Random discretization of O'Hara knot energy

TL;DR

This paper introduces a novel random discretization method for O'Hara energy, a knot energy used to define standard shapes of knots, and proves convergence properties using optimal transport theory.

Contribution

It presents a new random discrete approximation of O'Hara energy and establishes local uniform convergence and compactness results, advancing the analysis of knot energies.

Findings

Established local uniform convergence of the discretization

Proved compactness of the discrete energy in a transport-based space

Extended understanding of discretization methods for knot energies

Abstract

We considered random discrete approximation of O'Hara energy. O'Hara energy is the energy defined for a knot, and O'Hara energy was introduced for defining the standard shape for each knot class (equivalence class by ambient isotopy) by variational method. In the case of a specific exponent, due to energy invariance under Moebius transformation, this energy is called Moebius energy. Although discretization for various Moebius energies has been defined to analyse the shape of the minimizer so far, only Gamma-convergence to the original energy has been shown for a conventional discretization. In this study, we are successful to show locally uniform convergence and compactness of discrete energy in a space based on optimal transport theory, by introducing random discrete approximation of O'Hara energy using random variable.

Equations

$${\mathcal{E}}^{\alpha ,p}(\gamma ):={\int }_{(\mathbb{R}/\mathcal{L}\mathbb{Z}{)}^2}{\mathcal{M}}^{\alpha ,p}(\gamma )dxdy,$$

$${\mathcal{M}}^{\alpha ,p}(\gamma )(x,y)={\left(\frac{1}{|\gamma (x)-\gamma (y){|}^{\alpha }}-\frac{1}{\mathcal{D}(\gamma (x),\gamma (y){)}^{\alpha }}\right)}^p(x,y)\in (\mathbb{R}/\mathcal{L}\mathbb{Z}{)}^2$$

$$\mathcal{D}(\gamma (x),\gamma (y))=\min \{\mathcal{L}-|x-y|,|x-y|\}.$$

$${\mathcal{E}}_{\rho }^{\alpha ,p}(\gamma ):={\int }_{(\mathbb{R}/\mathcal{L}\mathbb{Z}{)}^2}{\mathcal{M}}^{\alpha ,p}(\gamma )\rho (x)\rho (y)dxdy.$$

$${\mathcal{E}}_n(P):=\sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n\left(\frac{1}{|P(a_j)-P(a_i){|}^2}-\frac{1}{d(a_j,a_i{)}^2}\right)d(a_{i+1},a_i)d(a_{j+1},a_j),$$

$${\mathcal{E}}_s^m(P)={\tilde{\mathcal{E}}}_s^m(P)-{\tilde{\mathcal{E}}}_s^m(R_m)+4$$

$${\tilde{\mathcal{E}}}_s^m(P)=\sum _{|i-j|>1}\frac{|X_i||X_j|}{\mathrm{dist}(X_i,X_j{)}^2},$$

$${\mathcal{E}}_{\cos }^m(P)=\sum _{d_m(i,j)>1}\frac{|{\Delta }_{i}P||{\Delta }_{j}P|}{|{\Delta }_i^{j}P||{\Delta }_{i+1}^{j+1}P|}\left(1-\frac{1}{2}(\cos ({\alpha }_{ij})+\cos ({\tilde{\alpha }}_{ij}))\right),$$

$$R_{n,\rho }{\mathcal{E}}^{\alpha ,p}(\gamma ):=\frac{1}{n^2}\sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n{\left(\frac{1}{|\gamma (X_i)-\gamma (X_j){|}^{\alpha }}-\frac{1}{\mathcal{D}(\gamma (X_i),\gamma (X_j){)}^{\alpha }}\right)}^p.$$

$$T{L}^q(\mathbb{R}/\mathcal{L}\mathbb{N}):=\left\{(\mu ,f)\mid \mu \in \mathcal{P}(\mathbb{R}/\mathcal{L}\mathbb{N}),f\in L^q(\mathbb{R}/\mathcal{L}\mathbb{Z};\mu )\right\},$$

$$d_{TL^{q}}\bigl{(}(\mu,f),(\nu,g)\bigr{)}:=\inf_{\pi\in\Gamma(\mu,\nu)}\left(\int_{(\mathbb{R}/\mathcal{L}\mathbb{Z})^{2}}\left(|x-y|^{q}+|f(x)-g(y)|^{q}\right)d\pi(x,y)\right)^{1/q},$$

$${\nu }_n:=\frac{1}{n}\sum _{i=1}^n{\delta }_{X_i},$$

$$\underset{n\in \mathbb{N}}{\sup }\left(R_{n,\rho }{\mathcal{E}}^{\alpha ,p}({\gamma }_n)+\Vert {\gamma }_n{\Vert }_{L^1(\mathbb{R}/\mathbb{Z},{\nu }_n)}\right)<\infty .$$

