F Barthe

D Cordero-Erausquin

A Gaussian correlation inequality for plurisubharmonic functions

A positive correlation inequality is established for circular-invariant plurisubharmonic functions, with respect to complex Gaussian measures. The main ingredients of the proofs are the Ornstein-Uhlenbeck semigroup, and another natural semigroup associated to the Gaussian ∂-Laplacian.

Introduction

The motivation of the present work comes from a Gaussian moment inequality in C n due to Arias de Reyna [START_REF] Arias-De-Reyna | Gaussian variables, polynomials and permanents[END_REF]. We will show that his result is in fact a very particular case of a new correlation inequality, which can be seen as the complex analogue of the following correlation inequality for convex function in R n due to Hu [START_REF] Hu | Itô-Wiener chaos expansion with exact residual and correlation, variance inequalities[END_REF]: if µ is a centered Gaussian measure on R n and if f, g : R n → R are convex functions in L 2 (µ) and f is even, then f g dµ ≥ f dµ g dµ.

We will say that a function on C n is circular-symmetric if it is invariant under the action of S 1 (i.e. multiplication by complex numbers of modulus one) ; in other words a function f defined on C n is circular-symmetric if

f (e iθ w) = f (w) ∀θ ∈ R, ∀w ∈ C n .
A function u : C n → [-∞, +∞) is plurisubharmonic (psh) if it is upper semi-continuous and for all a, b ∈ C n the function z ∈ C → u(a + zb) is subharmonic. Classically, a twice continuously differentiable function u : C n → R is psh if and only if for all w, z ∈ C n j,k

∂ 2 zj z k u(z) w j w k ≥ 0,
where

∂ zj = 1 2 ∂ xj -i∂ yj , ∂ zj = 1 2 ∂ xj + i∂ yj , z = x + iy with x, y ∈ R n .
The later condition means that the complex Hessian D 2 C u is pointwize Hermitian semi-definite positive. We refer e.g. to the textbook [START_REF] Hörmander | Notions of convexity[END_REF] for more details.

We consider the standard complex Gaussian measure γ on C n , dγ(w) = dγ n (w) = π -n e -w•w dℓ(w) = π -n e -|w| 2 dℓ(w),

where ℓ denotes the Lebesgue measure on C n ≃ R 2n and w • w ′ = w j w ′ j for w, w ′ ∈ C n . For convenience, let us introduce the following class of L 2 (γ) functions with controlled growth at infinity:

G := f : C n → C ; f ∈ L 2
loc (λ) and ∃ǫ, c, C > 0 such that |f (w)| ≤ e c|w| 2-ǫ , ∀|w| ≥ C .

In particular any function (locally L 2 ) dominated by a polynomial function on R 2n belongs to G.

Our main result reads as follows:

Theorem 1 (Correlation for psh functions). Let f, g : C n → [-∞, ∞) be two plurisubharmonic functions belonging to G. If f is circular-symmetric, then f g dγ ≥ f dγ g dγ.

One can extend the result by approximation to more general psh functions in L 2 (γ). The inequality also extends to arbitrary centered complex Gaussian measure, which are images of γ by C-linear maps. Indeed composing a psh function with a C-linear map gives another psh function.

Let us give some direct consequences of this theorem. First, we see that when F , G are holomorphic functions, or simply complex polynomial functions, belonging to L 2 (γ) and F is homogeneous, then for any α, β ≥ 0 we have

|F | α |G| β dγ ≥ |F | α dγ |G| β dγ.
Indeed, if F is holomorphic f = |F | β is psh, and if F is homogeneous, then f is also circularsymmetric. This argument can also be used for products of the form

f := |F 1 | α1 . . . |F k | α k
where the F j are holomorphic and the α j 's are nonnegative real numbers, so that f is log-psh, in the sense that

log f (w) = k ℓ=1 α ℓ log |F ℓ (w)|
is psh. This implies that f is also psh, and if the holomorphic functions F j are homogeneous then f is also circular-symmetric.

