A note on Monge–Kantorovich problem

Statistics and Probability Letters 84 (2014) 204–211

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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

A note on Monge–Kantorovich problem Pengbin Feng ∗ , Xuhui Peng Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, PR China

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Article history: Received 2 September 2013 Received in revised form 18 October 2013 Accepted 18 October 2013 Available online 24 October 2013 Keywords: Monge–Kantorovich problem Optimal transportation Partial differential equations

abstract Shen and Zheng (2010) and Xu and Yan (2013) considered the Monge–Kantorovich problem in the plane and proved that the optimal coupling for the problem has a form (X1 , g (X1 , Y2 ), h(X1 , Y2 ), Y2 ), and then they assumed (X1 , Y2 ) has a density p and gave the equation which p should satisfy. In this article, we prove that (X1 , Y2 ) naturally has a density under more weak conditions. We again prove a similar result in dimension 3 and give an exact form (X1 , g1 (X1 , Y2 , Y3 ), g2 (X1 , Y2 , Y3 ), h(X1 , Y2 , Y3 ), Y2 , Y3 ) depending on a certain convex function. © 2013 Elsevier B.V. All rights reserved.

1. Introduction and notations Let L(F , G) be the set of all 2n-dimensional random variables whose marginal distribution functions are F and G, respectively. The so-called Monge–Kantorovich problem is to find an optimal coupling of (X , Y ) ∈ L(F , G) such that E|X − Y |2 attains the minimum. x In this article, we assume that F has a density f that means F (x) = −∞ f (x′ )dx′ , G has a density f˜ . µ, ν are two measures on Rn such that µ(dx) = f (x)dx, ν(dx) = f˜ (x)dx. In dimension 2 (n = 2), Shen and Zheng (2010) and Xu and Yan (2013) proved that the optimal coupling of F and G has the following form:

(X1 , g (X1 , Y2 ), h(X1 , Y2 ), Y2 ); here g, h are some functions depending on f , f˜ and the law of (X1 , Y2 ). Then, Shen and Zheng (2010) and Xu and Yan (2013) assumed that Z = (X1 , Y2 ) has a density and gave the equation which p(., .) should satisfy. In this article, in Section 2, we consider the situation with dimension 2 (n = 2) and prove that if

(X1 , X2 , Y1 , Y2 ), is the optimal coupling of F and G, then the law of (X1 , Y2 ) is naturally absolutely continuous with the Lebesgue measure on R2 . In Section 3, we consider the situation with dimension 3 (n = 3). First, we prove that if (X1 , X2 , X3 , Y1 , Y2 , Y3 ) is the optimal coupling of F and G, then (X1 , Y2 , Y3 ) has a density naturally. Then we prove that the optimal coupling of F and G can be assumed to have the following form:

(X1 , g1 (X1 , Y2 , Y3 ), g2 (X1 , Y2 , Y3 ), h(X1 , Y2 , Y3 ), Y2 , Y3 );

(1)

here the functions g1 , g2 , h depend on f , f˜ and the law of (X1 , Y2 , Y3 ). Let p be the density of (X1 , Y2 , Y3 ), and then we give the expression of p in some sense.

∗

Corresponding author. Tel.: +86 15010221272. E-mail addresses: [email protected] (P. Feng), [email protected] (X. Peng).

0167-7152/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.spl.2013.10.011

P. Feng, X. Peng / Statistics and Probability Letters 84 (2014) 204–211

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In the following particular Monge–Ampère equation, F (x) = det(D2 ϕ(x)),

(2)

