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Log-concave measures

We study the log-concave measures, their characterization via the Pr\'ekopa-Leindler property and also define a subset of it whose elements are called super log-concave measures which have the property of satisfying a logarithmic Sobolev inequality. We give some results about their stability. Certain relations with measure transportation of Monge-Kantorovitch and the Monge-Amp\'ere equation are also indicated with applications.


Introduction
The importance of logarithmically concave functions and measures has been discovered in the 70's (cf. [10,11,9,2]) and they have found applications immediately in physics (cf. [13]). This notion has gained further importance when its close relations to Monge-Ampère equation and more generally to measure transportation has been realized (cf. [16] and the references there). In this work we give a general treatment of the subject beginning from the finite dimensional case and going towards to the infinite dimensions. A preliminary definition of log-concave measures is given using the Lebesgue measure then we extend this definition to the measures without density by using the Prékopa-Leindler property which characterizes them, which is also equivalent to Brunn-Minkowski property. We also introduce the notion of super log-concave measures, namely these are the measures which decrease very rapidly at infinity, in fact faster than some Gaussian measure and they always satisfy the logarithmic Sobolev inequality. Their definition uses the Euclidean structure of the underlying space on which they are defined, consequently, we need a special structure if we want to extend this notion to the infinite dimensional case. This is done by adjoining a rigged Hilbert space structure to the Fréchet space supporting the measure under question, whose typical example is an abstract Wiener space with its Cameron-Martin space. Afterwards we concentrate ourselves to the case of Wiener space, first we give some complementary results about the Jacobians corresponding to the image of the Wiener measure under a general perturbation of identity with lacking regularity and show that although each term of the Jacobian is not properly defined, their multiplication may create a renormalization, which is the typical case with monotone shifts. Even in this case, the log-concave character of the Jacobian is preserved and we use this observation to give another proof of the Prékopa-Leindler property for the Gaussian measure with "less log-concave" functions (called 1-log concave) and we give also another proof of this property using the reverse martingale convergence theorem, in particular we prove that the property of the functions used to test Prékopa-Leindler property is preserved under the conditional expectations and the Ornstein-Uhlenbeck semigroup.

Preliminaries and notations
Generally each section contains its notational conventions with the exception of the Gaussian case; in fact we denote by (W, H, µ) an abstract Wiener space, namely, W is a separable Fréchet space, H ⊂ W is a separable Hilbert space densely injected into W , µ is the unit Gauss measure supported by W which is quasi-invariant under the translation by the elements of H. One defines the usual Gateaux derivative of the nice functions on W along the subspace H, due to the quasi-invariance, this derivative has a unique (closed) extension to all the L p (µ)-spaces with p ≥ 1 and it is denoted by ∇ and called Sobolev derivative (cf. [14] for instance). Hence, for a nice function f on W , ∇f defines a linear functional on H µ-almost surely, consequently it can be identified with an element of H ⋆ = H. Since the Sobolev derivative maps the scalar functions to H-valued functions, its adjoint, called the divergence operator, maps the vector valued functions to scalar ones and denoted by δ, in particular, we have for ξ : W → H cylindrical, well-known as the integration by parts formula. The very remarkable property of the divergence is that in the case of classical Wiener space W = C([0, 1], IR n ), if ξ has an adapted Lebesgue density ξ ′ (t, w), then it holds that where the integral with dW denotes the Itô stochastic integral.

