogorov's theorem.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 Then by Bourgain's estimate on the solutions of NLS [B], there exists C(t 0 ) such that where By basic results about Gaussian series, the first factor on the right-hand side is bounded by the eighth power of so we obtain (4.43).Also, by rotation invariance of the Gibbs measure, we have ) is 1/4-Hölder continuous along solutions in the support of the Gibbs measure.

Hasimoto transform
We recall the Hasimoto [H] transform, which associates with a solution u ∈ C 2 of (1.3) a space curve in R 3 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 κ = ∂T ∂x .We have a polar decomposition u = κe iσ where σ(x, t) = x 0 τ (y, t)dy and τ is the torsion.Then the Serret-Frenet formula is so the frame develops along the space curve.Let X = [T ; N ; B] ∈ SO(3), and Ω 1 (x, t) the matrix in (5.1).The space with pointwise multiplication is a loop group, and its Lie algebra may be regarded as The frame also evolves with respect to time, so that with µ = − ∂σ ∂t − βκ 2 , we have (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 ∈ 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 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 (5.4) The total curvature of the space curve is which is an invariant under the flow associated with the NLS.
Proposition 5.2.Let (5.6) (i) Then −H 2 is convergent almost surely and is invariant under the flow associated with N LS, (ii) −H 2 represents the area that is enclosed by the contour {u(x) : x ∈ [0, 2π]} in the complex plane, and (5.7) 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 which converges almost surely.This follows since where the final integral involves the series lim which is a martingale; by Fatou's Lemma, we have (5.10) so the series in (5.9) is marginally exponentially integrable.Hence the integrals in (5.8) converge by the L p martingale maximal theorem for all 1 < p < ∞.
(ii) One can write H 2 in terms of P + iQ = κe iσ , and make a change of variables to obtain ) and To interpret this as an area, We write θ ∈ [0, 2π] for the space variable and extend functions on [0, 2π] 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 in (5.11)This is similar to Lévy's stochastic area, as discussed in Example 5.1 of [IW].
), so that T (x, t) represents the spin of the particle at (x, t) and let which corresponds to our 5.5.One can consider infinitesimal variations T → T + T × V and thereby compute ∂E ∂T .In the focusing case β = −1, Ding [D] introduces a symplectic structure on the space of such maps such that the Hamiltonian flow is 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 (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.

Gibbs measure transported to the frames
The compact Lie group SO(3) of real orthogonal matrices with determinant one is a subset of M 3×3 (R), which has the scalar product X, Y = trace(XY † ) and associated The Lie group SO(3) has tangent space at the identity element give by the skew symmetric matrices so(3), so the tangent space We consider the differential equation where t ∈ [0, 1] is the evolving time.Let ν t for t ∈ [0, 1] be a weakly continuous family of probability measures on SO(3), such that converges and ΩX is locally bounded so X → ΩX is locally Lipschitz, and such that the family (ν t ) satisfies the weak continuity equation The weak continuity equation is equivalent to for all f ∈ C(SO(3); R), where X 0 → 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 ΩX gives an optimal transportation plan, but it does an upper bound on the Wasserstein transportation cost Then by Theorem 23.9 of [V2], the path (ν t ) of probability measures is absolutely continuous, so there exists t 1 (t)dt and 1/2-Hölder continuous, so there exists . 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 .
(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 S 2 .
We now consider the case in which Ω is a so(3)-valued random variable over (M ∞ , µ K,β , L 2 ).
Proposition 6.2.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).
Proof.Each solution 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 converges.Hence the condition (6.2) is satisfied, for almost all u, and we can invoke Theorem 23.9 of [V2].
(ii) 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 P n + iQ n = κ n e iσn be the corresponding solution of the NLS truncated in Fourier space, giving matrix 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).
We have 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 τ n to be locally bounded.Then by applying Cauchy-Schwarz inequality to the integral where the integral is finite by (6.12). Also We have ) and the inner product on M 3×3 (R), we have where κ n → κ in H 1 , so with norm convergence, we have ∂κn ∂x → ∂κ ∂x in L 2 , and κ n Ω (n) → κΩ 1 as n → ∞, and with weak convergence in L 2 , we have X The simulation of this differential equation computes X x ∈ S 2 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 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 × Ω x X x is perpendicular to the plane that contains the great circle.
(ii) 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 P n + iQ n = κ n e iσn be the corresponding solution of the NLS truncated in Fourier space, giving matrix be a solution of 6.10).Then X x norm to X t as n → ∞ where X t gives a weak solution of (5.2).
Proof.(i) The essential estimate is Also, we can control the κτ term via which is a bounded martingale transform of Wiener loop.
We have where the final integral is part of the Hamiltonian.With Z ∈ C ∞ (T; M 3×3 (R)), we have the integral equation for the pairing where Then the variational equation has solution x (6.25) so from this differential inequality we have x du, (6.27) so we have a weak solution of the ODE.

