Free Diffusions, Free Entropy and Free Fisher Information

FREE DIFFUSIONS, FREE ENTROPY AND FREE FISHER INFORMATION

Philippe BIANE a, Roland SPEICHER b a CNRS, DMA, École Normale Supérieure, 45, rue d’Ulm, 75005 Paris, France b Department of Mathematics and Statistics, Queen’s University, Jeffery Hall, Kingston, ON K7L 3N6, Canada

Received 26 January 2000, revised 17 November 2000

ABSTRACT. – Motivated by the stochastic quantization approach to large N matrix models, we = − 1 study solutions to free stochastic differential equations dXt dSt 2 f(Xt )dt where St is a free brownian motion. We show existence, uniqueness and Markov property of solutions. We define a relative free entropy as well as a relative free Fisher information, and show that these quantities behave as in the classical case. Finally we show that, in contrast with classical diffusions, in general the asymptotic distribution of the free diffusion does not converge, as t →∞,towards the master field (i.e., the Gibbs state).  2001 Éditions scientifiques et médicales Elsevier SAS

RÉSUMÉ. – Nous étudions des équations différentielles stochastiques du type dXt = dSt − 1 2 f(Xt )dt où St est un mouvement brownien libre, suggérées par la quantification stochastique des modèles matriciels de grande taille. Nous établissons l’existence et l’unicité des solutions, ainsi que leur caractère Markovien. Nous définissons une entropie libre relative et une information de Fisher relative, adaptées à ces diffusions et montrons que ces quantités se comportent comme leurs analogues classiques. Enfin nous montrons qu’en général, contrairement à ce qui se passe dans le cas classique, la distribution de la diffusion ne converge pas vers l’état de Gibbs.  2001 Éditions scientifiques et médicales Elsevier SAS

1. Introduction

The purpose of this paper is to start the study of diffusion equations where the driving noise is a free brownian motion. Reasons for considering such equations will be explained in the next sections of this introduction.

E-mail addresses: [email protected] (P. Biane), [email protected] (R. Speicher). Part of this work was completed while the second author was visiting the IHP in Paris. This author gratefully acknowledges hospitality from this institution. 582 P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606

1.1. Gibbs states and diffusion theory

Let V be a C2 function on Rd , with Z = e−V(x)dx < ∞. Rd

The probability measure 1 µ(dx) = e−V(x)dx (1.1.1) Z is called the Gibbs state associated with the potential V . A well known way of obtaining the Gibbs state (useful for example in a Monte-Carlo simulation), is to construct the Rd − 1 ∇ diffusion process on , with drift 2 V , which is the solution to the stochastic differential equation 1 dX = dB − ∇V(X)dt. (1.1.2) t t 2 t d Here Bt is a brownian motion on R . Then, for any initial distribution of the diffusion, the distribution of Xt converges, as t →∞to the Gibbs state. More precise statements can be given if one introduces the following quantities. For two mutually absolutely continuous probability measures let ν(dx) H(ν| µ) = log ν(dx) (1.1.3) µ(dx) R be the relative entropy of ν with respect to µ. Recall that H(ν | µ) 0 with equality = = ν(dx) only if µ ν. If the density p µ(dx) is differentiable, let 2 ∇p(x) I(ν| µ) = ν(dx) (1.1.4) p(x) R be the relative free Fisher information. When µ is the Gibbs state (1.1.1), and ν(dx) = q(x)dx, one has H(ν| µ) = q(x)log q(x)dx + V (x)q(x)dx + log Z R R = H(ν)+ V (x)q(x)dx + log Z, (1.1.5) R H(ν)being the entropy of ν,and 2 ∇q(x) I(ν| µ) = +∇V(x) µ(dx). (1.1.6) q(x) R P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 583

For µt (dx), the distribution of X at time t, one has ∂ 1 H(µ | µ) =− I(µ | µ) ∂t t 2 t so that the relative entropy is nonincreasing. Furthermore one has

H(µt | µ) → 0ast →∞.

Exponential rate of convergence to 0 is obtained when µ satisﬁes a Logarithmic Sobolev Inequality, i.e. 1 H(ν| µ) I(ν| µ) 2ρ for some positive constant ρ, and for all measures ν such that the right hand side is ﬁnite, see, e.g., [15] for a discussion.

1.2. Matrix models

In these models one considers the limit, as N →∞, of the quantities 1 1 −NtrN (P (M)) trN Q(M) e dM, (1.2.1) ZN N k (HN ) where P and Q are non-commutative polynomials in k non-commuting indeterminates, k trN denotes the trace over N × N matrices, the integral is over the set (HN ) of k-tuples M = (M1,...,Mk) of Hermitian N × N matrices and −NtrN (P (M)) ZN = e dM.

k (HN )

For fixed P , if the limit exists for all Q, then it can be put (via the GNS construction) in the form 1 1 − lim tr Q(M) e NtrN (P (M)) dM = τ Q(X) , (1.2.2) →∞ N P N ZN N k (HN ) where X = (X1,...,Xk) is a k-tuple of self-adjoint elements in some von Neumann algebra AP , equipped with a normal tracial state τP .Thek-tuple of operators X is called the master field, see [7,11]. Proving its existence seems to be a difficult problem, with implications in the physics of quantum fields. In one dimension (i.e., for a one-matrix model), the situation is well understood, and the master field is then characterized by its distribution, which is the probability measure on the real line whose moments are given n by τP (X ); n 0. This probability measure achieves the unique global maximum of the functional

P (ν) = log |x − y| ν(dx)ν(dy) − P(x)ν(dx) R R 584 P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 on probability measures ν on R, and thus it is uniquely determined by the polynomial P . It exists if and only if the polynomial P is non-constant and bounded below on R. In fact potentials more general than polynomials can be considered, see [16] where such maximization problems are thoroughly considered. In the multi-matrix case, very little is known unless the polynomial P splits as a sum

P(M1,...,Mk) = P1(M1) +···+Pk(Mk) for some one-variable polynomials P1,...,Pk. Then the master ﬁeld (X1,...,Xk) is known to consist of free random variables in the sense of Voiculescu [22]. The distribution of each of the variables is obtained by resolving the corresponding one- matrix model, while their joint distribution, i.e., the computation of all moments τP (Q(X)) is obtained by Voiculescu’s freeness prescription. As soon as there is a nontrivial interaction between the components of the matrix model, we do not know any way to prove the existence of the master ﬁeld, and no explicit formula for the joint moments.