Theorem 2. Let F 1, . . . , F N be a family of homogeneous polymomial functions on C n . Then for any α, . . . , α N ≥ 0 and k ≤ N -1 we have

N j=1 |F j | αj dγ ≥ k j=1 |F j | αj dγ N j=k+1 |F j | αj dγ ≥ N j=1 |F j | αj dγ.
A standard complex Gaussian vector in C n is a random vector taking values in C n according to the distribution γ = γ n . A random vector X = (X 1 , . . . , X N ) ∈ C N is a centered complex Gaussian vector if there is an n, a standard complex Gaussian vector G in C n and a C-linear map A : C n → C N such that X = AG. It turns out that the law for X is then characterized by its complex covariance matrix E(X k X ℓ ) 1≤k,ℓ≤N . Denoting by a 1, . . . a N ∈ C n the rows the matrix of A in the canonical basis, X j = G • a j . Applying the later theorem to the complex linear forms F j (w) = w • a j yields the following result. Theorem 3. Let (X 1 , . . . , X N ) ∈ C N be a centered complex Gaussian vector, and let α 1 , . . . , α N ∈ R + . Then, for any k ≤ N -1

E N j=1 |X j | αj ≥ E k j=1 |X j | αj E N j=k+1 |X j | αj (1)
and in particular

E N j=1 |X j | αj ≥ N j=1 E |X j | αj . (2)
In other words, among centered complex Gaussian vectors (X 1 , . . . , X N ) ∈ C N with fixed diagonal covariance (i.e. (E|X j | 2 ) j≤N fixed) the expectation of N j=1 |X j | αj is minimal when the variables are independent. Inequality (2) is an extension of an inequality of Arias de Reyna [START_REF] Arias-De-Reyna | Gaussian variables, polynomials and permanents[END_REF], who established the particular case where all the α j = 2p j are even integer by rewriting the left hand side in terms of a permanent of a 2m matrix (m = p j ) and using an inequality for permanents due to Lieb. Actually, Inequality (1) in the case where the α j are even integers is equivalent to Lieb's permanent inequality, so in particular we are giving a new proof of this inequality.

In the next section we will introduce the tools that will be used in the proof, that is two semigroups: the usual Ornstein-Uhlenbeck semi-group and another natural semi-group associated to the ∂ operator (the generator of which could be called, depending from the context, Landau or magnetic Laplacian). In the last section we give the proof of our correlation inequality.

Semi-groups

To get the result, we will let the circular-symmetric psh function evolve along the Ornstein-Uhlenbeck semi-group on C n = R n + iR n ≃ R 2n associated with the measure γ and the scalar product : w, w ′ = ℜ(w • w ′ ). We recall that its generator is given, for smooth functions f , writing w = x + iy, x, y ∈ R n , by

L ou f (w) := 1 4 ∆f (w) - 1 2 w, ∇ f (w) = n j=1 1 4 ∂ 2 xj xj f (w) + ∂ 2 yjyj f (w) - 1 2 x j ∂ xj f (w) + y j ∂ yj f (w) .
Note that the normalization differs slightly from the usual one on R 2n because our Gaussian measure has complex covariance Id n but real covariance equal to 1 2 Id 2n . Accordingly, the spectrum of -L ou on L 2 (γ) is here 1 2 N = {0, 1 2 , 1, . . .}. The Ornstein-Uhlenbeck semi-group P ou t = e tL ou admits the representation, for suitable functions f : C n → C,

P ou t f (z) = f (e -t/2 z + √ 1 -e -t w) dγ(w) (3)
= π -n (1 -e -t ) -n f (w) e -1 1-e -t |w-e -t/2 z| 2 dℓ(w) (4)
As usual in semi-group methods, it is convenient to work with a nice stable space of functions. Here, we can for instance consider

G ∞ := f ∈ C ∞ (C n ) ; f
and all its derivatives belong to G .

Note that for f ∈ G, we can define P ou t f by ( 3) and then we have that (t, z) → P ou t f (z) is smooth on (0, ∞) × C n , with P ou t f ∈ G ∞ and ∂ t P ou t f = L ou P ou t f for every t > 0, and P ou t f → f in L 2 (γ) as t → 0. We refer to [START_REF] Bakry | Analysis and geometry of Markov diffusion operators[END_REF] for details. Let us also mention here that with formula (3) it is readily checked that properties like convexity, subharmonicity, pluri-subharmonicity are preserved along

P ou t .
Another natural operator will be used. Indeed, since pluri-subharmonicity is characterized through a "∂ 2 zz operation", we shall also use the following differential operator on smooth functions

f on C n : Lf = n j=1 ∂ 2 zjzj f -z j ∂ zj f = n j=1 e |z| 2 ∂ zj e -|z| 2 ∂ zj f Note that Lf = 0 if (and only if, see below) f is holomorphic. Formally L = -∂ * ∂ on L 2 (γ)
equipped with the Hermitian structure (f, g) = f g dγ. More precisely, denoting for a differentiable function

∂ z f = (∂ z1 f, . . . , ∂ zn f ) and ∂ z f := (∂ z1 f, . . . , ∂ zn f )
we have the following standard fact.