$$\underset{n\to \infty }{\mathrm{liminf}}{F}_n(x_n)\ge F(x),$$

$$\underset{n\to \infty }{\mathrm{limsup}}{F}_n(x_n)\le F(x).$$

$$\underset{n\to \infty }{\lim }\underset{\gamma \in K}{\sup }|F_n(\gamma )-F(\gamma )|=0.$$

$$\underset{n\to \infty }{\lim }{F}_n({\gamma }_n)=F(\gamma ).$$

$$r$$

$$\displaystyle=d_{TL^{q}}\bigl{(}(\nu_{n},\gamma_{n}),(\nu_{m},\gamma_{m})\bigr{)}.$$

$$K_0:=\left\{\{{\gamma }_i^m{\}}_{m=1}^{N_m}\in K|\bigcup _{i=1}^{N_m}\mathcal{B}({\gamma }_i^m,{\epsilon }_m)\supset K\right\}.$$

$$|F_n({\gamma }_1)-F_n({\gamma }_2)|$$

$$\epsilon$$

$$|F_n(\gamma )-F_n({\gamma }_i^m)|<\eta /3.$$

$$d_{TL^{q}}\bigl{(}(\nu_{n},\gamma_{i}^{m}),(\nu,\gamma)\bigr{)}<\varepsilon_{m}.$$

$$|F_{n_k}(\gamma )-F_{n_l}(\gamma )|$$

$$\le \eta .$$

$$|F_n({\gamma }_n)-F(\gamma )|$$

$$\displaystyle\leq\sup_{y\in\{\gamma_{n}\}_{n\in\mathbb{N}}\cup\{\gamma\}}\bigl{(}|F_{n}(y)-F(y)|\bigr{)}+|F(\gamma_{n})-F(\gamma)|\xrightarrow{n\rightarrow\infty}0.$$

$$T_{\#}\mu (A):=\mu (T^{-1}(A)),A\in \mathcal{B}(\mathbb{R}/\mathcal{L}\mathbb{Z}).$$

$$\underset{n\to \infty }{\lim }{\int }_{(\mathbb{R}/\mathcal{L}\mathbb{Z}{)}^2}|x-y{|}^{q}d{\pi }_n(x,y)=0.$$

$$\underset{n\to \infty }{\lim }{\int }_{\mathbb{R}/\mathcal{L}\mathbb{Z}}|x-T_n(x){|}^{q}d\mu (x)=0.$$

$${\int }_{(\mathbb{R}/\mathcal{L}\mathbb{Z}{)}^2}|\gamma (x)-{\gamma }_n(x){|}^{q}d{\pi }_n(x,y)\to 0.$$

Full text

1 Introduction

Let $\mathcal{A}$ be a set of all closed regular curve that is parametrised by arc length in $\mathbb{R}^{n},$ with no self-intersections, and with total length $\mathcal{L}$ i.e. $\mathcal{A}:=\{\gamma\in C^{0,1}(\mathbb{R}/\mathcal{L}\mathbb{Z},\mathbb{R}^{n})\mid|\gamma^{\prime}(x)|=1\ \ \mbox{a.e.}~{}x\in\mathbb{R}/\mathcal{L}\mathbb{Z}\},$ and $\alpha,p\in(0,\infty),$ the O’Hara $(\alpha,p)$-energy $\mathcal{E}^{\alpha,p}:\mathcal{A}\rightarrow\mathbb{R}\cup\{+\infty\}$ is defined as follow:

[TABLE]

where

[TABLE]

and $\mathcal{D}$ is the length of shortest arc of the curve $\gamma$ connecting the two points $\gamma(x)$ and $\gamma(y),$ i.e.

[TABLE]

This energy introduced and investigated by O’Hara in [1]-[4] for defining the standard shape for each knot class by variational method.

In the case of $\alpha=2,p=1$, due to energy invariance under Möbius transformation, this energy called ”Möbius energy”. It is possible to show the existence of minimizer in the ”prime knot”. R. Kusner and J. Sullivan conjectured the minimizer in composite knot class may not exist [5]. This conjecture was established by numerical calculation with discretization of Möbius energy.

In this paper, we introduce the weighted O’Hara energy $\mathcal{E}^{\alpha,p}_{\rho}:\mathcal{A}\rightarrow\mathbb{R}\cup\{+\infty\}$ with weight $\rho:\mathbb{R}/\mathcal{L}\mathbb{Z}\rightarrow\mathbb{R}_{\geq 0}$ defined

[TABLE]

1.1 Known results

Various discretization of O’Hara energy has been considered mainly due to the purpose of numerical calculation.

First, D. Kim and R. Kusner introduced the discretization of Möbius energy for polygons in [6].

This energy, defined on the class of arc length parametrizations of polygons of length $\mathcal{L}$ with $n$ segments, is given by

[TABLE]

where the $a_{i}$ are consecutive points on $\mathbb{R}/\mathcal{L}\mathbb{Z}$, or interval $[0,\mathcal{L}]$ if we consider the polygon parametrised over an interval. This energy is scale invariant. A slight variant would be to take $2^{-1}(d(a_{k-1},a_{k})+d(a_{k},a_{k+1}))$ instead of $d(a_{k+1},a_{k})$.