F (x) plays simultaneously the role of right hand side and coefficients due to the structure of its nonlinearity. The standard argument shows that this equation is elliptic only when D2 ϕ(x) is a positive definite matrix, equivalently, (2) is elliptic only for functions ϕ that is strictly convex in its domain. To ensure (2) exists a solution, F (x) must be positive. Let C α be the collection of functions which are Hölder continuous of order α . In this article, we make the following assumptions on f and f˜ : (H) f , f˜ ∈ C α , α ∈ (0, 1) and for any x, f (x) and f˜ (x) are positive. Hence we do not need f ; f˜ are as smooth as that in Xu and Yan (2013). From Theorem 11 in Villani (2002), there exists unique mappings ∇ϕ and ∇ϕ ∗ from Rn to Rn such that ∇ϕ #µ = ν and ∇ϕ ∗ #ν = µ, here ϕ and ϕ ∗ are convex functions from Rn to R. Then ϕ and ϕ ∗ are convex functions satisfying the following specific Monge–Ampère equations: f (x) = f˜ (∇ϕ(x)) det(D2 ϕ(x)),

x ∈ Rn

f˜ (x) = f (∇ϕ ∗ (x)) det(D2 ϕ ∗ (x)),

(3)

x∈R . n

(4) 2,α

Similar to the result in linear elliptic equations, it can be obtained that ϕ ∈ C by using standard but more complicated continuity methods. Furthermore, by bootstrapping argument, there is the following proposition. Proposition 1.1 (Cf. Caffarelli (2002)). If f and f˜ never vanish or if the supports of f and f˜ are convex, then the regularity of ∇ϕ(x) is ‘‘one derivative better’’ than f and f˜ . Remark 1.1. The solution is not unique for the general Monge–Ampère equation in a full space since it has a very rich family f (x) , because of the special structure on the right hand side, the of invariants. But for our particular case, det(D2 ϕ(x)) = ˜ f (∇ϕ(x))

uniqueness of ∇ϕ(x) can be proved (cf. Villani, 2002, Theorem 11); here we require ϕ(x) to be a convex function. For any function φ : Rn → R, we denote by ∇1 φ = ∇x1 φ(x1 , . . . , xn ), . . . , ∇n φ = ∇xn φ(x1 , . . . , xn ), ∇φ = (∇1 φ, . . . , ∇n φ). If X , Y are two random variables, X ∼ Y means X and Y have the same distribution. If µ0 is a measure, then X ∼ µ0 means the distribution function of X is given by F (x) = µ0 ((−∞, x]). 2. Case with dimension 2 When n = 2 (in the plane), Shen and Zheng (2010) and Xu and Yan (2013) proved that the optimal coupling of F and G has the following form:

(X1 , g (X1 , Y2 ), h(X1 , Y2 ), Y2 ), for some functions g, h depending on f , f˜ and the law of (X1 , Y2 ). They assumed that the 2-dimensional random vector

Z = (X1 , Y2 ) has a density p(., .) and gave the equation p should satisfy. In this section, if

(X1 , X2 , Y1 , Y2 ), is the optimal coupling of F and G, we prove that the law of (X1 , Y2 ) is naturally absolutely continuous with the Lebesgue measure on R2 . Define mapping Q : (x1 , x2 ) → (Q1 (x1 , x2 ), Q2 (x1 , x2 )) = (x1 , ∇2 ϕ(x1 , x2 )) and Range(Q ) := {(x1 , ∇2 ϕ(x1 , x2 )) : (x1 , x2 ) ∈ R2 }. To prove the main theorem in this section, we need the following two lemmas. Lemma 2.1. If E ⊆ Rm is a measurable set, T : Rm → Rm , if (1) T −1 : T (E ) → E exists, (2) T and T −1 map the measurable set to measurable set, then there exists a integrable function JT such that

 T (E )

f (x)dx =



f (Ty)JT (y)dy,

∀f ∈ L1 (T (E )).

E

Lemma 2.2. The mapping Q is injective. Proof. From (3), det(D2 ϕ(x)) = ∇22 ϕ(x)∇11 ϕ(x) − (∇12 ϕ)2 = f (x)/f˜ (∇ϕ(x)) > 0.

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So,

∇22 ϕ(x)∇11 ϕ(x) > 0. From the above inequality, the mapping Q is injective.