Fact 4 (Integration by parts). For regular enough functions f, g : C n → C, for instance for functions in G ∞ , we have

(Lf ) g dγ = -∂ z f • ∂ z g dγ (5)
and in particular (Lf )

f dγ = -|∂ z f | 2 dγ ≤ 0.
We can also write

(Lf ) g dγ = -∂ z f • ∂ z g dγ = f (Lg) dγ (6)
where

Lf := n j=1 ∂ 2 zj zj f -z j ∂ zj f .
Indeed, it suffices to sum over j the equations

e |z| 2 ∂ zj e -|z| 2 ∂ zj f g dγ = π -n ∂ zj e -|z| 2 ∂ zj f g dλ(z) = -∂ zj f ∂ zj g dγ = -∂ zj f ∂ zj g dγ.
The assumption that f, g ∈ G ∞ guarantees that the boundary terms (at infinity) in the integration by parts vanish.

As a consequence, we see that the Gaussian measure γ is invariant for L, and actually that L is a symmetric nonpositive operator on L 2 (γ) with the above-mentioned Hermitian structure. The kernel of L is the Bargmann space H 0 formed by the holomorphic functions on C n that belong to L 2 (γ).

We want to work with the semi-group P t = e tL which is also Hermitian (formally):

(P t f ) g dγ = f P t g dγ (7)
Although we will not explicitly use it, let us discuss a bit the (well known) spectral analysis of L on the complex Hilbert space L 2 (γ). This analysis is indeed fairly standard using the ideas introduced by Landau. Following for instance the presentation given in [4, Section 4], consider the "annihilation" operators a j = ∂ zj and their adjoints, the "creation" operators b j := a * j = z j -∂ zj . Then L = -j≤n b j • a j , with [a j , b j ] = 1, and all these operators commute for distinct indices j. Plainly, if a function f and a scalar λ ∈ C satisfy -Lf = λf , then -L(a j f ) = (λ -1)a j f and -L(b j f ) = (λ+1)b j f . This implies that the spectrum of -L is N and that the eigenspace associated to the eigenvalue k ∈ N is given by the sum of the spaces b

m H 0 with m = (m 1 , . . . , m n ) ∈ N n , |m| := j≤n m j = k and the convention b m := b m1 1 • . . . • b mn n . Moreover, if we introduce the classical projection Π 0 : L 2 (γ) → H 0 onto holomorphic functions Π 0 f (z) := f (w) e z•w dγ(w) = f (z + w) e -z•w dγ(w),
then the projector Π k on the k-eigenspace can be expressed in terms of Π 0 and the creation and annihilation operators. This allows to compute the reproducing kernel of Π k , in terms of classical families of orthogonal polynomials. Next, one can sum over k and obtain the kernel K t (z, w) for e tL = k e -kt Π k . Only the formula for K t will be useful in the sequel and we shall actually check below that this suggested formula is indeed the kernel of e tL .

An explicit formula for K t can be found in [START_REF] Askour | Explicit formulas for reproducing kernels of generalized Bargmann spaces of C n[END_REF]: setting

K t (z, w) := 1 π n (1 -e -t ) n exp z • w - e -t |z -w| 2 1 -e -t -|w| 2 , (8)
then

P t f (z) = f (w) K t (z, w)dℓ(w) (9) = (1 -e -t ) -n f (w) e z•w-e -t |z-w| 2 1-e -t dγ(w)
Next, let us note that by performing the change of variable w = z + √ 1 -e -t ξ for fixed z we find

P t f (z) = f (z + √ 1 -e -t ξ) e - √ 1-e -t z•ξ dγ(ξ). (10)
On this formula, we see immediately that P 0 = Id and P ∞ = Π 0 .