S. Scholtes proved that this discretization $\Gamma$-converges to the Möbius energy in [13]. He furthermore showed that this energy is minimized by regular n-gons.

Second, J. Simon [7] defined the so-called minimal distance energy for a polygon $P$ by

[TABLE]

with

[TABLE]

where $R_{n}$ is the regular $n$-gon. Note, that this energy is scale invariant. Third, in [14] is established Möbius invariant discrete energy and show $\Gamma^{-}$-convergence in $W^{1,q}$-metric sense. The definition of that energy is as follows:

[TABLE]

where $P(\theta_{i})$ is a vertices of closed polygon, $i=1,2,...,m$ and $\Delta^{j}_{i}P:=P(\theta_{j})-P(\theta_{i})$, $\Delta_{i}P=\Delta^{i+1}_{i}P$, $\alpha_{ij}$ be the angle of crossing of the circles through at the points $P(\theta_{i}),P(\theta_{i+1}),P(\theta_{j})$ and $P(\theta_{j}),P(\theta_{j+1}),P(\theta_{i})$ and $\tilde{\alpha}_{ij}$ be the angle of crossing of the circles through at the points $P(\theta_{i}),P(\theta_{i+1}),P(\theta_{j+1})$ and $P(\theta_{j}),P(\theta_{i+1}),P(\theta_{j+1})$.

Our result is to construct random discretization of O’Hara energy, and to show $\Gamma$-convergence in that space based on optimal transport theory. We even show locally uniform convergence and compactness.

1.2 Main results

We defined new discretization of O’Hara $(\alpha,p)$-energy using random variable on $\mathbb{R}/\mathcal{L}\mathbb{Z}$.

Definition 1.1** (Random O’Hara Energy).**

Let $\left\{X_{i}\right\}_{i\in\mathbb{N}}$ be a sequence of i.i.d. random variable on $\mathbb{R}/\mathcal{L}\mathbb{Z}$ with probability density function $\rho$.

Random O’hara energy $R_{n,\rho}\mathcal{E}^{\alpha,p}:\mathcal{A}\rightarrow\mathbb{R}\cup\{+\infty\}$ is defined as follow:

[TABLE]

Remark 1**.**

Since $\{X_{i}\}_{i\in\mathbb{N}}$ has probability density function $\rho$, we always have

$\mathbb{P}(X_{i}=X_{j})=0,$ for any $i\neq j$. Therefore we can defined $R_{n,\rho}\mathcal{E}^{\alpha,p}$ almost surely.

Then we introduce the space for comparing continuous model and discrete model as follows.

Definition 1.2** (The $TL^{q}$ metric space [8]).**

$TL^{q}$ metric is defined on particular spaces of the family

[TABLE]

where $1\leq q<\infty$ and $\mathcal{P}(\mathbb{R}/\mathcal{L}\mathbb{N})$ denotes the set of Borel probability measure on $\mathbb{R}/\mathcal{L}\mathbb{N}.$ For $(\mu,f)$ and $(\nu,g)$ in $TL^{q}$ we define the distance

[TABLE]

where $\Gamma(\mu,\nu)$ is the set of all coupling (or transportation plans) between $\mu$ and $\nu$, that is, the set of all Borel probability measures on $(\mathbb{R}/\mathcal{L}\mathbb{Z})^{2}$ for which the marginal on the first variable is $\mu$ and the marginal on the second variable is $\nu$.

It is shown in [9] the $d_{TL^{q}}$ is actually a metric.

The distance $d_{TL^{q}}$ called a transportation distance between functions defined on graph. The $TL^{q}$ topology provides a general and versatile way to compare functions in a discrete setting with functions in a continuum setting. It is a generalization of the weak convergence of measures and of $L^{q}$ convergence of functions.

Definition 1.3**.**

Let $\{X_{i}\}_{i\in\mathbb{N}}$ be a sequence of i.i.d. random variable and let us denote by $\nu_{n}$ the empirical measure of $\{X_{i}\}_{i\in\mathbb{N}}$:

[TABLE]

where $\delta_{X}$ is Dirac measure of $X$.

Definition 1.4**.**

Let $\{X_{i}\}_{i\in\mathbb{N}}$ be a sequence of i.i.d. random variable on $\mathbb{R}/\mathcal{L}\mathbb{Z}$ and $\nu_{n}$ be a empirical measure of $\{X_{i}\}_{i\in\mathbb{N}}$ and $\nu$ be a distribution measure of $\{X_{i}\}_{i\in\mathbb{N}}$ then, we use a slight abuse of notation and write $(\nu_{n},\gamma_{n})\xrightarrow{TL^{q}}(\nu,\gamma)$ as $\gamma_{n}\xrightarrow{TL^{q}}\gamma.$

The main results of the paper is

Theorem 1.1** ($\Gamma$-convergence and locally uniform convergence).**

Let $\{X_{i}\}_{i\in\mathbb{N}}$ be a sequence of i.i.d. random variable with probability density function $\rho$ on $\mathbb{R}/\mathcal{L}\mathbb{Z}$. Then $R_{n,\rho}\mathcal{E}^{\alpha,p}$ $\Gamma$-converge to $\mathcal{E}^{\alpha,p}_{\rho}$ as $n\rightarrow\infty$ in the $TL^{1}$ sense.