1.3. Free stochastic quantization

A stochastic quantization approach to the master ﬁeld has been proposed in the physical literature, see [12,6,7]. At the level of the matrix models this means looking, as in (1.1.2), at the solution to the diffusion equation (also called Langevin equation)

1 dM = dB − N∇(tr P )(M )dt, (1.3.1) t t 2 N t

k 2 where B denotes a brownian motion on (HN ) , normalized so that E[trN (Bi(t) )]= 2 N t. In terms of the components M1,...,Mk this gives

1 dM (t) = dB (t) − N∂ P M (t) dt, (1.3.2) i i 2 i i where ∂i is the ith partial cyclic derivative on polynomials in k non-commuting indeterminates (X1,...,Xk),givenby

n ∂ia1Xi a2Xi ···an−1Xian = akXiak+1 ···Xiana1Xi ···Xi ak−1, k=2 when a1,...,an are polynomials in the other variables. Remark that if %P is the map (M1,...,Mn) → trN P(M1,...,Mn) then the cyclic derivative satisﬁes ∇i%P (M).H = trN H∂iP(M) . P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 585

By the ﬁnite dimensional Gibbs state result, we know that the measure

1 − e NtrN (P (M)) dM ZN is the large t limit of the distribution of Mt . As explained above, the master field should be obtained by taking the large N limit of this Gibbs state. Free diffusions arise when one tries exchanging the large t and large N limits, so that one considers first the large N limit of (1.3.1). In taking the large N limit, one first rescales the time by 1/N,sothat the brownian motion Bt/N converges as N →∞towards the free brownian motion (see, e.g., [1]), namely one looks at the equation

1 dM (t) = dB (t/N) − ∂ P M(t) dt i i 2 i and when N →∞the equation becomes

1 dX (t) = dS (t) − ∂ P(X )dt, (1.3.3) i i 2 i t where Xi (t) are the unknown non-commutative random variables, and Si(t) are free brownian motions. The problem is to understand the large t limit of the solution to Eq. (1.3.3). One hopes that when t →∞the Xi(t) will converge to the master field. As we shall see, in general this procedure fails to recover the master fields, but in an interesting way. We shall give a rigourous mathematical treatment of Eq. (1.3.3), with special emphasis on the case of the one-matrix models. In this case, we shall consider the equation 1 dX(t) = dS(t)− f(X )dt (1.3.4) 2 t for some class of drift f , and prove under regularity assumptions on f that Eq. (1.3.4) with given initial value admits a unique solution. We shall define quantities analogous to (1.1.3) and (1.1.4) which play the same role for free diffusions. We shall then show that in general the distribution of the solution Xt fails to converge to the master field distribution as t →∞.

1.4. Large matrix heuristics

In this section we shall give a quick heuristic derivation, based on the large matrix approximation, of some of the results we prove rigorously below. Let f : R → R be a smooth function, and let us consider the stochastic differential equation on N × N Hermitian matrices 1 dM = dB − f(M)dt, t t/N 2 t where f is meant to act by functional calculus for Hermitian operators. Because of the time scaling, the brownian motion Bt/N is associated with the rescaled Hilbert–Schmidt 1 ∗ scalar product N trN (AB ). It follows from the expression of the Laplace operator in 586 P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 polar coordinates on HN (see, e.g., [8]) that the eigenvalues (λ1(t), . . . , λN (t)) of the matrix Mt satisfy a stochastic differential equation 1 1 1 1 dλ (t) = √ dβ (t) + dt − f λ (t) dt, i i N λ − λ i N 1jN i j 2 i=j where the βi are independent one dimensional brownian motions. The generator of the eigenvalue diffusion process is thus the sum of the diffusive term √1 + (where + is the 2 N usual Laplace operator), an entropic term 1 N 1 ∂ , − N i=1 1jN λi λj ∂λi i=j N 1 ∂ and the drift term = f(λi) . We now argue that when N is large, the brownian i 1 2 ∂λi term in this equation becomes small in front of the other terms, and in finite time, the process behaves like a dynamical system with a small random perturbation (see, e.g., [9] for the theory of such dynamical systems). In the large N limit, the trajectory of the eigenvalue vector behaves as that of the flow of the deterministic vector field N 1 1 1 ∂ − f(λ ) . − i (1.4.1) i=1 N 1jN λi λj 2 ∂λi i=j 1 N →∞ Assume that the empirical distribution N i=1 δλi (t) converges, as N ,towards some limit distribution λt (dx) = pt (x) dx, then for any smooth test function g one has ∂ 1 N 1 N 1 1 g λ (t) = g λ (t) − f λ (t) i 2 i − i ∂t N i=1 N i=1 j=i λi(t) λj (t) 2 and taking the large N limit, one gets ∂ 1 g(x)p (x) dx = g(x) Hp (x) − f(x) p (x) dx, ∂t t t 2 t R where u(y) Hu(x):= p.v. dy x − y R is (up to a multiplicative constant) the Hilbert transform. The flow equation gives then the following “free Fokker–Planck equation” for pt ∂p ∂ 1 t =− p Hp − f . (1.4.2) ∂t ∂x t t 2 P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 587