To avoid discussions regarding unbounded operators and existence of semi-groups, we will proceed in the opposite direction and use the previous formula to define P t f . Actually, to be fair, we should mention that later, in the proof of our result, we only need to work with smooth functions f ∈ G ∞ ; these functions provide nice initial data, ensuring existence and uniqueness of strong solutions for the semi-group equation. Nevertheless, we feel it is of independent interest to start from the integral formula (9) or (10) and derive from it the semi-group properties ; we will first check that (9) solves (11) indeed. The drawback is that some properties that are obvious (formally) for e tL f need to be checked thoroughly when using this kernel representation, in particular because the kernel (8) is not Markovian.

Formulas ( 9)-(10) make sense pointwise, for z ∈ C n fixed, as long as f ∈ L 2 (γ); actually we have the pointwise estimate

|P t f (z)| ≤ C t (z) f L 2 (γ)
for some constant C t (z) > 0 depending on t and z. In order to derive some properties of P t f , a stronger integrability condition (as f ∈ G) will be assumed.

Lemma 5. Given f ∈ G, let us define P t f using the formula (9) or (10). Then one has that (t, z) → P t f (z) is smooth on (0, ∞) × C n , with for any t > 0, P t f ∈ G ∞ and

∂ t P t f = LP t f. (11)
Moreover we have

P t f L 2 (γ) ≤ f L 2 (γ)
and

P t f → f in L 2 (γ) as t → 0.
Proof. It is readily checked that, for any fixed w ∈ C n one has

∂ t K t (•, w) = L K t (•, w).
Moreover, for T, R, k > 0 fixed, there exists constants c = c(T, R, k), C = C(T, R, k) such that for F = ∂K t , or F = K t or else F being any of the first kth partial derivatives of K t , it holds that |F (z, w)| ≤ Ce -c|w| 2 for all w ∈ C n , all t ∈ (0, T ) and all |z| ≤ R. From this and the definition of G, we can call upon dominated convergence to conclude to the smoothness of (t, z) → P t f (z) and to the fact that

P t f ∈ G ∞ with ∂ t P t f = LP t f.
Regarding the contraction property, we want to avoid direct computations or spectral arguments, and so we make a detour and use some obvious but important properties of P t .

Fist, we will use the semi-group property: for t, s > 0 that P t • P s = P t+s . This can be seen in two ways. One can invoke that for a smooth function F := P t0 f ∈ G ∞ then there is existence and uniqueness for the equation ∂ t F t = LF t with initial condition F 0 = F and we have seen that (P t0+t f ) is a solution; from there one can conclude. Or else, in a more pedestrian way, one can check that

K t (z, w)K s (ξ, z) dℓ(z) = K t+s (ξ, w).
For this, one may use that given z, ξ ∈ C n and c ∈ C with ℜ(c) > 0,

e z•w+ξ•w e -c|w| 2 dℓ(w) = π n c -n e z•ξ /c . (12)
Next, recall that P t is Hermitian, in the sense [START_REF] Malicet | Squared chaotic random variables: new moment inequalities with applications[END_REF], on G ⊂ L 2 (γ); this can be seen directly from the integral formula since K t (z, w) = K t (w, z). Finally, we will use that P t f L 2 (γ) decreases for t ∈ (0, ∞). This is immediate from the non-positivity of L since for t > 0 we have

P t f ∈ G ∞ and d dt |P t f | 2 dγ = 2 (LP t f )P t f dγ = -2 |∂ z P t f | 2 dγ ≤ 0.
So for our f ∈ G and t > 0 we have

P t f 2 L 2 (γ) = (P 2t f ) f dγ ≤ P 2t f L 2 (γ) f L 2 (γ) ≤ P t f L 2 (γ) f L 2 (γ)
which implies the contraction property in L 2 (γ).

To prove the continuity at t = 0 in L 2 (γ) we first assume that f is smooth and compactly supported. Using (10) we see that P t f converge point-wise to f and that for t ∈ (0, 1) we have |P t f (z)| ≤ C e c|z| for some constant c, C > 0; so we can conclude by dominated convergence. For f ∈ G and ǫ > 0, introduce g smooth compactly supported such that f -g L 2 (γ) ≤ ǫ and let δ > 0 be such that t ≤ δ ensures that P t g -g L 2 (γ) ≤ ǫ holds. For t ≤ δ,

P t f -f L 2 (γ) ≤ P t f -P t g L 2 (γ) + P t g -g L 2 (γ) + g -f L 2 (γ) ≤ 2 f -g L 2 (γ) + ǫ ≤ 3ǫ.
This establishes the desired continuity.

Remark 6 (Contraction property).