Log-concave and super log-concave measures in finite dimension
We begin by an initial concept: Definition 1. A (positive) measure ρ on IR d is said to satisfy the Prékopa-Leindler property if for any positive, continuous functions of compact support, say a, b, c such that for any x, y ∈ IR d , s + t = 1, one has Theorem 1. Assume that θ is a log-concave function, denote by ρ the measure dρ(x) = θ(x)dx.
Here are two other well-known results which are due to Prékopa, [10], that we derive using the above considerations: Corollary 1. Let f and g be two integrable, log-concave functions, then their convolution f ⋆ g is log-concave. Moreover, for any c ∈ IR, the function f ⋆ c g which is defined as is again a log-concave function. Proof: ) t , hence we can apply the theorem and obtain the inequality which characterizes the log-concavity. The proof of the second part is similar.
Then F is log-concave on IR d .
Proof: Let f x (ξ) denote f (x, ξ), then for s + t = 1, x, y ∈ IR d , ξ, η ∈ IR m , we have The results above indicate that the following definition is reasonable Definition 2. A (positive) measure on IR d is called log-concave if any of its convolutions with log-concave continuous functions of compact support has a log-concave density.
Remark: Let us note that, for a measure ρ to be log-concave, it suffices the existence of just one continuous, log-concave function θ such that θ ε ⋆ ρ has a log-concave density for any ε > 0 where θ ε denotes the d-dimensional rescaling of θ. In fact using the commutativity of convolutions, we obtain also thatθ ⋆ ρ has again a log-concave density, for any other continuous log-concave function of compact supportθ.
Proposition 1. Let ρ 1 and ρ 2 be two log-concave measures on IR n and IR m respectively. Then the product measure ρ 1 ⊗ ρ 2 is log-concave on IR n × IR m .
Proof: Let θ be a log-concave function as described above, then the density of θ ⋆ ρ is the limit of the densities of (θ ⋆ ρ n ).
The following theorem is the first pavement to extend the definition of log-concavity to the infinite dimensional case where the Lebesgue measure does not exist: Theorem 2. ρ is a log-concave measure if and only if it satisfies the Prékopa-Leindler property.
Proof: Suppose that ρ has a continuous density F with respect to the Lebesgue measure of IR d . Let f = 1 A , g = 1 B , where A and B are two balls whose centers are located at a and b respectively and whose diameter will tend to zero. Let C = 1 2 (A + B) and h = 1 C then h(αx + βy) ≥ f (x) α g(y) β for any α + β = 1, x, y ∈ IR d . Hence for α = 1 2 , by taking the limit, we obtain which is a sufficient condition for the log-concavity of F . The general case follows by taking the convolution of ρ. The necessity follows from the transport argument that we have used in the proof of Theorem 1. Proof: If the relation (3.2) is satisfied by ρ, then it is also satisfied by ρ ⋆ θ where θ is a log-concave function, since ρ(θ ⋆ a) = (ρ ⋆ θ)(a).
Corollary 4. Let ρ be a log-concave measure and let F be a convex function, then the measure ν defined as Proof: The new measure obviously satisfies the Prékopa-Leindler property.
A very close characterization of the log-concave measures can be given by the Brunn-Minkowski inequality whose proof is similar to that of Theorem 2 Theorem 3. The measure ρ is log-concave if and only if for any measurable A, B and s + t = 1.
Assume that a measure ρ can be represented as where V is a locally integrable, lower bounded function such that Proof: Evidently the condition implies the convexity of the function Remark 2. If ρ i are finitely many α i -s.l.c. measures on IR ni , then their product is an min i α i -s.l.c. measure.
The proof of the following lemma follows from that of Theorem 2: if and only if for any a, b, c continuous, positive functions of compact support such that, for any x, y ∈ IR n , s + t = 1, a(sx where, for a given function f , f α is defined as The proof of the following is obvious: is a sequence of measures converging weakly to ρ. Assume that ρ n is α n -s.l.c. for any n ≥ 1, then ρ is α 0 = inf n α n -s.l.c.
Lemma 3. Assume that dρ is an α-s.l.c. measure and denote by p σ the Gaussian density exp − 1 2σ |x| 2 . Then, for any δ > 0 satisfying Proof: We want to determine the set of δ's for which the function x → e δ 2 |x| 2 (ρ ⋆ p σ )(x) is log-concave. Let us denote by ρ α the measure defined by We can write which is a Gaussian kernel provided that is a log-concave function from Corollary 2.
Proof: Assume first that ρ has a density w.r. to the Lebesgue measure l. We can write IR m = Im(F ) + ker(F ), then Since, by Corollary 1, Theorem 4. Assume that ρ is an α-s.l.c. measure on IR n , then it satisfies the logarithmic Sobolev inequality: for any smooth function f with ρ(f 2 ) = 1.
Consequently, applying the logarithmic Sobolev inequality for the Gaussian measure (cf. [8]) The general case now follows from Lemma 3 and a limit procedure.