1 ∇ On the other hand, the vector ﬁeld (1.4.1) is 2 0N for the function

1 N N 0N (λ) = log |λi − λj |− F(λi), N i=1 i=j i=1 where F is a primitive of f . It follows that along the trajectories of the ﬂow, the quantity 1 |∇ |2 0N (λ(t)) is increasing with derivative 2 0N (λ(t)) . One has 1 lim 0N λ(t) = log |x − y|pt (x)pt (y) dx dy − F(x)p(x)dx := F (pt ) N→∞ N R R (1.4.3) and 2 1 2 1 lim ∇0N λ(t) = 4 Hpt (x) − f(x) pt (x) dx := IF (pt ). (1.4.4) N→∞ N 2 R

We obtain from this the differential relation ∂ 1 (p ) = I (p ) (1.4.5) ∂t F t 2 F t which, as one can check at least formally, follows from (1.4.2). The two expres- sions (1.4.3) and (1.4.4) are closely related to the free entropy and the free Fisher information measure defined by Voiculescu [18] (which correspond to the case where f = 0). We shall prove, using the free Ito’s formula, that the distribution of the solution of the free stochastic differential equation (1.3.4) indeed satisfies, in a weak sense, the free Fokker–Planck equation (1.4.2), and that (1.4.5) holds. As we are dealing with gradient flows, we know that in general the trajectory of such a gradient flow converges to a local maximum of the function, but depending on the initial distribution, and may not converge towards the global maximum. We shall see that such a phenomenon occurs for the free stochastic differential equation, and this explains why we should not expect, in general, to get the master field as the large t limit distribution of the free diffusion. It should be possible to make our arguments rigourous and prove the formulas above for the free diffusions, starting from the matricial approximations. However such an approach, which relies on an understanding of the eigenvalues of the approximating matrices is restricted to the case of one matrix models, whereas we want to derive methods applicable to multimatrix models. Therefore we shall follow another route and work directly on the limiting objects, i.e., semi-circular systems. ∂p = The stationary (i.e., ∂t 0) Fokker–Planck equation reads 1 p Hp − f = 0. (1.4.6) 2 This is the Schwinger–Dyson equation of the matrix model (cf. [7,11]). As we see, this does only give us the equation for a local maximum of the relative free entropy (1.4.3), and so it may have solutions which are not given by the master field. 588 P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606

Although the preceding arguments give us some insight into the free diffusions, they do not provide a complete picture. Indeed the one dimensional distributions of the free diffusion are not enough to determine the process, in particular as we shall see the free diffusion is a non-commutative Markov process with probability transition functions satisfying a linearized version of the free Fokker–Planck equation. This cannot be seen if one only looks at the behaviour of the eigenvalues of the matrix models, indeed the Markov property really comes from the behaviour of the eigenspaces of the diffusing matrix.

1.5.

This paper is organized as follows: in Section 2 we recall some preliminary facts from free probability theory. In Section 3 we introduce the free diffusion equation and obtain existence and uniqueness results, with bounds on the solutions. We also establish the Markov property of the solution. We specialize to the one-dimensional case in Section 4, where we derive the Fokker–Planck equation, and discuss more thoroughly the Markov property of the free diffusion. An Euler scheme for the approximation of the solution is described in Section 5, and used to prove regularity properties of the distribution of the free diffusion. These regularity properties are then used in Section 6 to prove the relation between relative free entropy and free Fisher information. Finally in Section 7 we consider the asymptotic behaviour of the free diffusion.

2. Preliminaries and notations

2.1. Non-commutative probability spaces and free random variables

In this paper we will consider non-commutative probability spaces which are von Neumann algebras with a faithful, normal tracial state. We refer to [22,20,14,3], for further information on the basics of free probability. We shall recall some facts about freeness with amalgamation. Let (A,τ)be a non-commutative probability space, and let B be a unital weakly closed subalgebra of A, then we denote by τ(.| B) the conditional expectation onto B. One deﬁnes B-free independence of subalgebras of A containing B, in a similar way as free independence, using the conditional expectation τ(.| B) in place of the state τ, see, e.g., [14].

LEMMA 2.1. – Let (A,τ) be a von Neumann non-commutative probability space, ∗ let B1, B2 ⊂ A be free von Neumann-subalgebras, and let X = X ∈ B1, then the algebras (B2 ∪{X}) and B1 are {X} -free, and for any Y ∈ (B2 ∪ X) one has ∗ ∗ τ(Y | B1) = τ(Y | X). Furthermore τ(.| X) maps C (B2,X)onto C (X).

Proof. – Let b1,...,bn ∈ B2, then it follows from the moment cumulant formula [14], that for any b ∈ B1 the expression τ(b1Xb2X ···bn−1Xbnb) can be expressed as a linear combination of products of the form k v ls l τ bi τ X τ X b , r=1 i∈Ir s=1 P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 589 where {1,...,n}=I1 ∪···∪Ik, n = l + l1 +···+lv. We deduce that the conditional expectation of b1X ···bn−1Xbn onto B1 is of the form P(X) for some polynomial P . The assertions in the lemma follow easily from this observation. ✷

2.2. Free brownian motion

Let (A,τ) be a von Neumann non-commutative probability space. We shall assume A A ∗ that is ﬁltered, so that there exists a family ( t )t∈R+ of unital, weakly closed - subalgebras of A, such that As ⊂ At for all s,t with s t. Further we shall assume A that there exists an ( t )t∈R+ -free brownian motion (St )t∈R+ , i.e., each St is a self adjoint element of A with semi-circular distribution of mean zero and variance t, one has Xt ∈ At for all t 0, and for all s,t with s t, the element St − Ss is free with As, and has semi-circular distribution of mean zero and variance t − s. Once this brownian motion exists, one can deﬁne stochastic integrals of biprocesses with respect to S, as in [5]. The main results about stochastic integrals that we shall use are the free Burkholder–Gundy inequality (Theorem 3.2.1 of [5]), and the free Itô’s formula (Theorem 4.1.2 of [5], or the functional form, see Section 4.3).