1. We observe that some results, which can be deduced from the spectral decomposition and Hilbertian analysis, may be derived in a soft way thanks to flexible semigroup techniques.

We have proved that starting from formula (9) we have

P t f L 2 (γ) ≤ f L 2 (γ)
on the dense subspace G, which together with the pointwise estimate given just before the previous Lemma implies by density that

P t L 2 (γ)→L 2 (γ) ≤ 1.
Formally, by letting t → ∞ in P t L 2 (γ)→L 2 (γ) ≤ 1 we recover that

Π 0 L 2 (γ)→L 2 (γ) ≤ 1.
Actually, the convergence of P t towards Π 0 can be quantified rigorously through Hörmander's inequality,

ϕ -Π 0 ϕ 2 L 2 (γ) ≤ ∂ z ϕ 2 L 2 (γ) = (-Lϕ)ϕ dγ
valid for any suitable ϕ, for instance for ϕ ∈ G ∞ . Note that from formula (10), P t acts as the identity on holomorphic functions, so P t Π 0 = Π 0 P t = Π 0 . A classical Grönwall type argument (using the previous Lemma to justify the computation of

d dt |P t (f -Π 0 f )| 2 dγ) ensures that for f ∈ G and t ≥ 0 P t f -Π 0 f 2 L 2 (γ) ≤ e -2t f -Π 0 f 2 L 2 (γ) .
2. In analogy with the Markovian case P ou t we may wonder if P t is also a contraction on some L p (γ). However, for any p = 2 we have

P t L p (γ)→L p (γ) = +∞,
as it can be seen by taking, in dimension n = 1, for a ∈ R,

f a (w) := e aw+w , w ∈ C.
Indeed,repeated applications of (12) with c = 1 show, setting s t := √ 1 -e -t and using (10), that P t f a (z) = e s 2 t a e az+(1-s 2 t )z and that

P t f a p L p (γ) f a p L p (γ) = C(t, p) e a s 2 t (p-p 2 /2) .
The next result describes how derivatives and P t commute, an important issue in semi-group methods.

Lemma 7 (Commutation relations). For any suitable f , say f ∈ G ∞ , and t > 0 we have for every

1 ≤ j ≤ n and z ∈ C n , ∂ zj (P t f (z)) = P t (∂ zj f )(z) and ∂ zj (P t f (z)) = e -t P t (∂ zj f )(z).
Proof. We use (10). The first equality is obvious. For the second one, setting

s t = √ 1 -e -t , we have ∂ zj (P t f )(z) = P t (∂ zj f )(z) -s t f (z + s t ξ) e -st zξ ξ j dγ(ξ), and
π -n f (z + s t ξ) e -st zξ ξ j e -ξ•ξ dℓ(ξ) = -π -n f (z + s t ξ) e -st zξ ∂ ξj (e -ξ•ξ ) dℓ(ξ) = s t (∂ zj f )(z + s t ξ) e -st zξ dγ(ξ) = s t P t (∂ zj f )(z).
Now, and for the rest of this section, we focus on the case of circular-symmetric functions. Given θ ∈ R and a function f we denote f θ the function f θ (w) = f (e iθ w). Note that

P t (f θ ) = (P t f ) θ . ( 13
)
Recall that a function f is said to be circular-symmetric if f θ = f for every θ. It is worth noting that a holomorphic function on C n is necessarily constant when circular-symmetric. Indeed if h : C → C has both properties then invariance and the Cauchy formula give h(1)

= 2π 0 h(e iθ )dθ/(2π) = h(0); next if f : C n → C is holomorphic and circular symmetric, then for any (z 1 , . . . , z n ) ∈ C n , the function h : z ∈ C → f (zz 1 , . . . , zz n ) is also holomorphic and circular-symmetric, hence f (z 1 , . . . , z n ) = h(1) = h(0) = f (0). Accordingly if f ∈ L 2 (γ) is circular-symmetric then Π 0 f ≡
f dγ since the gaussian density is also circular-symmetric. Actually, much more can be said, as we shall see below.