Moreover, we set $\mathcal{F}:=\{\gamma\in\mathcal{A}\mid\mathcal{E}^{\alpha,p}_{\rho}(\gamma)<\infty\}$, then $R_{n,\rho}\mathcal{E}^{\alpha,p}\lfloor_{\mathcal{F}}$ locally uniformly converge to $\mathcal{E}^{\alpha,p}_{\rho}\lfloor_{\mathcal{F}}$ as $n\rightarrow\infty$ in the $TL^{1}$ sense a.s. $\omega\in\Omega$.

Theorem 1.2** (Compactness).**

Let $\rho$ be a bounded from below by a positive constant and let $2\leq\alpha p<2p+1$ and $1\leq q<\infty$,

Assume $\{\gamma_{n}\}_{n\in\mathbb{N}}\subset TL^{q}(\mathbb{R}/\mathcal{L}\mathbb{Z})$ satisfying

[TABLE]

Then $\{\gamma_{n}\}_{n\in\mathbb{N}}$ is relatively compact in the $TL^{q}(\mathbb{R}/\mathcal{L}\mathbb{Z})$ sense a.s. $\omega\in\Omega$.

The metric used in the $\Gamma$-convergence and locally uniform convergence is the $TL^{1}$ sense, which will be defined in Section 2.

2 Preliminaries

2.1 $\Gamma$-convergence on metric spaces and locally uniformly convergence in the $TL^{q}$ sense

We recall notation of general $\Gamma$-converge and locally uniform convergence in the $TL^{q}$ sense.

Definition 2.1** ($\Gamma$-convergence on metric spaces).**

Let $(X,d)$ be a metric space. Let $F_{n}:X\rightarrow[0,\infty]$ be a sequence of functionals. The sequence $\{F_{n}\}_{n\in\mathbb{N}}$ $\Gamma$-converges with respect to metric $d$ to the functional $F:X\rightarrow[0,\infty]$ as $n\rightarrow\infty$ if the following inequality hold:

• i)

For every $x\in X$ and every sequence $\{x_{n}\}_{n\in\mathbb{N}}$ converging to $x$

[TABLE]

• ii)

For every $x\in X$ there exists a sequence $\{x_{n}\}_{n\in N}$ converging to $x$ satisfying

[TABLE]

Definition 2.2** (Locally uniform convergence in the $TL^{q}$ sense).**

A set $K\subset\mathcal{A}$ is sequentially compact in the $TL^{q}$ sense if it satisfies following conditions.

For all sequence $\{\gamma_{n}\}_{n\in\mathbb{N}}\subset K$ , there is a subsequence $\{\gamma_{n_{k}}\}_{k\in\mathbb{N}}$ and a $\gamma\in K$ such that $\gamma_{n_{k}}\xrightarrow{TL^{q}}\gamma$ as $k\rightarrow\infty$.

Let $X$ be a space containing $L^{q}(\nu_{n})$ and $L^{q}(\nu)$,

and let $F_{n}:X\rightarrow\mathbb{R}$ and $F:X\rightarrow\mathbb{R}.$

The sequence $\{F_{n}\}_{n\in\mathbb{N}}$ locally uniformly converges to $F$ in the $TL^{q}$ sense if it satisfies following conditions.

For any sequentially compact set $K\subset X$ in the $TL^{q}$ sense,

[TABLE]

We first discuss an equivalent condition for locally uniform convergence in the $TL^{q}$ sense.

Proposition 2.1**.**

Let $F:X\rightarrow\mathbb{R}$ is continuous and a sequence $\{F_{n}\}_{n\in\mathbb{N}}$ locally uniformly converge to $F$ in the $TL^{q}$ sense.

if and only if

For any sequence $\{\gamma_{n}\}_{n\in\mathbb{N}}\subset X$ with $\gamma_{n}\xrightarrow{TL^{q}}\gamma$,

[TABLE]

This can be proved in a similar way to prove Ascoli-Arzelà theorem [18, Theorem 7.25.].

Proof.

For $\gamma\in\mathcal{A}$ and $r>0$, we set $\displaystyle\mathcal{B}(\gamma,r):=\{\tilde{\gamma}\in X\mid\mbox{There exists an }n\in\mathbb{N}$ such that d_{TL^{q}}\bigl{(}(\nu_{n},\gamma),(\nu,\tilde{\gamma})\bigr{)}<r\}.