2.3. Operator Lipschitz function

Let f : R → C be a locally bounded measurable function, it is called an operator Lipschitz function if there exists a constant K>0suchthat

f(X)− f(Y) KX − Y (2.3.1) for all self-adjoint operators X, Y on a Hilbert space. The function f is called locally operator Lipschitz if for every A>0 there exists a constant KA > 0 such that (2.3.1) holds for all self-adjoint operators X, Y , of norm less than A. Clearly, an operator Lipschitz function is a Lipschitz function, but the converse is not true, and in fact being a C1 function does not insure that the function is locally operator Lipschitz. Examples of operator Lipschitz functions are functions of the form f(x)= eixyµ(dy), R where µ is a bounded complex measure such that R |x||µ|(dx) < ∞, (this follows from Duhamel’s formula, see, e.g., Section 1.2 in [5]). From this one can infer easily that C2 functions are locally operator Lipschitz. More precise description of the classes of operator Lipschitz and locally operator Lipschitz functions can be given in terms of Besov spaces, see, e.g., Peller [13]. If (A,τ) is a non-commutative probability space, we shall also consider more Ak → A A A generally functions Q : sa sa (where sa is the self-adjoint part of ), and call such functions locally Lipschitz if there exists constants C(K); K>0 such that for all Xi,Yi with norms K, one has

Q(X1,...,Xk) − Q(Y1,...,Yk) C(K) Xi − Yi . i=1 590 P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606

Examples of such functions include self-adjoint polynomials in non-commuting indeterminates, but also many functions not given by functional calculus.

3. Existence of multidimensional free diffusions

Let (A,τ)be a ﬁltered non-commutative probability space, as in Section 2.2, in which S1(t), . . . , Sk(t) (t 0,) a k-dimensional free brownian motion, is deﬁned. Each Si(t) is an At -free brownian motion, and {S1(t) | t 0},...,{Sk(t) | t 0} are free in (A,τ). Ak → A Let Q1,...,Qk : sa sa be k locally Lipschitz functions, such that each Qi maps Ak A s,sa to s,sa for all s 0. Consider the system of stochastic differential equations dXi (t) = Qi X1(t), . . . , Xk(t) dt + dSi(t) (1 i k) (3.1.1) which means that we are looking for maps t → Xi(t) with values in A, such that

t Xi (t) = Xi(0) + Si(t) + Qi X1(s), . . . , Xk(s) ds for all t>0, (3.1.2) 0 where Xi (0) are the initial data.

THEOREM 3.1. – Assume the following condition is satisﬁed for some constants a ∈ R and b 0:

k k + + 2 + Qi (X1,...,Xk)Xi Xi Qi(X1,...,Xk) 1 a Xi b (3.1.3) i=1 i=1

∈ A = 0 ∈ A = for all X1,...,Xk sa. Then, given arbitrary initial conditions Xi (0) Xi 0 (i 1,...,k), the system (3.1.1) of stochastic differential equations has a unique solution (X1(t), . . . , Xk(t)) for all t 0. Furthermore, we have Xi(t) ∈ At for all i = 1,...,k and all t 0, and the maps t → Xi(t) are norm continuous. Proof. – We will construct the solution by the Picard iteration method. In order to keep the Lipschitz constants bounded we have to truncate the polynomials for big norms. This will be done by a function h : [0, ∞) →[0, 1] which has the following properties: there exists R>0 such that h is identically 1 on [0,R] and identically 0 on [2R,∞]; h is continuous, 0 h(t) 1forallt 0, and h has a ﬁnite Lipschitz constant, i.e., there exists a C>0suchthat h(t) − h(s) C|t − s| for all t,s 0.

Then we truncate a given locally Lipschitz function Q by going over to

k f(X1,...,Xk) := Q(X1,...,Xk)h Xi . i=1 P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 591

LEMMA 3.2. – There exists a constant c>0 such that for X1,...,Xk,Y1,...,Yk ∈ A, we have the estimate:

f(X1,...,Xk) − f(Y1,...,Yk) c Xi − Yi. i=1

Proof. – Put mX := max X1,...,Xk and mY := max Y1,...,Yk .

Note that mX 2R implies f(X1,...,Xk) = 0. Thus the estimate is trivially satisﬁed in the case mX 2R and mY 2R. Consider now the case mX 2R and mY < 2R.Thenwehave

f(X1,...,Xk) − f(Y1,...,Yk) = f(Y1,...,Yk) k = · Q(Y1,...,Yk) h Yi . i=1

Now note that Q(Y1,...,Yk) is bounded on the set given by mY < 2R and that k k k = − h Yi h Yi h Xi = = = i 1 i 1 i 1 k k k − − C Yi Xi C Yi Xi . i=1 i=1 i=1

The case mX < 2R and mY 2R is analogous. So assume ﬁnally that both mX < 2R and mY < 2R.Thenwehave

f(X1,...,Xk) − f(Y1,...,Yk) k k · − Q(X1,...,Xk) h Xi h Yi i=1 i=1 k + − · Q(X1,...,Xk) Q(Y1,...,Yk) h Yi i=1 k