Let us first investigate the relation between L and L ou . Note that one can write

Lf = L ou f + i 2 n j=1 y j ∂ xj f -x j ∂ yj f . (14)
So we have

L ou = ℜ(L) = L+L 2
where Lf = n j=1 ∂ 2 zj zj f -z j ∂ zj f has a kernel formed by the anti-holomorphic functions. The operators L and L are Hermitian symmetric, whereas L ou is symmetric for the real and the Hermitian product, and preserves the subspace of real valued functions. As we said, its spectrum is -1 2 N, as can be seen also from the formula L ou = L+L 2 . Let us illustrate this on two examples, obtained by applying the creation operator b 1 = z 1 -∂ z1 to the holomorphic functions z → 1, and z → z 1 . The function z → z 1 is an eigenfunction for L with eigenvalue -1, for L with eigenvalue 0, and for L ou with eigenvalue -1/2. The function z → |z 1 | 2 -1 is an eigenfunction for L with eigenvalue -1, for L with eigenvalue -1, and for L ou with eigenvalue -1.

The special role played by circular-symmetric functions is due to the fact that these operators, and the associated semi-groups, coincide for them.

Lemma 8 (Action of L and P t on circular-symmetric functions). If f is a smooth circularsymmetric function, then we have Lf = Lf = L ou f.

In particular we have, when f, g ∈ G ∞ and f is circular-symmetric,

(Lf )g dγ = f Lg dγ = -∂ z f • ∂ z g dγ.
Also, if f ∈ L 2 (γ) is circular-symmetric then we have

P t f = P ou t f
for every t ≥ 0.

Proof. Writing w = x + iy, x, y ∈ R n , the symmetry rewrites as f ((cos(θ)x -sin(θ)y) + i(cos(θ)y + i sin(θ)x)) = f (x + iy). Taking the derivative at θ = 0 we find

n j=1 -y j ∂ xj f (x + iy) + x j ∂ yj f (x + iy) = 0,
and this for every x, y ∈ R n . This implies in view of ( 14) that Lf = L ou f = Lf . Next, for any smooth function g we have, using that Lf = Lf and ( 6),

(Lf )g dγ = (Lf )g dγ = f Lg dγ = -∂ z f • ∂ z g dγ.
Although it is formally trivial that equality of L and L ou on circular-symmetric functions implies equality of the semi-groups P t and P ou t , a bit more should be said since we defined the semi-group using the explicit formula (9). And it is anyway instructive to compute the kernels on circularsymmetric functions. Denote by K ou t the kernel of the Ornstein-Uhlenbeck semi-group that we recalled above:

K ou t (z, w) = π -n (1 -e -t ) -n e - 1
1-e -t |w-e -t/2 z| 2

. So we have, setting c t := e -t/2 and s t := √ 1 -e -t ,

K ou t (z, w) = π -n s -2n t e -s -2 t |w| 2 -s -2 t c 2 t |z| 2 e s -2 t ct(w•z+w•z) and K t (z, w) = π -n s -2n t e -s -2 t |w| 2 -s -2 t c 2 t |z| 2 e s -2 t (c 2 t w•z+w•z) .
Note that only the last exponentials differ in these two formulas. When f is circular-symmetric, in order to check that P t f = P ou t f it suffices to check that for fixed w, z, t one has Observe that for a, b ∈ C, we have

1 2π 2π 0 e ae iθ +be -iθ dθ = 1 2π 2π 0 p,q∈N a p b q p!q! e i(p-q)θ dθ = n≥0 (ab) n (n!) 2 = B(ab)
with B(x) := n≥0

x n (n!) 2 . Therefore, we have

1 2π 2π 0 K t (z, e iθ w) dθ = π -n s -2n t e -s -2 t |w| 2 -s -2 t c 2 t |z| 2 B(s -4 t c 2 t |w • z| 2 ) = 1 2π 2π 0 K ou t (z, e iθ w) dθ,
as wanted.

3 Proof of Theorem 1

First, let us note that we can assume that g is smooth, and actually that g ∈ G ∞ . Indeed, if g ∈ G then P ou t g ∈ G ∞ and we mentionned that P ou t g converges to g in L 2 (γ) and therefore also in L 1 (γ), as t → 0. Consequently, if we know the conclusion for a function in G ∞ , then f P ou t g dγ ≥ f dγ P ou t g dγ and by passing to the limit when t → 0 we know it also for g ∈ G. For the same reason, we can assume that f ∈ G ∞ , recalling that P ou t f is circular-symmetric when f is. So in the sequel, we are given two psh functions f, g ∈ G ∞ , with f circular-symmetric. As in the proof of the correlation for convex functions [START_REF] Hu | Itô-Wiener chaos expansion with exact residual and correlation, variance inequalities[END_REF], we will compute some kind of second derivative in t for integrals involving P ou t f ; recall that ∂ t P ou t f = L ou P ou t f . The novelty is that, along the way, we will also use P t f which satisfies ∂ t P t f = LP t f (Lemma 5).