Infinite dimensional case
The following is basic: for any u, v ∈ E and s + t = 1. Then ρ satisfies the Prékopa-Leindler property: Proof: We can suppose that ρ has convex, compact support K and replace E by IR IN by injection. Then we can replace K by a product of compact intervals J = ∞ 1 J n . If f and g are continuous, cylindrical functions on J, for x ∈ J, define The function k is then continuous on J and we have, from the finite dimensional case, If f and g are upper semi-continuous on J, there exist two sequences of continuous and cylindrical functions (f n ) and (g n ), decreasing to f and g respectively. Hence, (k n , n ≥ 1), where k n is defined as above, converges to k as defined above and we again have Finally, if f and g are only Borel measurable, there exist two monotone, increasing sequences (f n ), (g n ) whose elements are upper semi-continuous such that lim n ρ(f n ) = ρ(f ) and lim n ρ(g n ) = ρ(g). Since we have h ≥ k n , it follows that The proof of the above theorem contains also the proof of the following Lemma 5. In order the Prékopa-Leindler to hold it is necessary and sufficient that it holds only for the continuous functions f, g, h such that h(sx + ty) ≥ f (x) s g(y) t for any x, y ∈ E and s + t = 1.
The proof of the following theorem is quite similar to the proof of Theorem 5: Theorem 6. Assume that E and F are two separable Fréchet spaces with ρ and ν satisfying the Prékopa-Leindler property on E and F respectively. Then the product measure ρ ⊗ ν satisfies also the Prékopa-Leindler property on E × F .

The following definition is now justified:
Definition 4. A Radon measure on a locally convex space E is called log-concave if it satisfies the Prékopa-Leindler property.
The following result is immediate: Proposition 4. The image of a log-concave measure under any linear, continuous map is again log-concave.
From Lemma 5 we get at once Corollary 5. Let (ρ n , n ≥ 1) be a sequence of log-concave measures converging weakly to a measure ρ, then ρ is also log-concave. Corollary 6. Let ρ be a bounded measure on IR IN , let us denote by (π n , n ≥ 1) the canonical finite dimensional projections. The measures (π n (ρ), n ≥ 1) are log-concave if and only if ρ is log-concave.
Proof: We can write ρ as the weak limit of the sequence of measures (π n (ρ) ⊗ δ n , n ≥ 1) where δ n is the image under I − π n of the Dirac δ 0 measure on IR IN .
Assume now that E is a separable Fréchet space and assume that H is a separable Hilbert space continuously and densely injected into E. We identify H with its continuous dual, hence E ⋆ ⊂ H ⊂ E. Choose a sequence (ẽ i , i ≥ 1) from E ⋆ such that its image under the injection E ⋆ ֒→ H, denoted as (e i , i ≥ 1) is a complete, orthonormal base of H. Define π n on E as (1) if lim n π n (x) = x ρ-almost everywhere, (2) if π n (ρ) is α-s.l.c. on the Euclidean space spanned by {e 1 , . . . , e n }, for any n ≥ 1.
We have the following result which is the immediate consequence of the finite dimensional case (cf. Theorem 4): Theorem 7. If ρ is an α-s.l.c. measure on E, then it satisfies the logarithmic Sobolev inequality: for any smooth, cylindrical function f with ρ(f 2 ) = 1.

The case of abstract Wiener space
While working in this frame one encounters often the difficulty of defining a proper Jacobian due to the lack of regularity of the corresponding transformation. This happens especially in the case of the measure transportation theory. Consequently it is reasonable to push the ways to extend as much as possible the notion of Jacobian of a transformation with unsufficient regularity. Lemma 6. Assume that U = I W + u is a monotone shift with u ∈ ID p,1 (H), p > 1. Then for any positive f ∈ C b (W ), where and det 2 (I H + ∇u) denotes the modified Carleman-Fredholm determinant.
Proof. The necessary background about the subject and the proof follows from [15], Theorem 6.3.1.