Theorem 2.1. Let (H) hold and (X1 , X2 , Y1 , Y2 ) be the optimal coupling of F and G. Then (a) the law of (X1 , Y2 ) is absolutely continuous with the Lebesgue measure λ on R2 ; (b) the density of (X1 , Y2 ) is given by

 

p(u, v) =

f (Q −1 (u, v)) ·

0,

1

∇22 ϕ(Q −1 (u, v))

,

if (u, v) ∈ Range(Q ), if (u, v) ̸∈ Range(Q ).

Proof. (a) For (X1 , X2 , Y1 , Y2 ) is the optimal coupling of F and G, by Theorem 11 in Villani (2002)

(X1 , X2 , Y1 , Y2 ) ∼ µ(dx)δ[y=∇ϕ(x)] . So for any bounded Borel measurable function k,

Ek(X1 , Y2 ) =

  R

k(x1 , ∇2 ϕ)f (x1 , x2 )dx1 dx2 . R

Let A be a Borel measurable set with λ(A) = 0 and k(x) = 1A (x), and then

E1A (X1 , Y2 ) =

  R

1B (x1 , x2 )f (x1 , x2 )dx1 dx2 , R

where B = {(x1 , x2 ), (x1 , ∇2 ϕ) ∈ A}. From Lemma 2.2 and the fact Q is bijective between R2 and Range(Q ), λ(B) = 0. So (a) follows. (b) Assume Q −1 (x1 ′ , x2 ′ ) = (q1 (x1 ′ , x2 ′ ), q2 (x1 ′ , x2 ′ )), ∀(x1 ′ , x2 ′ ) ∈ Range(Q ), and then,

Ek(X1 , Y2 ) =

  R

  = R 

k(x1 , ∇2 ϕ)f (x1 , x2 )dx1 dx2 R

 ∇ q (x ′ , x ′ ) k(x1 , x2 )f (q1 (x1 , x2 ), q2 (x1 , x2 ))  1 1 1 ′ 2 ′ ∇1 q2 (x1 , x2 ) R ′

′

k(u, v)f (Q −1 (u, v))

= R

′

R

′

′

′

1

∇22 ϕ(Q −1 (u, v))

 ∇1 q1 (x1 ′ , x2 ′ ) ′ ′ dx dx . ∇2 q2 (x1 ′ , x2 ′ ) 1 2

dudv. 

Remark 2.1. It is easy to show that Range(Q ) is not R2 , for example ϕ = α x + β , where α, β ∈ R2 are constants. However, if we assume that ϕ is strictly convex, i.e., Hessian (ϕ) ≥ α I2 for some α > 0, then it is easy to show that the mapping Q is injective and surjective. Proposition 2.2. p satisfies the following equations:

 p(u, ∇2 ϕ(u, v))∇22 ϕ(u, v) = f (u, v), f (u, v) 2 . det D ϕ(u, v) = ˜ f (∇ϕ(u, v)) Proof. It directly comes from Theorem 2.1.



Remark 2.2. If f ∈ C α (R2 , R) with f > 0, α ∈ (0, 1), f˜ is the density function of uniform distribution on domain [0, 1] × [0, 1], and then the results of Theorem 2.1 are also satisfied and Eq. (3) becomes f (x) = det(D2 ϕ(x)),

x ∈ R2 .

(5)

Furthermore, if f has the following special forms eλx1 h(ax1 + bx2 ), eα x1 h(eβ x1 x2 ),

β

xα1 h(x1 x2 ), ekx2 /x1 h(x1 ),

h(ax21 + bx1 x2 + cx22 + kx1 + sx2 ), h(x1 )q(x2 ),

4 −1 −1 −2 x− 2 exp(α x2 )h(x1 x2 + β x2 ),

here h(.), q(.) are arbitrary functions and a, b, c , s, k, α, β, λ are arbitrary constants, and then the exact solutions to (5) are outlined in Polyanin and Zaitsev (2004). That means, in these special cases, we can know ϕ and p exactly.

