A connection of the Brascamp-Lieb inequality with Skorokhod embedding

We reveal a connection of the Brascamp-Lieb inequality with Skorokhod embedding. Error bounds for the inequality in terms of variance are also provided.


Introduction
The Brascamp-Lieb moment inequality plays an important role in statistical mechanics, such as in the analysis of gradient interface models; see, e.g., [8,7,10]. It asserts that centered moments of a distribution with log-concave density relative to a Gaussian distribution do not exceed those of that Gaussian's; it is used to derive the tightness of finite-volume Gibbs measures describing the static interface, strict convexity of the associated surface tension, and so on.
The Skorokhod embedding problem is to find a stopping time T for one-dimensional Brownian motion B such that B(T ) is distributed as a given probability measure on R. The problem was proposed by Skorokhod [14] and a number of solutions have been constructed since then ( [12]); they are applied to the proof of Donsker's invariance principle, robust pricings of options in mathematical finance (see, e.g., [11]), and so on.
In this paper, we reveal a connection between the Brascamp-Lieb inequality and the Skorokhod embedding of Bass [1]; as a by-product, we also provide error bounds for the inequality in terms of variance by applying the Itô-Tanaka formula. Let Y be an n-dimensional Gaussian random variable defined on a probability space (Ω, F , P ) with law ν. Let X be an n-dimensional random variable on (Ω, F , P ), whose law µ is given in the form with V a convex function on R n such that In what follows, we fix v ∈ R n (v = 0) arbitrarily. For a one-dimensional random variable ξ, we denote its variance by var(ξ): Here a · b denotes the inner product of a, b ∈ R n . We also set The result of this paper is stated as follows.
More precisely, we have where ψ ′′ (dx) denotes the second derivative of ψ in the sense of distribution.
(ii) For every p > 1, it holds that Here C(a, ψ, q) ∈ [0, ∞] is given by with q the conjugate of p: Taking ψ(x) = |x| and some manipulation show that for any convex V .
The rest of the paper is organized as follows: In Section 2 we prove Theorem 1.1. The Brascamp-Lieb inequality (1.2) is proved in Subsection 2.1; we devote Subsection 2.2 to the proof of (1.3) and (1.4); in Subsection 2.3 we prove Lemma 2.1, which plays an essential role in the proof of Theorem 1.1. In the appendix we discuss an extension of the Brascamp-Lieb inequality to the case with V not necessarily convex.
For every function f on R and x ∈ R, we denote respectively by f ′ + (x) and f ′ − (x) the right-and left-derivatives of f at x if they exist. For each x, y ∈ R, we write x ∧ y = min{x, y} and x + = max{x, 0}. Other notation will be introduced as needed.
2 Proof of Theorem 1.1 In this section we give a proof of Theorem 1.1. Without loss of generality, we may assume that ν is centered: E[Y ] = 0. Moreover, Theorem 4.3 of [4] reduces the proof to the case n = 1; that is, the density of the law P • (v · X) −1 relative to the onedimensional Gaussian measure P • (v · Y ) −1 is log-concave. Therefore in what follows, we take the Gaussian measure ν in (1.1) as and V as a convex function on R. We accordingly write X and Y for v · X and v · Y , respectively; that is, X is distributed as µ and Y as ν.

Proof of (1.2)
In this subsection we prove the inequality (1.2) in Theorem 1.1. We denote by F µ the distribution function of µ: We also set Apparently g is strictly increasing. By convexity of V we have moreover We postpone the proof of this lemma to Subsection 2.3. Once this lemma is shown, the proof of (1.2) is straightforward from the Skorokhod embedding of Bass [1]; for other types of embeddings, we refer the reader to the detailed survey [12] by Ob lój. Let {W t } t≥0 be a standard one-dimensional Brownian motion on (Ω, F , P ).
Proof of (1.2). Note that g(W 1 ) is distributed as µ. Applying Clark's formula to g(W 1 ) yields where for 0 ≤ s ≤ 1 and y ∈ R, By the Dambis-Dubins-Schwarz theorem (see, e.g., [13, Theorem V.1.6]), there exists a Brownian motion We know from [1] that T := 1 0 a(s, W s ) 2 ds is a stopping time in the natural filtration of B. Moreover, by (2.2) and Lemma 2.1, we have T ≤ a P -a.s. We denote by {L x t } t≥0,x∈R the local time process of B. For every x ∈ R, Tanaka's formula yields which is equal, by (2.3), (2.4) and E [B(T )] = 0, to the right-hand side of (2.5) with ψ(0) subtracted. Hence (2.5) holds. As T ≤ a a.s. and ψ ′′ ≥ 0, it is immediate from and B(a) , the inequality (2.6) follows readily from the optional sampling theorem applied to the submartingale {ψ(B(t))} 0≤t≤a .