First, we show that suppose $K\subset X$ be a sequentially compact set in the $TL^{q}$ sense, then for any $\varepsilon>0$, there is a sequence $\{\gamma_{i}\}_{i=1}^{N_{\varepsilon}}\subset K,$ such that $\displaystyle\bigcup_{i=1}^{N_{\varepsilon}}\mathcal{B}(\gamma_{i},\varepsilon)\supset K.$

If not, there is a $r>0$ such that for all $\{\gamma_{i}\}_{i=1}^{m}\subset K$, $\displaystyle K\setminus\bigcup_{i=1}^{m}\mathcal{B}(\gamma_{i},r)\neq\emptyset.$

We choose $\gamma_{1}\in K$ and inductively choose $\displaystyle\gamma_{n}\in K\setminus\bigcup_{i=1}^{n-1}\mathcal{B}(\gamma_{i},r),$ then for all $m,n\in\mathbb{N},$

[TABLE]

This is a contradiction to the fact that $K$ is sequentially compact set in the $TL^{q}$ sense.

Let $\varepsilon_{m}>0$ with $\varepsilon_{m}\searrow 0$. and we set

[TABLE]

Second, we show that for all $\varepsilon>0$, there exists a $\delta>0$ such that for all $\gamma_{1},\gamma_{2}\in K$ and for all $n\in\mathbb{N}$, if \displaystyle d_{TL^{q}}\bigl{(}(\nu_{n},\gamma_{1}),(\nu,\gamma_{2})\bigr{)}<\delta then

[TABLE]

If not, there exists an $\varepsilon>0$ such that for all $n\in\mathbb{N}$, there exists $\gamma_{1}^{n},\gamma_{2}^{n}\in K$ and $m_{n}\in\mathbb{N}$ such that \displaystyle d_{TL^{q}}\bigl{(}(\nu_{n},\gamma_{1}^{n}),(\nu,\gamma_{2}^{n})\bigr{)}<1/n and $\displaystyle|F_{m_{n}}(\gamma_{1}^{n})-F_{m_{n}}(\gamma_{2}^{n})|\geq\varepsilon.$

By the $K$ is sequentially compact, there exists a subsequence $\{\gamma_{1}^{n_{k}}\}_{k\in\mathbb{N}},\{\gamma_{2}^{n_{k}}\}_{k\in\mathbb{N}}$ and $\gamma\in K$ such that $\gamma_{1}^{n_{k}}\xrightarrow{TL^{q}}\gamma$ , $\gamma_{2}^{n_{k}}\xrightarrow{TL^{q}}\gamma.$ Then

[TABLE]

This is a contradiction.

Third, we show that there exists a subsequence $\{F_{n_{k}}\}_{k\in\mathbb{N}}$ such that for all $j$ , $\{F_{n_{k}}(\gamma_{j})\}_{k\in\mathbb{N}}$ is convergence sequence by a diagonal argument.

By $\displaystyle\lim_{n\rightarrow\infty}F_{n}(\gamma_{1})=F(\gamma_{1})$, $\{F_{n}(\gamma_{1})\}_{n\in N}$ is bounded sequence on $\mathbb{R}.$

Therefore there exists a subsequence $\{F_{n(1,k)}\}_{k\in\mathbb{N}}$ such that $\{F_{n(1,k)}(\gamma_{1})\}_{k\in\mathbb{N}}$ is convergence sequence on $\mathbb{R}.$

In the same way, there exists a subsequence $\{F_{n(2,k)}\}_{k\in\mathbb{N}}$ such that $\{F_{n(2,k)}(\gamma_{2})\}_{k\in\mathbb{N}}$ is convergence sequence on $\mathbb{R}.$

Further in the same way, we construct subsequence $\{F_{n(p,k)}\}_{k\in\mathbb{N}},p=3,4,...,$ and we set $F_{n_{k}}=F_{n(k,k)}$.

Finally, let any $\eta>0$, for sufficient large $m$ such that for all $n\in\mathbb{N}$, if \displaystyle d_{TL^{q}}\bigl{(}(\nu_{n},\gamma_{1}),(\nu,\gamma_{2})\bigr{)}<\varepsilon_{m} then

[TABLE]

Since $\{F_{n_{k}}(\gamma_{i}^{m})\}_{k\in\mathbb{N}}$ is a convergence sequence on $\mathbb{R}$, there exists a number $N$ such that if $k,l>N$ then $|F_{n_{k}}(\gamma_{i}^{m})-F_{n_{l}}(\gamma_{i}^{m})|<\eta/2$ for $i=1,2,...,N_{m}$.

Now, let $\gamma\in K$, by (5) there exist an $i$ and $n$ such that

[TABLE]

By (6) and (7) if $k,l>N$ then

[TABLE]

This indicates that the uniform convergence on $K$.

Assume that $F$ is continuous and $F_{n}$ locally uniformly converge to $F$ in the $TL^{q}$ sense and $\gamma_{n}\xrightarrow{TL^{q}}\gamma.$

Clearly $\{\gamma_{n}\mid n\in\mathbb{N}\}\cup\{\gamma\}\subset X$ is sequentially compact in the $TL^{q}$ sense.