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 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 = κe iσ where, κ = P 2 + Q 2 and σ is such that τ = ∂σ ∂s .Define σ (P, Q) := tan −1 ( P Q P 2 + 2 ) as the regularised Ito integral of τ .The Ito differential dσ can be written as dσ = f 1 (P, Q)dP + f 2 (P, Q)dQ + f 3 (P, Q)ds, where 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 dX s =AX s ds + BX s dP + CX s dQ (7.1) where, As justified above P and Q are each a Brownian bridge with period T = 2π, thus they can be expressed in terms of Brownian motions W 1 and W 2 ; that is, P (s) = W 1 (s) − sW 1 (2π)/2π and likewise for Q.Equation (7.1) is now written as a standard Ito SDE, where A, B and C are defined as in Equation ( 7.2).The resulting stochastic process 3) is then used to rotate the unit vector y 0 = [0, 0, 1] T on 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 s , y s ×y s }.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.The sample paths start off on the great circle perpendicular to the y-axis, and so have constant binormal y s × y s .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 R 3 .Let ŷs,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 → 0, (7.4) for some ε > 0 [PS].A value of ε = 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 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(N T /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 × 10 6 the algorithm takes around 1 week to run on our system.
Since the sample paths are constrained to S 2 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.
Wasserstein distance between measures on S 2 .We start by calculating the Wasserstein distance W 1 (ν 1 , ν 2 ) between probability measures ν 1 and ν 2 on S 2 , which are absolutely continuous with respect to area and have disintegrations where 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 φ dφ.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 Hence the Wasserstein distance can be bounded in terms of the cumulative distribution functions by 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.
For each s ∈ [0, 10], let F θs and G φs be the marginal CDFs of θ s and φ s respectively.For N ∈ N, denote by F θs N and G φs N the empirical CDFs of θ s and φ s .We generate empirical CDFs F θs N and G φs N with s = 0.3, 0.6, 0.9, . . ., 6.0, and N = 10 5 .Figure 3 demonstrates that W 1 (F 1 , F θs N ) and W 1 (G 1 , G φs N ), each decreases with increasing s.As a consequence

1
arXiv:2205.12868v1 [math.AP] 25 May 2022 Consider the discrete metric δ on [0, 1], and observe that I A gives a 1-Lipschitz function on [0, 1] for all open A ⊆ [0, 1].Then we have I A (x)(dµ(x) − dν(x)) = µ(A) − ν(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 4.4 and 4.5 depend upon the choice of cost function as well as the measures.

W 2 (
D n µ, µ) be the Wasserstein transportation distance between D n µ and free Brownian loop measure µ for the cost function u − v 2 L 2 , where D n is the Dirichlet projection.

Figure 1 .
Figure 1.One sample path of the stochastic process X s , over [0, 10] with step size of h = 10 −5 .

Figure 2 .
Figure 2. Left: A histogram of (θ s , φ s ) over one hemisphere at s = 1.Right: A histogram of (θ s , φ s ) over the sphere at s = 10.