Q(X1,...,Xk) · C Yi − Xi+ Q(X1,...,Xk) − Q(Y1,...,Yk) . i=1

The assertion follows now by noticing that Q(X1,...,Xk) remains bounded on the set given by mX < 2R and that we have on the set given by mX < 2R and mY < 2R an estimate of the form

Q(X1,...,Xk) − P(Y1,...,Yk) c˜ Xi − Yi i=1 for some constant c˜. ✷ 592 P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606

We can now approximate the solution of the system (3.1.1) by replacing the functions Qi by their truncated versions

k fi (X1,...,Xk) = Qi (X1,...,Xk)h Xj j=1 for a function h as above with some ﬁxed R>0. Thus we consider now the system of stochastic differential equations dXi (t) = fi X1(t), . . . , Xk(t) dt + dSi(t) (1 i k) (3.1.4) with the initial conditions = 0 Xi (0) Xi . This can be solved by Picard iteration applied to the integral equations

t = 0 + + Xi (t) Xi fi X1(s), . . . , Xk(s) ds Si(t), 0 which gives in the usual way the existence and uniqueness of the solution of the system (3.1.4). Since the whole construction above depends via the function h on the R R parameterR we will denote this solution by (X1 (t), . . . , Xk (t)). It is also clear that as k R R R long as i=1 Xi (t) R, this solution (X1 (t), . . . , Xk (t)) is also a solution of the original problem (3.1.1). To ensure that for R →∞we get a solution of (3.1.1), we thus need an argument ensuring that the norms of our solutions do not explode in ﬁnite time. To see this let us consider the function

k 2 Z(t) := Xi(t) . i=1 An application of the free Ito’s formula yields

k k −aτ −aτ 2 −aτ 2 d e Z(τ) =−ae Xi(τ) dτ + e d Xi (τ) i=1 i=1

k −aτ 2 =−ae Xi(τ) dτ i=1 k −aτ + e fi X1(t), . . . , Xk(t) Xi(τ) + Xi(τ)fi X1(t), . . . , Xk(t) + 1 dτ i=1 k −aτ + e dSi (τ)Xi(τ) + Xi(τ) dSi(τ) . i=1 By the norm continuity of stochastic integrals with respect to their upper bounds, the R norms Xi (t) are continuous functions of the time t. Let us consider P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 593 = { | k R } TR inf t i=1 Xi (t) >R ,thenTR > 0andfort TR, by the hypothesis on the functions Qi ,wehave

t k t t at −aτ at −aτ −aτ Z(t) Z(0) + e b e dτ + e dSi(τ)Xi (τ)e + Xi(τ)e dSi (τ) . = 0 i 1 0 0 By using the Burkholder–Gundy inequality in operator norm for stochastic integrals with respect to free brownian motion [5] we obtain b Z(t) Z(0)+ eat − 1 a

t t k + at −aτ + −aτ e dSi(τ) Xi (τ)e Xi (τ)e dSi(τ) = i 1 0 0 t 1/2 k √ b at 2 −2aτ at Z(0)+ e − 1 + 2 · 2 2 Xi (τ) e dτ e a = i 1 0 t 1/2 b √ Z(0)+ eat − 1 + k · 4 2max Z(s) e−2aτ dτ eat a 0st 0 b e2at − 1 Z(0)+ eat − 1 + k · 4max Z(s) . a 0st a Put now ϕ(t) := max Z(s). 0st

Then we have, for all t TR b ϕ(t) ϕ(0) + eat − 1 + 4k ϕ(t) e2at − 1 /a. a

2 2 2 At time TR one has max Xi (TR) R/k, therefore R /k ϕ(TR) R , hence 2 2 b aT R /k ϕ(0) + e R − 1 + 4kR e2aTR − 1 /a. a

From this we deduce a lower bound for TR,sothatTR →∞as R →∞if a>0, =∞ R = and TR for R large enough if a<0. Thus for ﬁxed t, one has Xt Xt for all R R0 and if a<0, then there exists a uniform bound on the solution for all times, i.e., maxi,t0 Xi(t) < ∞. ✷ 3.2. Markov property of the free diffusion

Deﬁne for t 0 the following C∗ and von Neumann subalgebras of A; F 0 = ∗ { ; ; = ; } ; X 0 = ∗ ; = ; t C X0 Si(s) i 1,...,k s t t C Xi(t) i 1,...,k 594 P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 G0 = ∗ { ; − ; = ; } t C Xi (t) Si(s) Si(t) i 1,...,k s t , F = F 0 ; X = X 0 ; G = G0 t t t t t t which represent respectively the past, present and future of the process. Observe X 0 ⊂ F 0 ∩ G0 that t t t . The following proposition is an immediate consequence of the uniqueness of the solution of (3.1.1), and of Lemma 2.1.

PROPOSITION 3.3. – The algebras Ft and Gt are Xt -free, furthermore the conditional | F G0 X 0 expectation τ(. t ) maps t onto t .