Therefore

[TABLE]

$\Box$

2.2 The property of $TL^{q}$ space and empirical measure

In this subsection, we consider the space based on optimal transport theory to compare discrete and continuous model.

Given a Borel map $T:\mathbb{R}/\mathcal{L}\mathbb{Z}\rightarrow\mathbb{R}/\mathcal{L}\mathbb{Z}$ and $\mu\in\mathcal{P}(\mathbb{R}/\mathcal{L}\mathbb{Z})$ the push-forward of $\mu$ by $T$, denoted by $T_{\#}\mu\in\mathcal{P}(\mathbb{R}/\mathcal{L}\mathbb{Z})$ is given by:

[TABLE]

Definition 2.3** ([8]).**

We say that a sequence of transportation plans $\left\{\pi_{n}\right\}_{n\in\mathbb{N}}\subset\Gamma(\mu,\mu_{n})$ is stagnating if it satisfies

[TABLE]

$\mu,\mu_{n}\in\mathcal{P}(\mathbb{R}/\mathcal{L}\mathbb{Z}),T_{n}:\mathbb{R}/\mathcal{L}\mathbb{Z}\rightarrow\mathbb{R}/\mathcal{L}\mathbb{Z}:$ transportation maps with ${T_{n}}_{\#}\mu=\mu_{n}$.

We say that a sequence of transportation maps $\{T_{n}\}_{n\in\mathbb{N}}$ is stagnating if it satisfies

[TABLE]

Accept the following proposition introduced by [8]

Proposition 2.2** ([8]).**

Let $(\mu,\gamma)\in TL^{q}$ and let $\left\{(\mu_{n},\gamma_{n})\right\}_{n\in\mathbb{N}}\subset TL^{q}$．

The following statements are equivalent;

• i)

$(\mu_{n},\gamma_{n})\rightarrow(\mu,\gamma)$* in the $TL^{q}$.*

• ii)

$\mu_{n}\rightharpoonup\mu$* and for every stagnating sequence of transportation plans $\left\{\pi_{n}\right\}_{n\in\mathbb{N}}\subset\Gamma(\mu,\mu_{n})$*

[TABLE]

• iii)

$\mu_{n}\rightharpoonup\mu$* and there exists a stagnating sequence of transportation plans $\left\{\pi_{n}\right\}_{n\in\mathbb{N}}\subset\Gamma(\mu,\mu_{n})$ such that*

[TABLE]

Moreover, if the measure $\mu$ is absolutely continuous with respect to the Lebesgue measure, the following are equivalent to the previous statement:

• iv)

$\mu_{n}\rightharpoonup\mu$* and there exists a stagnating sequence of transportation maps $\left\{T_{n}\right\}_{n\in\mathbb{N}}$ $($ with ${T_{n}}_{\#}\mu=\mu_{n})$ such that*

[TABLE]

• v)

$\mu_{n}\rightharpoonup\mu$* and for any stagnating sequence of transportation maps $\left\{T_{n}\right\}_{n\in\mathbb{N}}$ with ${T_{n}}_{\#}\mu=\mu_{n})$，*

[TABLE]

Remark 2**.**

Thanks to Proposition 2.2 when $\mu$ is absolutely continuous with respect to the Lebesgue measure $\gamma_{n}\xrightarrow{TL^{p}}\gamma$ as $n\rightarrow\infty$ if and only if for every (or one) $\{T_{n}\}_{n\in\mathbb{N}}$ stagnating sequence of transportation maps (with ${T_{n}}_{\#}\mu=\mu_{n}$) $\gamma_{n}\circ T_{n}\xrightarrow{L^{p}(\mu)}\gamma$ as $n\rightarrow\infty.$ Also $\{u_{n}\}_{n\in\mathbb{N}}$ is relatively compact in $TL^{p}$ if and only if for every (or one) $\{T_{n}\}_{n\in N}$ stagnating sequence of transportation maps (with $T_{\#}\mu=\mu_{n}$) $\{u_{n}\circ T_{n}\}_{n\in\mathbb{N}}$ is relatively compact in $L^{p}(\mu)$.

We recall the following proposition.

Proposition 2.3** **(Glivenko-Cantelli’s Theorem).

Let $\{X_{i}\}_{i\in\mathbb{N}}$ be a sequence of i.i.d. random variable on $\mathbb{R}/\mathcal{L}\mathbb{Z}$, and let $\nu$ is distribution measure of $\{X_{i}\}_{i\in\mathbb{N}}$, and $\nu_{n}$ is empirical measure of $\{X_{i}\}_{i\in\mathbb{N}}$.