Remark: If
A is a nuclear operator on a Hilbert space, then det 2 (I H + A) is defined as and this function has an analytic extension to the space of Hilbert-Schmidt operators, consequently, the log-concavity of the ordinary determinant implies the log-concavity of the map A → det 2 (I H +A).
Lemma 7. Assume that U = I W + u is a monotone shift and u ∈ ID p,0 (H) = L p (µ, H) for some p > 1. Then there exists some λ(u) ≥ 0 a.s., E[λ(u)] ≤ 1 and for any positive and measurable f . In particular, U µ is absolutely continuous w.r.to µ on the set on which λ(u) > 0.
Proof. Let (P t , t ≥ 0) be the Ornstein-Uhlenbeck semigroup, let U n = I W + u n , with u n = P 1/n u. Then apply Lemma 6 to U n . Define λ(U ) does exist and, from the Fatou lemma, it is integrable and satisfies the claim.
Corollary 7. The map U → λ(U ) is almost surely log-concave on the set of monotone shifts.
Proof. If U i = I W + u i , with u i ∈ L p (µ, H), i = 1, 2 are two monotone shifts, define u n i = P 1/n u i , U n i = I W + u n i , i = 1, 2. Then, by the log-concavity of the Carleman-Fredholm determinant, for s, t ∈ [0, 1] with s + t = 1, we have Λ(sU n 1 + tU n 2 ) ≥ Λ(U n 1 ) s Λ(U n 2 ) t a.s. If we take the lim inf of both sides as n → ∞, the inequality is preserved.
Proposition 5. Assume that U n = I W + u n , n ≥ 1, is a sequence of shifts, where u n ∈ ID p,1 (H), for some p > 1. Assume that U n µ ≪ µ for all n ≥ 1 and denote Assume further that L n • U n Λ(U n ) = 1 a.s. for all n ≥ 1, also that L n → L in L 1 (µ) and finally that U n → U = I W + u in L 0 (µ, W ). Then L n • U n → L • U , Λ(U n ) → λ(U ) in L 0 (µ) and we have a.s. Besides, under the additional hypothesis: the sequence (Λ(U n ), n ≥ 1) is uniformly integrable, hence it converges to λ(U ) also in L 1 (µ) and we have Proof. Since L n → L in L 1 (µ) and U n → U in probability, it follows from the Lusin theorem that L n • U n → L • U in probability, hence (Λ(U n ), n ≥ 1) converges in probability and hence The rest is obvious, since the last hypothesis implies precisely the uniform integrability of the sequence (Λ(U n ), n ≥ 1). Remark 3. Proposition 5 is astonishing in the sense that we do not make any assumption about the convergence of the H-valued parts of the shifts at all. Assume now that u n → u in L 0 (µ, H), then, although δu either ∇u do not exist, the sequence (det 2 (I + ∇u n )e −δun , n ≥ 1) converges in probability to a non-trivial limit that we denote by J(U ), hence λ(U ) can be represented as