This means that the process Xt is a free Markov process with respect to the ﬁltrations F G X 0 → X 0 and . It follows that for all times s

4. One dimensional free diffusions and the Fokker–Planck equation

4.1. The free Fokker–Planck equation

Let f : R → C be a locally operator Lipschitz function, such that

−xf (x) ax2 + b for all x ∈ R, (4.1.1) and let X be the solution to 1 dX = dS − f(X )dt (4.1.2) t t 2 t which exists according to Theorem 3.1. Let ϕ be a C2 function on R, then by the free Itô’s formula ([5, Section 4.3]), one has

t t t 1 1 ϕ(X ) = ϕ(X ) + ∂ϕ(X )>dS − ϕ(X )f (X )ds+ + ϕ(X )ds, t 0 s s 2 s s 2 s s 0 0 0 where one deﬁnes ∂ g(x) − g(y) + g(x) = 2 µ (dy) s ∂x x − y s R

2 for a C function g, µs being the distribution of Xs . Note that the factor 2 in the deﬁnition of +s above was overlooked in [5]. Taking the trace of both sides of the equation, one P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 595 has, since the stochastic integral has expectation 0,

t t 1 1 τ ϕ(X ) = τ ϕ(X ) − τ ϕ(X )f (X ) ds + τ + ϕ(X ) ds t 0 2 s s 2 s s 0 0 or t 1 ϕ(x)µ (dx) = ϕ(x)µ (dx) − ϕ(x)f (x) µ (dx) ds t 0 2 s R R 0 R t ∂ ϕ(x) − ϕ(y) + µ (dy) µ (dx) ds. (4.1.3) ∂x x − y s s 0 R R

As we shall see later, the distribution of Xt has a bounded (non smooth in general) density pt (x),forallt>0. Making formal integration by parts in this formula, Eq. (4.1.3) means that µt (dx) = pt (x) dx is a weak solution of the free Fokker–Planck equation (1.4.2). It is known that even for a smooth initial distribution, if f = 0, then the distribution of Xt can develop singularities, i.e., for t large enough, there will exist points where the density pt is not differentiable, hence in general Eq. (1.4.2) cannot be taken in a pointwise sense.

4.2. Free Markov property of the diffusion

In the one-dimensional case, the results from Section 3.2 yield an operator X 0 → X 0 Ps,t : t s which is a Markov operator and is given by a Feller kernel of probability measure Ps,t(x, dy) on Spec(Xs ) × Spec(Xt ). The free Markov property of the diffusion now implies that for all times t1

4.3. The stationary case

Suppose that the initial distribution of Xt is chosen so that the distribution µt (dx) = pt (x) dx is constant. In this case one has, from the free Fokker–Planck equation, that = 1 Hp 2 f on the support of p(x)dx. As we shall see, this can be achieved by taking X0 to be distributed as the measure maximizing the relative free entropy. In this case, the Markov operators Ps,t become time homogeneous, indeed they depend only on t − s. As it is easy to check, these correspond to the classical Markov transition operator associated with the Dirichlet form 2 1 ϕ(x) − ϕ(y) Q(ϕ) = µ(dx) µ(dy). 2 x − y R

See, e.g., [10]. Observe that this Dirichlet form can be put in the form

1 Q(ϕ) = µ ⊗ µ |∂ϕ|2 , 2 where, e.g., on polynomials, ∂ : C[X]→C[X]⊗C[X] is the non-commutative derivation (for the natural C[X] bimodule structures of C[X] and C[X]⊗C[X]) such that ∂X = 1 ⊗ 1. More generally ∂ can be deﬁned on functions by

ϕ(x) − ϕ(y) ∂ϕ(x,y)= . x − y

This is to be compared with the expression of the stationary diffusion (1.1.2),whichis associated with the Dirichlet form on L2(p(x) dx) 1 Q(ϕ) = ∇ϕ(x)2 µ(dx). 2 R

Thus, going from the classical diffusions to the free diffusions means replacing the gradient by its non-commutative analogue.

5. Euler scheme for the solution of the free diffusion equation

5.1. Convergence of the Euler scheme

In order to obtain regularity results on the free diffusion, we shall develop in this section an Euler scheme for the solution of our diffusion equation. This Euler scheme could be defined in the multidimensional case, in the setting of Theorem 3.1, and the arguments presented below would imply its convergence without difficulty, however for notational simplicity, and since we shall only use the one dimensional case, we stick to this case here. We continue with the hypotheses of Section 4.1. Let us fix n ∈ N and define a process X(n), first in the interval [0, 1/n] by P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 597

(n) = X0 X0, 1 X(n) = X + S for u< , u 0 u n (n) = + − 1 + X 1 X0 S 1 f(X0 S 1 ), n n 2n n (n) k and then continue by induction on k, assuming Xu has been deﬁned for u n ,deﬁne k k + 1 (n) = (n) + − Xu X k Su S k for u< , and n n n n (n) (n) 1 (n) X k+1 = X k + S k+1 − S k − f X k + S k+1 − S k . n n n n 2n n n n This will give an Euler approximation to the solution of Eq. (4.1.2).

THEOREM 5.1. – Let t>0, then there exists a positive constant C (depending on f , t and X0), such that C (n) − √ sup Xs Xs for all n 0. (5.1.1) st n ∞ Proof. – We know that supst Xs < , hence we can always assume that f ∞ < ∞,andf is operator Lipschitz with some constant K (depending on t). Deﬁne for k 1, (n) (n) (n) (n) (n) = − = − − − − ak X k X k and uk X k X k X k−1 X k 1 , n n n n n n then, clearly (n) = (n) (n) + (n) a0 0,ak ak−1 uk for k 1 and

(n) (n) 1 X − X a − + f ∞ t t k 1 2n k−1 k for n t

k n s K 1 − − + (n) S k Ss f(Xu)du ak−1 ds 2 n 2 k−1 k−1 n n k n − K k 1 k 1 (n) 2 − s + f ∞ s − + a − ds 2 n 2 n k 1 k−1 n K (n) −1 −3/2 1 −2 a − n + 2n + f ∞n 2 k 1 4

K (n) −3/2 a − + χn 2n k 1 for some constant χ. Finally we get (n) (n) K −3/2 a a − 1 + + χn . k k 1 2n

(n) = (n) + K −k Let bk ak (1 2n ) , then one has

k (n) (n) K −3/2 (n) K k −3/2 b b − + χ 1 + n b − + χe 2n n . k k 1 2n k 1 Summing over k gives

(n) (n) K k k K k −1/2 a b e 2n χ e n n . k k n Estimation (5.1.1) follows from this. ✷

5.2. Regularity of the free diffusion

1/2 1/2 2 Let us denote by D g the half derivative of a function g,thenD g = 2 R |x|ˆg(x)|| dx,wheregˆ is its Fourier transform. We assume that f has a derivative, and we are in the situation of Theorem 3.1, with either f a Lipschitz function, or f locally Lipschitz, satisfying (4.1.1), with a<0. In these latter case, since Xt remains uniformly bounded, we can as well assume that f is in fact a Lipschitz operator function, and let 2K be a Lipschitz constant for f .