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3. Case with dimension 3 In this section, we consider the situation with dimension 3 (n = 3) and the main theorem is Theorem 3.1. In Theorem 3.1, we prove that the optimal coupling of F and G can be assumed to have the following form:

(X1 , g1 (X1 , Y2 , Y3 ), g2 (X1 , Y2 , Y3 ), h(X1 , Y2 , Y3 ), Y2 , Y3 ), (6) ˜ for some functions g1 , g2 , h depending on f , f and the law of (X1 , Y2 , Y3 ). Theorem 3.1 also tells us that (X1 , Y2 , Y3 ) in (6) can be assumed to have a density p. In this theorem, we give the expression of p in some forms. The mapping Q in this section is defined by Q : (x1 , x2 , x3 ) → (Q1 (x1 , x2 , x3 ), Q2 (x1 , x2 , x3 ), Q3 (x1 , x2 , x3 ))

= (x1 , ∇2 ϕ(x1 , x2 , x3 ), ∇3 ϕ(x1 , x2 , x3 )). If the law of (X1 , Y2 , Y3 ) is absolutely continuous with Lebesgue measure, its density is denoted by p. For any y2 , y3 fixed, define functions

 y1 FY1 |Y2 ,Y3 (y1 |y2 , y3 ) : = P(Y1 ≤ y1 | Y2 = y2 , Y3 = y3 ) = 

−∞



R x1

f˜ (t , y2 , y3 )dt

f˜ (t , y2 , y3 )dt

,  x1

p(t , y2 , y3 )dt = −∞ . ˜ p(t , y2 , y3 )dt f (t , y2 , y3 )dt R R

FX1 |Y2 ,Y3 (x1 |y2 , y3 ) : = P( X1 ≤ x1 | Y2 = y2 , Y3 = y3 ) = 

−∞

p(t , y2 , y3 )dt

1 For each y2 , y3 , denote the inverse function of FY1 |Y2 ,Y3 ( .| y2 , y3 ) by F − Y1 |Y2 ,Y3 ( .| y2 , y3 ). Before we give the main theorem, we need the following lemmas.

Lemma 3.1. Let (H) hold, then for any x ∈ R3 ,

∇22 ϕ∇33 ϕ − (∇32 ϕ)2 > 0, ∇11 ϕ > 0, ∇22 ϕ > 0, ∇33 ϕ > 0, ∇11 ϕ ∗ > 0. Proof. Since ϕ ∈ C 2,α and ϕ is convex, Hessian (ϕ ) is positive definite in a classical definition. ϕ is convex as a function of x3 when x1 , x2 are fixed, so

∇33 ϕ ≥ 0.

(7)

Furthermore,

∇11 ϕ ≥ 0,

∇22 ϕ ≥ 0.

(8)

Denote by A := ∇22 ϕ∇33 ϕ(x) − (∇32 ϕ)2 (x), then det(D2 ϕ(x)) = ∇11 ϕ · A − ∇12 ϕ(∇12 ϕ∇33 ϕ − ∇23 ϕ∇31 ϕ) + ∇13 ϕ(∇12 ϕ∇23 ϕ − ∇22 ϕ∇31 ϕ)

= ∇11 ϕ · A − ∇33 ϕ(∇12 ϕ)2 − ∇22 ϕ(∇13 ϕ)2 . From the above equality and (3),

∇11 ϕ · A > ∇33 ϕ(∇12 ϕ)2 + ∇22 ϕ(∇13 ϕ)2 . Due to (7), (8) and the above inequalities, A > 0,

∇11 ϕ > 0.

For the same reason,

∇22 ϕ > 0,

∇33 ϕ > 0,

∇11 ϕ ∗ > 0. 

Lemma 3.2. The mapping Q is injective. Proof. If

(x1 , ∇2 ϕ(x1 , x2 , x3 ), ∇3 ϕ(x1 , x2 , x3 )) = (x1 ′ , ∇2 ϕ(x1 ′ , x2 ′ , x3 ′ ), ∇3 ϕ(x1 ′ , x2 ′ , x3 ′ )), then x1 = x1 ′ . From Theorem 11 in Villani (2002), we know that ∇ϕ is a one to one mapping from R3 to R3 (here we notice that µ, ν are equivalent to the Lebesgue measure). So, if ∇1 ϕ(x1 , x2 , x3 ) = ∇1 ϕ(x1 , x2 ′ , x3 ′ ), then x2 = x2 ′ , x3 = x3 ′ . If ∇1 ϕ(x1 , x2 , x3 ) ̸= ∇1 ϕ(x1 , x2 ′ , x3 ′ ), we will prove it is impossible. Without loss of generality, we can assume

∇1 ϕ(x1 , x2 , x3 ) > ∇1 ϕ(x1 , x2 ′ , x3 ′ ).