Proof of (1.3) and (1.4)
In this subsection we prove the inequalities (1.3) and (1.4) in Theorem 1.1. We keep the notation in the previous subsection. By (2.5), the proof is reduced to showing the following proposition.
for all x ∈ R.
(2) For every p > 1, it holds that for all x ∈ R.
To prove these estimates, we prepare a lemma.
Lemma 2.2. For every t > 0 and x ∈ R, we have Proof. The first equality is seen from the occupation time formula. The second is due to the identity for every x ∈ R, which is deduced from Lévy's theorem for Brownian local time. The third one follows from change of variables.
The proof of the proposition then proceeds as follows. Recall T ≤ a a.s.
Proof of Proposition 2.1.
(1) By the strong Markov property of Brownian motion, .
(2.13) By (2.12), this is rewritten as (2.14) Using Fubini's theorem and Jensen's inequality, we bound this from below by By the optional sampling theorem and Schwarz's inequality, Plugging this and using the identity between (2.12) and (2.10) lead to where the equality follows from Wald's identity and from (2.7). This proves (2.8).
(2) First we show that for every t > 0 and x ∈ R, We note the identity (y − |x − z|) + = (z − x + y) + ∧ (x + y − z) + for z ∈ R, to bound the expectation in the integrand from above by Here for the inequality, we used the optional sampling theorem; the equality follows from the monotonicity of E (B(a) − x + y) + in x and the symmetry in the sense that E (B(a) − (−x) + y) + = E (x + y − B(a)) + . Therefore (2.17) is dominated by where we changed variables with u = z+y √ a+t and v = tz−ay √ at(a+t) for the equality. Now (2.16) follows from the identity between (2.12) and (2.10). By (2.13), (2.10) and Hölder's inequality, Moreover, by (2.16), Combining these leads to (2.9) and ends the proof of Proposition 2.1.

Proof of Lemma 2.1
We conclude this section with the proof of Lemma 2.1; the assertion itself is nothing but that of [5,Theorem 11]. Here we give a different proof. To begin with, note that we only need to to consider the case a = 1; indeed, setting . Therefore the assertion of Lemma 2.1 is equivalent to Note that V remains convex. From now on we let a = 1. We start with These also hold true with which is nothing but the first inequality. The latter is proved similarly.
We also utilize the following: 1), is concave and symmetric with respect to ξ = 1/2, and satisfies Proof. A simple calculation shows Since Φ −1 : (0, 1) → R is increasing, the concavity follows. The symmetry and values at boundary are obvious.
Using the above two lemmas, we prove Lemma 2.1.

Proof of Lemma 2.1.
Since , the assertion of the lemma with a = 1 is equivalent to By Lemma 2.3, we have b(x) ≥ 1 and for all x ∈ R. We take ξ ∈ (0, 1) sufficiently small so that Since Φ ′ • Φ −1 is increasing on (0, 1/2] as seen from Lemma 2.4, it then holds that by (2.20). Therefore, for ξ sufficiently small, which is nonnegative since for every fixed c ≥ 1, we have by Lemma 2.4. We thus obtain the former inequality in (2.19). By considering V (−x) and using the symmetry of Φ ′ • Φ −1 , we also have the latter. Note that G is both rightand left-differentiable since F ′ µ is and since F −1 µ is monotone. Suppose now that G has a local minimum at some ξ 0 ∈ (0, 1).

Appendix
In this appendix we discuss an extension of the Brascamp-Lieb inequality (1.2) to the case with potential function V not necessarily convex. To avoid complexity, we restrict ourselves to one-dimension; generalizations to multidimension may be done by considering one-dimensional marginals. Recently, gradient interface models with nonconvex potential have been studied with great interest, see, e.g., [2,6,3]; we expect that the result presented here has a contribution to that study. A type of Brascamp-Lieb inequalities with nonconvex potential is also discussed by Funaki and Toukairin [9,Section 4] with some restriction on convex ψ.
For a given α > 0, suppose that the function k ∈ C 1 (R) satisfies and let the distribution µ on R be give in the form where the normalizing factor Z ′ = R e −U (x) dx is equal to √ 2π. Let X be a random variable distributed as µ, and Y a centered Gaussian random variable with variance 1/α. Under the above assumption, we have Proposition A.2. For every convex function ψ on R, it holds that Proof. Since the distribution function F µ of µ is written as the function g defined by (2.1) is equal to k −1 , the inverse function of k. Therefore by assumption (A.1), we have g ′ (x) ≤ 1/ √ α for all x ∈ R, hence the same proof as that of (1.2) applies.
Remark A.1. (1) Lemma 2.1 indicates that, by suitably adding a constant, the function of the form with V convex can be expressed as (A.2) for some k satisfying (A.1).
(2) In addition to (A.1), if we assume that for some β > α, then we also have the reverse inequality for every convex ψ. Here Y ′ is a centered Gaussian random variable with variance 1/β.
We conclude this paper with two examples of U.
Example A.1 (double-well type). Take α = 1 and k(x) = x + x 3 . Then This potential U has a double-well near the origin.
Example A.2 (log-mixture of centered Gaussians). For given p, q > 0 and 0 < a < b such that we take Then the corresponding U is expressed as This type of potentials is dealt with in [2,6,3]. The function k satisfies To prove the lower bound, it is sufficient to take x ≤ 0 by symmetry. Then, as the denominator of (A.8) is dominated by from which we obtain the upper bound in (A.7). We end this example with a remark that this upper bound also holds true in a general situation where k is given by for a positive measure ρ on (0, ∞) such that its support is included in (0, b] and ∞ 0 ρ(dκ) √ κ = 1.
The potential U corresponding to this k is given in the form which is referred to as a log-mixture of centered Gaussians in [3].
Remark A.2. For U given by (A.6), a concrete calculation shows that in fact (A.3) holds with α = a, which gives a better bound than the one discussed above because p 2 ≤ a by the relation (A.5).