THEOREM 5.2. – There exist constants K1,K2 depending only on f , such that the distribution of Xt has a density satisfying √ 1/2 K1 p ∞ K / t + K and D p + K for all t>0. t 1 2 t 2 t 2 Proof. – We shall prove only the first estimate, since this is the only one we shall use in the sequel. The second one can be obtained along similar lines, using the results of [19]. (n) Let us fix t, then by Theorem 5.1, we know that the Euler scheme approximation Xt (n) converges in norm towards Xt . In particular, this implies that the distribution of Xt P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 599 converges weakly towards that of Xt , hence it suffices to prove that for n large enough, (n) Xt has a density satisfying the estimates of Theorem 5.2, for some constants K1,K2, (n) independent of n and t. Let us prove first that Xt has a density for all t>0. For (n) 0 0and → − 1 bounded by 1/(π t).Forn large enough, the map x x 2n f(x)is a diffeomorphism, 1 −1 whose inverse has a derivative bounded above by (1 − f ∞) , hence the density √ 2n (n) − K of X1/n is bounded above by n/(π(1 n )). By induction on k, we see that for all k k+1 (n) t with n t< n the distribution of Xt has a bounded density, and the maximum (n) of this density is smaller than the maximum of the density of X k , since it is obtained n by a free convolution (see, e.g., [2]). It is therefore enough to prove the bounds for the = k (n) (n) times t n . Passing from X k−1 to X k consists in convolving freely with a semi-circular n n distribution of variance n−1 and then applying a diffeomorphism of derivative bounded − K (n) above by 1 n .Letvk be the supremum of the density of X k , then one has, by [2], n Lemma 6 and Proposition 5,

v − v k 1 . k + 2 2 − K (1 π arctan(2πvk−1/n))(1 n ) Let x ρ (x) = , n + 2 2 − K (1 π arctan(2πx /n))(1 n ) then the function ρn satisﬁes the following properties, for n large enough: (1) ρn is increasing on [0, +∞[. = n π K (2) ρn(x) > x for 0 xn. It follows that one has vk uk where uk is the recursive sequence √ u1 = n/π, uk+1 = ρn(uk),

This sequence is decreasing and uk → xn as k →∞. One has 2 2 u−2 − u−2 = u−2 (1 − K/n)2 1 + arctan 2πu2/n − 1 . k+1 k k π k

[ +∞[ 2 2 = −2 Since the function arctan is concave on 0, and uk/n u1/n π one has for all 2 2 2 k 1, π arctan(2πuk/n) > cuk/n for some universal constant c. Therefore 2 (1 − K/n) 1 + arctan 2πu2/n (1 − K/n) 1 + cu2/n π k k − + 2 − 2 1 K/n cuk/n cu1/n (K/n) − + 2 − −2 1 K/n cuk/n π cK/n. 600 P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 = 1 + 1 Let K2 K(π2 c ), then there exists a constant α such that if uk >K2 then one has 2 2 u−2 (1 − K/n)2 1 + arctan 2πu2/n − 1 α/n l π l for all 1 l k. In this case one has

k−1 −2 = −2 + −2 − −2 uk u1 ul+1 ul αk/n, l=1 n n + hence vk uk αk . Therefore for all k one has vk αk K2. This yields the required estimate. ✷ The continuity in Lp of the Hilbert transform yields the following (n) p COROLLARY 5.3. – The densities pt belong to all L spaces as well as their Hilbert (n) (n) p transforms for 1

log |x − y|pt (x)pt (y) dx dy

| − | (n) (n) → is deﬁned and continuous in t and one has log x y pt (x)pt (y) dx dy log |x − y|pt (x)pt (y) dx dy as n →∞.

6. Connection with free entropy and free Fisher information

6.1. Relative free entropy and free Fisher information

Let F be a locally bounded Borel function on R, let us introduce the following quantity

F (µ) := log |x − y| µ(dx) µ(dy) − F(x)µ(dx) (6.1.1) R2 R for a probability measure µ, with compact support in R. This quantity always makes sense in [−∞, +∞[. We shall also need the following relative version of free information, deﬁned for differentiable F with F = f by 1 2 I (µ) = 4 Hp(x) − f(x) p(x)dx (6.1.2) F 2 R

3 if µ has a density p ∈ L (R),andIF (µ) =∞if not. We shall see that the quantities above are the right analogues of the relative entropy given by (1.1.5), in the classical case and of the relative Fisher information (1.1.6). The limit eigenvalue distribution of P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 601 the matrix model with potential F is the unique measure µ which maximizes the quantity F (µ) (see [16] for some information about such maximization problems).

6.2. Nonincrease of the relative free entropy for the free diffusion

Let us consider again Eq. (4.1.2), where we assume now that f is a C2 function. As we shall see now, as in the case of classical diffusions, the quantity F (µt ) is nondecreasing with t, so that it converges to some value as t →∞, but in general this limit value is strictly smaller than the maximum value.