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P. Feng, X. Peng / Statistics and Probability Letters 84 (2014) 204–211

Due to Lemma 3.1 and Theorem 11 in Villani (2002), x1 = ∇1 ϕ ∗ ∇1 ϕ(x1 , x2 , x3 ), ∇2 ϕ(x1 , x2 , x3 ), ∇3 ϕ(x1 , x2 , x3 )





  > ∇1 ϕ ∗ ∇1 ϕ(x1 , x2 ′ , x3 ′ ), ∇2 ϕ(x1 , x2 ′ , x3 ′ ), ∇3 ϕ(x1 , x2 ′ , x3 ′ ) = x1 . This is a contradict. So x2 = x2 ′ and x3 = x3 ′ , which finishes the proof of this lemma.



Theorem 3.1. Let (H) hold and (X1 , X2 , X3 , Y1 , Y2 , Y3 ) be the optimal coupling of F and G. Then, (i) the law of (X1 , Y2 , Y3 ) is absolutely continuous with the Lebesgue measure; (ii)

E |X1 − Y1 |2 + |X2 − Y2 |2 + |X3 − Y3 |2 ≥ E|X1 − h(X1 , Y2 , Y3 )|2 + E|g1 (X1 , Y2 , Y3 ) − Y2 |2





+ E|g2 (X1 , Y2 , Y3 ) − Y3 |2 where

  h(x1 , y2 , y3 ) = FY−11|Y2 ,Y3 FX1 |Y2 ,Y3 (x1 |y2 , y3 )|y2 , y3 , g1 (x1 , y2 , y3 ) = ∇2 ϕ ∗ (h(x1 , y2 , y3 ), y2 , y3 ), g2 (x1 , y2 , y3 ) = ∇3 ϕ ∗ (h(x1 , y2 , y3 ), y2 , y3 ); (iii) (X1 , g1 (X1 , Y2 , Y3 ), g2 (X1 , Y2 , Y3 )) ∼ (X1 , X2 , X3 ) and (Y1 , Y2 , Y3 ) ∼ (h(X1 , Y2 , Y3 ), Y2 , Y3 ); (iv) the density of (X1 , Y2 , Y3 ) is given by

 1 f (Q −1 (u, v, w)) ·   , p(u, v, w) = ∇22 ϕ∇33 ϕ − (∇32 ϕ)2 (Q −1 (u, v, w))  0,

if (u, v) ∈ Range(Q ), if (u, v) ̸∈ Range(Q ).

Remark 3.1. From Lemmas 3.1 and 3.2, the expression of p in the above theorem is meaningful. Proof. The proof of Theorem 3.1. (i) For (X1 , X2 , X3 , Y1 , Y2 , Y3 ) is the optimal coupling of F and G, so by Theorem 11 in Villani (2002)

(X1 , X2 , X3 , Y1 , Y2 , Y3 ) ∼ µ(dx)δ[y=∇ϕ(x)] . Let λ be the Lebesgue measure on R3 and A be a Borel measurable set with λ(A) = 0. Then

E1A (X1 , Y2 , Y3 ) =

 3

R =

R3

1A (x1 , ∇2 ϕ, ∇3 ϕ)f (x1 , x2 , x3 )dx 1B (x1 , x2 , x3 )f (x1 , x2 , x3 )dx,

where B = {(x1 , x2 , x3 ), (x1 , ∇2 ϕ, ∇3 ϕ) ∈ A}. From Lemma 3.2, Q is bijective between R3 and Range(Q ). For λ(A) = 0 and Q is a bijective between R3 and Range(Q ), so λ(B) = 0 by Lemma 2.1. Then (i) follows. (ii) If (X1 , X2 , X3 ), (Y1 , Y2 , Y3 ) is the optimal coupling of F and G then   E|X1 − Y1 |2 = E E|X1 − Y1 |2 |Y2 = y2 , Y3 = y3 , and