$F_{n}(x):=\nu_{n}((-\infty,x])$* and $F$ is distribution function of $\{X_{i}\}_{i\in\mathbb{N}}$, then,*

[TABLE]

Theorem 2.1** ([10]).**

Let $D\subset\mathbb{R}^{d}$ be a bounded, connected, open set with Lipschitz boundary. Let $\nu$ be a probability measure on $D$ with density $\rho:D\rightarrow(0,\infty)$ which is bounded from below and from above by positive constants. Let $\{X_{i}\}_{i\in\mathbb{N}}$ be a sequence of independent random points distributed on $D$ according to measure $\nu$ and let $\nu_{n}$ be the associated empirical measures. Then there is a constant $C>0$ such that for $a.s.~{}\omega\in\Omega$ there exists a sequence of transportation maps $\{T_{n}\}_{n\in\mathbb{N}}$ from $\nu$ to $\nu_{n}$ $({T_{n}}_{\#}\nu=\nu_{n})$ and such that

if $d=2$ then

[TABLE]

and if $d\geq 3$ then

[TABLE]

3 $\Gamma$-convergence and locally uniformly convergence

Proof of Theorem 1.1

Proof.

Let $\nu_{n}$ be an empirical measure of $\mathcal{L}^{1}\lfloor\rho$ and $T_{n}:\mathbb{R}/\mathcal{L}\mathbb{Z}\rightarrow\mathbb{R}/\mathcal{L}\mathbb{Z}$ be a transportation maps with ${T_{n}}_{\#}\mathcal{L}^{1}\lfloor\rho=\nu_{n}$.

・liminf inequality

Assume that $\gamma_{n}\rightarrow\gamma$ in $TL^{1}$ as $n\rightarrow\infty$. Since ${T_{n}}_{\#}\mathcal{L}^{1}\lfloor\rho=\nu_{n}$, we change variables to get

[TABLE]

By the way we notice that

[TABLE]

and

[TABLE]

By Proposition 2.2 we deduce $\gamma_{n}\circ T_{n}\rightarrow\gamma$ in $L^{1}(\mathbb{R}/\mathcal{L}\mathbb{Z})$.

Thus, by taking an appropriate subsequence $\left\{\gamma_{n_{k}}\circ T_{n_{k}}\right\}_{k\in\mathbb{N}}$ of $\left\{\gamma_{n}\circ T_{n}\right\}_{n\in\mathbb{N}}$, we deduce $\gamma_{n_{k}}(T_{n_{k}}(x))\rightarrow\gamma(x)\ \ \mbox{a.e.}~{}x\in\mathbb{R}/\mathcal{L}\mathbb{Z}$ and therefore

[TABLE]

So that $\displaystyle\mathcal{M}^{\alpha,p}(\gamma_{n}\circ T_{n})>0$, using Fatou’s lemma we get

[TABLE]

・limsup inequality

Assume that $\gamma_{n}\rightarrow\gamma\ \ \mathrm{in}\ \ TL^{1}$ as $n\rightarrow\infty$.

If $\displaystyle\mathcal{E}^{\alpha,p}_{\rho}(\gamma)=\infty,$ we are done, and so we henceforth assume $\displaystyle\mathcal{E}^{\alpha,p}_{\rho}(\gamma)<\infty.$

We observe that

[TABLE]

and

[TABLE]

In the same way as liminf inequality, we get

$\displaystyle\limsup_{n\rightarrow\infty}\mathcal{M}^{\alpha,p}(\gamma_{n}\circ T_{n})\leq\mathcal{M}^{\alpha,p}(\gamma)\ \ \mbox{a.e.}~{}(x,y)\in(\mathbb{R}/\mathbb{Z})^{2}.$

Since $\mathcal{E}^{\alpha,p}_{\rho}(\gamma)<\infty$, using Fatou’s lemma to $\mathcal{M}^{\alpha,p}(\gamma)-\mathcal{M}^{\alpha,p}(\gamma_{n}\circ T_{n}),$ we get

[TABLE]

By Proposition 2.1, the proof is now complete.

$\Box$

4 Compactness

In this section, we would like to prove Theorem 1.2 we first recall several function space.

4.1 Function spaces

Definition 4.1** **(Sobolev-Slobodeckij spaces).

For $s\in(0,1)$ and $q\in[1,\infty)$ we set

[TABLE]

[TABLE]

and equip this space with the norm

[TABLE]

Furthermore, we let

[TABLE]

and

[TABLE]

Definition 4.2** **(Besov spaces).

For $s\in\mathbb{R}$, $1\leq p,q\leq\infty$ and $\mathcal{S}$ is the Schwartz space.

We set

[TABLE]

and we set $\varphi_{j}(\xi)=\varphi(2^{-j}\xi)$ and $\tau(D)f:=\mathcal{F}^{-1}[\tau\cdot\mathcal{F}f]$ for $\tau\in\mathcal{S}$ and $f\in\mathcal{S}^{\prime}$ we set

[TABLE]

[TABLE]

Note that $W^{s,p}$ agrees with $B^{s}_{p,p}$.

We recall the following theorem.

Proposition 4.1** **(Embedding Besov spaces).

Let $0<p_{0},p_{1}\leq\infty$ and $0<q_{0},q_{1}\leq\infty$ and $-\infty<s_{1}<s_{0}<\infty$. Assume that $s_{0}-\frac{n}{p_{0}}>s_{1}-\frac{n}{p_{1}}$, then

[TABLE]

Theorem 4.1** **(Kondrachov embedding theorem).