5.2.
The transport case and applications. Assume that L ∈ L log L(µ), E[L] = 1. Let T = I W + ∇φ, with φ in ID 2,1 , be the transport map which maps dµ to Ldµ whose properties are proved in [6]. Define L n = E[P 1/n L|V n ], where V n is the sigma algebra generated by {δe 1 , . . . , δe n }, (e i , i ≥) ⊂ W ⋆ being an orthonormal basis of H. Let T n = I W + ∇φ n , φ n ∈ ID 2,1 be the transport map which maps dµ to L n dµ. Recall that φ n is 1-convex, since it is V n -measurable, ∇ 2 φ and Lφ are lower bounded distributions, hence they are measures. We have for any f ∈ C b (W ). Besides, since L n > 0 a.s., we have where ∇ 2 a φ n and L a φ n denote respectively the absolutely continuous parts of the measures ∇ 2 φ n and Lφ n and det 2 denotes, as usual, the modified Carleman-Fredholm determinant (c.f. [15] for further information). It follows from [6] that φ n → φ in ID 2,1 , (L n , n ≥ 1) being uniformly integrable, L n • T n → L • T in probability, hence Λ(T n ) → λ(T ) also in probability, where λ(T ) can be represented as In [6], we have shown that the sequence (L a φ n , n ≥ 1) is a submartingale with respect to the increasing sequence of sigma algebras (V n , n ≥ 1) and the inequality L a φ n ≤ − log Λ n implies and Jensen inequality implies that which is a sufficient condition for the almost everywhere convergence of the submartingale (L a φ n , n ≥ 1) whose limit we denote as L(φ) ∈ L 1 (µ). Note that, as a consequence of this observation, combined with the convergence of (Λ(T n ), n ≥ 1), we deduce also the convergence of (det 2 (I H + ∇ 2 a φ n ), n ≥ 1) in probability. Moreover, we have also for any measurable, positive f and moreover, this inequality becomes an equality if log L is integrable. Proposition 6. Let L 1 , L 2 be as above, denote by T 1 , T 2 the corresponding transport maps and define M = aT 1 + bT 2 , where a + b = 1, a ≥ 0. Define M n = aT n 1 + bT n 2 as in Corollary 7 and define finally λ(M ) = lim inf n Λ(M n ). We then have M µ is absolutely continuous w.r.to µ and a.s. Hence λ(M ) > 0 a.s. This result combined with the following consequence of the Fatou lemma for any 0 ≤ f ∈ C b (W ), implies the absolute continuity M µ ≪ µ. Proof: First we shall prove the case q = 1, then the general case can be reduced to this particular case by replacing a, b and c by a q, b q and by c q respectively. Moreover, by replacing a, b, c by a ∧ n, b ∧ n and c ∧ n we may suppose that they are bounded. Finally, by multiplying them with adequate constants, we can also suppose that their integrals w.r.to µ are all equal to unity. Let T 1 = I + ∇φ 1 and T 2 = I + ∇φ 2 be the transport maps such that T 1 µ = b · µ, T 2 µ = c · µ, where l · µ denotes the measure with density l. It follows from above explanations, λ(T 1 ) and λ(T 2 ) are well-defined and b • T 1 λ(T 1 ) = c • T 2 λ(T 2 ) = 1 a.s. Let M = sT 1 + tT 2 , then M µ ≪ µ and as explained above Remark: Here is another proof of the theorem: let (π n , n ≥ 1) be a sequence of orthogonal projections, constructed from the elements of W ⋆ , of the Cameron-Martin space H increasing to identity such that lim π n w = w µ-a.s. Let w n = π n w and w ⊥ n = w − w n . For a measurable function f on W , denote the partial map w n → f (w n + w ⊥ n ) exp − 1 2 |w n | 2 by f w ⊥ n . The hypothesis above is equivalent to (cf. [5]) n ] t and the (second) proof follows from the (reverse) martingale convergence theorem. The following concept has been studied already in [5]: Definition 6. A measurable, IR + -valued function on W is called 1-log concave if, for any s + t = 1, s ≥ 0, f (x, y) µ 2 (dy) is 1-log concave on (W 1 , H 1 , µ 1 ).
For any k, k ′ ∈ H 2 , we have
Proof: We can suppose that a, b, c are bounded. Denote by a n , b n , c n the conditional expectations of a, b, c respectively w.r.to V n . Let now B and C be V n -measurable sets, hence sB + tC is also V n -measurable (recall that V n is completed!). It then follows from Theorem 8 sB+tC a n dµ = Since this true for any V n -measurable set, it follows from Lemma 8 that a n , b n and c n satisfy the inequality claimed in the first part of the theorem (with the Gaussian measure). To prove the second part, we can replace a, b, c by a n , b n , c n since P τ commutes with the conditional expectation w.r.to V n . Hence the problem is reduced to the finite dimensional case. Let us denote again by the same notation the Ornstein-Uhlenbeck semigroup on IR n . Then we can write sB+tC P τ a n (x)dµ(x) = sB+tC IR n a n (y)q τ (x, y)dydµ(x) , where q τ (x, y) = (2π(1 − e −2τ )) −n/2 exp − |y − e −τ x| 2 2(1 − e −2τ ) , which is a log-concave function in two variables. We can also write (sB + tC) × IR n = s(B × IR n ) + t(C × IR n ) .
It then follows from the Prékopa-Leindler inequality in IR n × IR n that sB+tC P τ a n dµ ≥ B P τ b n dµ s C P τ c n dµ t .
It follows then from Lemma 8 that we have P τ a n (w + sh + tk) exp − 1 2 |sπ n h + tπ n k| 2 H ≥ P τ b n (w + h) exp − 1 2 |π n h| 2 H s P τ c n (w + k) exp − 1 2 |π n k| 2 H t almost surely and we can pass to the limit as n → ∞ due to the martingale convergence theorem.