E |X1 − Y1 |2  Y2 = y2 , Y3 = y3 ≥







inf

(X ,Y )∈B

E (X − Y )2 ,





where B is the set of all 2-dimensional vectors whose marginal distributions are FX1 |Y2 ,Y3 ( .| y2 , y3 ) and FY1 |Y2 ,Y3 ( .| y2 , y3 ). If

(X˜ , Y˜ ) ∈ B and (X˜ , Y˜ ) are comonotonic, then (X˜ , Y˜ ) is an optimal coupling:     inf E (X − Y )2 = E (X˜ − Y˜ )2 . (X ,Y )∈B

It is an easy exercise to show that (X˜ , Y˜ ) ∈ B and (X˜ , Y˜ ) are comonotonic if and only if Y˜ = fˆ (X˜ ), where

     fˆ (x1 ) = F ˜−1 FX˜ (x1 ) = FY−11|Y2 ,Y3 FX1 |Y2 ,Y3 (x1 |y2 , y3 ) y2 , y3 = h(x1 , y2 , y3 ). Y

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Therefore,

E |X1 − Y1 |2 |Y2 = y2 , Y3 = y3 ≥





inf

E (X − Y )2



(X ,Y )∈B



= E|X˜ − Y˜ |2 = E[|X˜ − h(X˜ , y2 , y3 )|2 ]    = E (X1 − h(X1 , y2 , y3 ))2  Y2 = y2 , Y3 = y3    = E (X1 − h(X1 , Y2 , Y3 ))2  Y2 = y2 , Y3 = y3 ; then

E|X1 − Y1 |2 ≥ E|X1 − h(X1 , Y2 , Y3 )|2 . Denote by Y˜1 = h(X1 , Y2 , Y3 ). From (iii) below, (Y1 , Y2 , Y3 ) ∼ (Y˜1 , Y2 , Y3 ). From the above proof, if (X1 , X2 , X3 , Y1 , Y2 , Y3 ) is the optimal coupling of F and G, then also is (X1 , X2 , X3 , Y˜1 , Y2 , Y3 ). In the following, we will prove that

E|X2 − Y2 |2 = E|g1 (X1 , Y2 , Y3 ) − Y2 |2 . For (X1 , X2 , X3 , Y˜1 , Y2 , Y3 ) is the optimal coupling of F and G, by Theorem 11 in Villani (2002), the law of (X1 , X2 , X3 , Y˜1 , Y2 , Y3 ) is π (dx, dy) = µ(dx)δ[y=∇ϕ(x)] , so 2

E|X2 − Y2 | =

 R3

|x2 − ∇2 ϕ(x1 , x2 , x3 )|2 f (x1 , x2 , x3 )dx1 dx2 dx3 ,

(9)

and

2

E|g1 (X1 , Y2 , Y3 ) − Y2 |2 = E ∇2 ϕ ∗ (h(X1 , Y2 , Y3 ), Y2 , Y3 ) − Y2 



2    = E ∇2 ϕ ∗ (Y˜1 , Y2 , Y3 ) − Y2   = |∇2 ϕ ∗ (∇1 ϕ, ∇2 ϕ, ∇3 ϕ) − ∇2 ϕ|2 f (x1 , x2 , x3 )dx1 dx2 dx3 R3

 = R3

|x2 − ∇2 ϕ(x1 , x2 , x3 )|2 f (x1 , x2 , x3 )dx1 dx2 dx3 ;

(10)

the last equality comes from Theorem 11 in Villani (2002) which tells us that

∇ϕ ∗ ◦ ∇ϕ(x) = x, dµ.