Let $1\leq p,q<\infty$ and $1<k,s$.

Assume that $k-\frac{1}{p}>s-\frac{1}{q}$, then the Sobolev embedding

[TABLE]

is completely continuous.

Theorem 4.2** **(Gagliardo-Nirenberg interpolation inequality).

Suppose that $f\in W^{l,q}(\mathbb{R}/\mathcal{L}\mathbb{Z})\cap W^{j,r}(\mathbb{R}/\mathcal{L}\mathbb{Z})$ for some $j<l,q,r\geq 1.$

Assume that $j\leq k<l$ and for some $0<\Theta\leq 1$,

$k-\frac{n}{p}=\Theta\left(l-\frac{n}{q}\right)+(1-\Theta)\left(j-\frac{n}{r}\right),$* then $f\in W^{k,p}(\mathbb{R}/\mathcal{L}\mathbb{Z})$. Moreover, we have for all $f\in W^{l,q}(\mathbb{R}/\mathcal{L}\mathbb{Z})\cap W^{j,r}(\mathbb{R}/\mathcal{L}\mathbb{Z})$,*

[TABLE]

for some constant $C>0$ independent of $f$.

Proposition 4.2** **(Embedding $L^{1}$ space for Besov space).

Let $s\in\mathbb{R}$ and $0<q\leq\infty$, then

[TABLE]

Theorem 4.3**.**

Let $-\infty<s_{0}<\infty$ , $-\infty<s_{1}<\infty$ , $0<p_{0}\leq\infty$ , $0<p_{1}\leq\infty$ , $0<q_{0}\leq\infty$ , $s=(1-\Theta)s_{0}+\Theta s_{1}$. Assume that for some $1<\Theta\leq 1$, $\frac{1}{p}=\frac{1-\Theta}{p_{0}}+\frac{\Theta}{p_{1}}$ and $\frac{1}{q}=\frac{1-\Theta}{q_{0}}+\frac{\Theta}{q_{1}},$ then

[TABLE]

The following theorem is based on [11], and explain conditions for which O’Hara energy becomes finite.

Theorem 4.4** ([11]).**

Let $\gamma\in\mathcal{A}$ and $\alpha,p\in(0,\infty)$ with $\alpha p\geq 2$ and $s:=\frac{\alpha p-1}{2p}<1$ and $p\geq 1$, then $\mathcal{E}^{\alpha,p}(\gamma)<\infty$ if and only if $\gamma\in W^{1+s,2p}(\mathbb{R}/\mathcal{L}\mathbb{Z},\mathbb{R}^{n}).$ Moreover, there is a $C=C(\alpha,p)$ such that

[TABLE]

4.2 Proof of Theorem 1.2.

Proof.

Let $s:=\frac{\alpha p-1}{2p}$. By Theorem 4.4 we see

[TABLE]

By Theorem 4.3 we choose $\Theta>0$ such that $\Theta<\frac{(2p-1)q+2p}{(2p+\alpha p-2)q+2p}\leq 1$, and $t$ with $\frac{1}{2p}-1>t$， to get

[TABLE]

Since $L^{1}\hookrightarrow B^{0}_{1,\infty}\hookrightarrow B^{t}_{2p,2p}$ this implies yields

[TABLE]

Using Young’s inequality, for all $\varepsilon>0$, we get

[TABLE]

Therefore, for sufficient small $\varepsilon>0$, we conclude that

[TABLE]

Let $\{\gamma_{n}\}_{n\in\mathbb{N}}$ be a sequence of $TL^{q}(\mathbb{R}/\mathbb{Z})$ with

[TABLE]

and let $T_{n}:\mathbb{R}/\mathcal{L}\mathbb{Z}\rightarrow\mathbb{R}/\mathcal{L}\mathbb{Z}$ be a transportation maps with ${T_{n}}_{\#}\mu=\mu_{n}$, then

[TABLE]

Since $\rho$ is bounded from below by a positive constant, we deduce that

[TABLE]

Therefore by (18), we see

[TABLE]

Since $1+s-\frac{1}{2p}=\frac{2p+\alpha p-2}{2p}\geq 0>-\frac{1}{q}$, Theorem 4.1, yield a compact embedding

[TABLE]

Therefore there exists a $\{n_{k}\}_{k\in\mathbb{N}}\subset\mathbb{N}$ and $\gamma\in L^{q}(\mathbb{R}/\mathcal{L}\mathbb{Z},\rho)$ such that $\gamma_{n_{k}}\circ T_{n_{k}}\rightarrow\gamma$ in $L^{q}(\mathbb{R}/\mathbb{Z},\rho).$

By Proposition 2.2 we see $\gamma_{n_{k}}\rightarrow\gamma$ in $TL^{q}(\mathbb{R}/\mathcal{L}\mathbb{Z})$.

$\Box$