Consider α(t) := (P ou t f ) g dγ = (P t f ) g dγ ∈ R.

The function α is, by construction, smooth on (0, ∞) and continuous on [0, ∞) (see Lemma 5 for the continuity at zero). Since P ou t f tends to the constant f dγ when t → ∞, we have

α(t) → f dγ g dγ
In order to conclude, it suffices to show that α decreases. Actually we will prove that α is convex; which is enough, since a convex function with a bounded limit at +∞ cannot increase. It holds

α ′ (t) = L ou (P ou t f ) g dγ = LP t f g dγ. (15)
Since P t f is also circular-symmetric, we can invoke Lemma 8 and write

α ′ (t) = -∂ z P t f • ∂ z g dγ.
Next, using the first commutation relation from Lemma 7 we get

α ′ (t) = - n j=1 P t (∂ zj f )∂ zj g dγ
We stress that ∂ z f is no longer circular-symmetric, so we cannot exchange P t and P ou t . The second derivative of α is, using Fact 4,

α ′′ (t) = - n j=1 L(P t (∂ zj f ))∂ z j g dγ = n j=1 ∂ z (P t (∂ zj f ))•∂ z (∂ z j g) dγ = n j,k=1 ∂ z k P t (∂ zj f ) ∂ 2 z k zj g dγ.
Using the commutation relation from Lemma 7 we can write

α ′′ (t) = n j,k=1 ∂ 2 z k zj (P t f ) ∂ 2 z k zj g dγ = Tr (D 2 C P t f )(z) (D 2 C g)(z) dγ
where for a C 2 function h on C n the notation D 2 C h(z) refers to the Hermitian matrix ∂ 2 zj z k h(z) j,k≤n . Since P t f = P ou t f and g are psh, the corresponding matrices are nonnegative Hermitian matrices, which means that the trace of their product is still nonnegative. This shows that α ′′ ≥ 0 and finishes the proof of the theorem.

We would like to conclude with a discussion of the differences between the real case and the complex case. After all, we are computing second derivatives of the same object α(t) = (P ou t f ) g dγ along the Ornstein-Uhlenbeck semi-group exactly as in the case of convex functions, so what is going on?

In both cases we prove that α decreases by showing that α ′ ≤ 0 using the next derivative somehow, but we compute these derivatives differently. The argument for convex function goes as follows. A direct computation and usual commutation properties show that, if ∇f dγ = 0, which is the case when f is even, then α ′ (t) = -e -t/2 ∞ t Tr (D 2 P ou s f )(z) (D 2 g)(z) dγ(z) e s/2 ds where D 2 refers to the usual (real) Hessian on R 2n ; from this we conclude to the correlation for convex functions. On the other hand, we have proved, when f is circular-symmetric, that

α ′ (t) = - ∞ t
Tr (D 2 C P ou s f )(z) (D 2 C g)(z) dγ(z) ds.

Note that we have used here that α ′ (t) tends to 0 when t → +∞: this follows from the fact that α is convex and has a finite limit at +∞, and can also be seen from ( 15) since P ou t f tends to a constant when t → +∞. It is because we wanted to work with complex derivatives that we aimed at inserting L in place of L ou ; recall that ∂ zj f need not be circular-symmetric even when f is, although the second derivatives are again circular-symmetric.

Finally, let us observe that if we consider in dimension 1 the circular-symmetric psh functions f (w) = |w| 1/3 and g(w) = |w| 4 on C ≃ R 2 , then, as we already mentioned,

Tr (D 2 C f ) (D 2 C g) = ∆f ∆g ≥ 0, but a direct computation shows that Tr (D 2 f )(z) (D 2 g)(z) = - 4 3 |z| 1/3 ≤ 0 ∀z ∈ C.
Of course, this discrepancy cannot hold at all times for P ou t f in place of f (and moreover f is not smooth at zero, although this is not really an issue). But it suggests that the two formulas above for α ′ (t) are indeed quite different.

  t (z, e iθ w) dθ.

Acknowledgements: We thank Laurent Charles, Vincent Guedj and Pascal Thomas for useful discussions, and for pointing out to us relevent references.