(11)

Combining (9) with (10),

E|X2 − Y2 |2 = E|g1 (X1 , Y2 , Y3 ) − Y2 |2 . For the same reason,

E|X3 − Y3 |2 = E|g2 (X1 , Y2 , Y3 ) − Y3 |2 . (iii) The proof of (Y1 , Y2 , Y3 ) ∼ (h(X1 , Y2 , Y3 ), Y2 , Y3 ) is similar to that in Xu and Yan (2013), so we do not give a proof here. We will prove

(X1 , g1 (X1 , Y2 , Y3 ), g2 (X1 , Y2 , Y3 )) ∼ f (x)dx. Denote by Y˜1 = h(X1 , Y2 , Y3 ). For any bounded measurable function k,

Ek(X1 , g1 (X1 , Y2 , Y3 ), g2 (X1 , Y2 , Y3 )) = Ek(X1 , ∇2 ϕ ∗ (Y˜1 , Y2 , Y3 ), ∇3 ϕ ∗ (Y˜1 , Y2 , Y3 ))

 = R3

 = R3

k(x1 , ∇2 ϕ ∗ (∇ϕ), ∇3 ϕ ∗ (∇ϕ))f (x1 , x2 , x3 )dx k(x1 , x2 , x3 )f (x1 , x2 , x3 )dx;

the above second equality comes from the law of (X1 , X2 , X3 , Y˜1 , Y2 , Y3 ) that is π (dx, dy) = µ(dx)δ[y=∇ϕ(x)] by Theorem 11 in Villani (2002), and the above third equality comes from (11).

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P. Feng, X. Peng / Statistics and Probability Letters 84 (2014) 204–211

(iv) For (X1 , X2 , X3 , Y1 , Y2 , Y3 ) is the optimal coupling of F and G, so by Theorem 11 in Villani (2002),

(X1 , X2 , X3 , Y1 , Y2 , Y3 ) ∼ dπ (x, y) = µ(dx)δ[y=∇ϕ(x)] . Then, for any bounded Borel measurable function k,

Ek(X1 , Y2 , Y3 ) =

   R

R

k(x1 , ∇2 ϕ, ∇3 ϕ)f (x1 , x2 , x3 )dx1 dx2 dx3 R

  

k(u, v, w)f (Q −1 (u, v, w)) 

= R

R

R

which finishes the proof of (iv).

1

 dudv dw, ∇22 ϕ∇33 ϕ − (∇32 ϕ)2 (Q −1 (u, v, w))



Proposition 3.2. p satisfies the following equations:

   2  p(u, ∇2 ϕ(u, v, w), ∇3 ϕ(u, v, w)) ∇22 ϕ∇33 ϕ − (∇32 ϕ) (u, v, w) = f (u, v, w), f (u, v, w) 2  . det D ϕ(u, v, w) = ˜ f (∇ϕ(u, v, w)) Proof. It directly comes from Theorem 3.1.



Remark 3.2. Although the Monge–Ampère equation (dimension no less than 2) is fully nonlinear, it has a very rich family of invariants (Rotation, Rigid motions, Translations, Quadratic dialations, affine transformation...) from the point of Lie group. By Lie’s theorem (cf. Kumei and Bluman, 1982) and recent developments of symmetric group theory to nonlinear partial differential equations, many particular Monge–Ampére equations can be transformed 1–1 to second order elliptic equations or the group of linear partial differential equations (cf. Oliveri, 1998; Olver, 1993). Furthermore, the equation has inherent divergence structure and geometric properties, so many methods from other fields could be applied to simplify the equation in Propositions 2.2 and 3.2 (cf. Kushner, 2012). We will discuss these in our later articles. 4. Case with dimension n There are similar results with dimension n, our main result is as follows. Theorem 4.1. Let (H) hold and (X1 , . . . , Xn , Y1 , . . . , Yn ) be the optimal coupling of F and G, and then, for any 1 ≤ i ≤ n, the law of (X1 , . . . , Xi , Yi+1 , . . . , Yn ) is absolutely continuous with the Lebesgue measure λ on Rn . Proof. From Theorem 11 in Villani (2002),

(X1 , . . . , Xn , Y1 , . . . , Yn ) ∼ dµ(x)δ[y=∇ϕ(x)] . Define the mapping Q : x ∈ Rn → Q (x) = (x1 , . . . , xi , ∇i+1 ϕ(x), . . . , ∇n ϕ(x)) and the function G : Rn → R

  1      G(x) := det  ∇i+1,1 ϕ  .  .  . ∇ ϕ n ,1

..

. 1

··· .. .

∇i+1,i ϕ .. .

···

···

 ∇i+1,i+1 ϕ   .. = det  .   ∇n,i+1 ϕ

··· .. . ···

∇i+1,i+1 ϕ .. .

···  ∇i+1,n ϕ  ..  (x). .  ∇ n ,n ϕ 

From (3) and f (x) > 0, f˜ (x) > 0, ∀x ∈ Rn , det D2 ϕ(x) =

f ( x) f˜ (∇ϕ(x))

> 0,

∀x ∈ R n .

··· .. . ···

       ∇i+1,n ϕ  ..  .  ∇ ϕ  n,n

P. Feng, X. Peng / Statistics and Probability Letters 84 (2014) 204–211

211

n So, G(x) ̸= 0, ∀x ∈ Rn . Let {xk }∞ k=1 be a dense subset of R . From the implicit function theorem, for any k, there exists δk > 0 k n such that Q is injective from B(x , δk ) := {x ∈ R , |x − xk | < δk } to Rn . For any A ⊆ Rn with λ(A) = 0,

E IA (X1 , . . . , Xi , Yi+1 , . . . , Yn ) =





=



IA x1 , . . . , xi , ∇i+1 ϕ(x), . . . , ∇n ϕ(x) f (x)dx



Rn

∞   k=1

= 0;



IA∩B(xk ,δk ) x1 , . . . , xi , ∇i+1 ϕ(x), . . . , ∇n ϕ(x) f (x)dx



Rn



(12)

here, in the last equality, we have used Lemma 2.1 and the fact Q is bijective from B(xk , δk ) := {x ∈ Rn , |x − xk | < δk } to {Q (x) : x ∈ B(xk , δk )}. From (12), this theorem holds.  Remark 4.1. Actually, we can prove that if (X1 , . . . , Xn , Y1 , . . . , Yn ) is the optimal coupling of F and G, then the law of (Xi1 , . . . , Xik , Yik+1 , . . . , Yin ) is absolutely continuous with the Lebesgue measure on Rn , here, for any 1 ≤ k ≤ n, ik ∈ {1, . . . , n} and {i1 , . . . , in } = {1, . . . , n}. Acknowledgments The authors are very grateful to Professors Dong, Zhao and Zheng, Weian for their valuable discussions and suggestions. The second author was supported by ‘‘the applied mathematical research for the important strategic demand of China in information science and related fields’’ 2011CB808000 and NSFC, No. 11271356, 10721101. References Caffarelli, L.A., 2002. Nonlinear elliptic theory and the Monge–Ampère equation. ICM I, 179–187. Kumei, S., Bluman, G.W., 1982. When nonlinear differential equations are equivalent to linear differential equations. SIAM J. Appl. Math. 42, 1157–1173. Kushner, A.G., 2012. Three Lectures on Contact Geometry of Monge–Ampère Equations. Oliveri, F., 1998. Linearizable second order Monge–Ampère equations. J. Math. Anal. Appl. 218, 329–345. Olver, P.J., 1993. Application of Lie Groups to Differential Equations, second ed. Springer-Verlag, New York. Polyanin, A.D., Zaitsev, V.F., 2004. Handbook of Nonlinear Partial Differential Equations. CRC. Shen, Y.F., Zheng, W.A., 2010. On Monge–Kantorovich problem in the plane. C. R. Acad. Sci., Paris 348, 267–271. Villani, C., 2002. Topics in Mass Transportation. American Mathmatical Society. Xu, Z.Q., Yan, Jia-An., 2013. A note on the Monge–Kantorovich problem in the plane, preprint.