PROPOSITION 6.1. – Let µt (dx) = pt (x) dx be the distribution of the solution to (4.1.2), then one has for t>0 d 1 2 1 (µ ) = 2 Hp (x) − f(x) p (x) dx = I (µ ). (6.2.1) dt F t t 2 t 2 F t R

Proof. – We shall prove that for all s

t 1 2 (µ ) − (µ ) = 2 Hp (x) − f(x) p (x) dx du. F t F s u 2 u s R

Since, by Corollary 5.3 the quantity integrated is continuous in t, this will prove the k l (n) claim. Choose the nearest integers k,l such that n s t n . Denote by µt the (n) (n) (n) + − − distribution of Xt and by µ k − the distribution of X k−1 S k S k 1 . Since one obtains n n n n (n) (n) µ k+1 − from µ k by free convolution with a semi-circular distribution, one has, for all n n k 1, k+1 n (n) − (n) = (n) (n) 2 0 µ k+1 − 0 µ k 2 ps (x)Hps (x) dx ds n n k R n cf. [18], and from Eq. (4.1.3), applied to the case dXt = dSt ,

k+1 n (n) − (n) =− (n) (n) F(x)µk+1 −(dx) F(x)µk (dx) f(x)ps (x)Hps (x) dx ds, n n R R k R n hence (n) − (n) F µ k+1 − F µ k n n k+1 n = (n) (n) 2 − (n) ps (x) 2Hps (x) f(x)Hps (x) dx ds. (6.2.2) k R n 602 P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606

(n) (n) → − 1 The measure µ k is the image of µ k − by the map x x 2n f(x), consequently one n n has (n) − (n) = − − 1 + 1 − | − | F µ k+1 F µ k+1 − log x y f(x) f(y) log x y n n 2n 2n R × (n) (n) p k+1 −(x)p k+1 −(y) dx dy n n − − 1 − (n) F x f(x) F(x) p k+1 −(x) dx 2n n R − = 1 f(x) f(y) (n) (n) p k+1 −(x)p k+1 −(y) dx dy 2n x − y n n R 1 1 2 (n) 1 + f (x)p + (x) dx + O , k 1 − 2 2 n n n R (n) − = −1/2 k k+1 where the O is uniform in k. Moreover one has X k+1 Xs O(n ) for n

k+1 n f(x)− f(y) 1 = p(n)(x)p(n)(y)dxdyds+ O x − y s s n3/2 k R2 n k+1 n 1 = 2 f(x)p(n)(x) Hp(n)(x) dx ds + O . (6.2.3) s s n3/2 k R n Comparing (6.2.2) and (6.2.3), we get the convergence result. ✷

7. Asymptotic behaviour of the free diffusion

7.1. Nonconvergence towards the master ﬁeld

We shall now give examples of potentials F for which there exist initial distributions such that the distribution of Xt does not converge towards that of the master ﬁeld. The examples we give have several potential wells, and if these wells are deep enough then no mass can escape from them. Let us consider a function f such that f has isolated zeros at x1,...,xn ∈ R, and there exist positive constants a,b such that f >aon the interval ]xj − b,xj + b[, and these intervals are two by two disjoint. Then there exists a 2 = − 2 ] − + [ nonnegative C function G such that G(x) (x xj ) on the interval xj b,xj b , = n ] − + [ 2 and J j=1 xj b,xj b is exactly the set where G(x) < b . Let us apply Ito’s 2at formula to e G(Xt ),weget P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 603

t t 2at 2as 2as e G(Xt ) = G(X0) + e ∂G(Xs)>dSs − e G (Xs)f (Xs )ds 0 0 t t 1 + 2a e2asG(X )ds+ e2as+ G(X )ds. s 2 s s 0 0 2 2 Suppose that G(X0)

 −G (Xt )f (Xt ) + 2aG(Xt ) 0 as self adjoint operators,

∂ G(x) − G(y) +t G(Xt ) sup 2, x,y∈J ∂x x − y so that

t t 1 e2atG(X ) G(X ) + e2as∂G(X )>dS + e2as+ G(X )ds, t 0 s s 2 s s 0 0 therefore, using the free Burkholder–Gundy inequality 2at 4at 2at e G(Xt ) G(X0)+4b 2 e − 1 /4a + 2 e − 1 /2a.

Assume that √ 2 G(X0)+4b/ 2a + 1/(2a) < b , then we see that t0, i.e., T =∞, and thus the support of the distribution of Xt always remains in J .SinceXt is norm continuous, it follows that the mass put by the distribution of Xt on each of the intervals ]xj − b,xj + b[ remains constant in time. Therefore, the distribution of Xt cannot converge towards the master ﬁeld, unless one puts the right masses in the wells at the initial distribution. = 1 2 + g 4 Let us consider the quartic model with P(X) 2 X 4 X . Then the limit distribution of the matrix model has a Cauchy transform given by 1 1 G(z) = z + gz3 − 1 + 2ga2 + gz2 z2 − 4a2, 2 2 where 3ga4 + a2 = 1 (see, e.g., [7, Eq. (5.3)]). In particular, this solution exhibits an analytic continuation for negative g, with g>−1/12, although then one has ZN =∞ for all N. These solutions are interpreted as coming from the local minimum at 0 of the potential P . Indeed if we look at the free diffusion equation 1 dX = dS − X + gX3 dt t t 2 t t 604 P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 then starting at X0 = 0, and using the same technique as above with the free Burkholder– Gundy inequality, one can see that for g negative, but close enough to zero there is indeed a solution deﬁned for all times t, which moreover remains uniformly bounded in norm. This suggests that the distribution of Xt converges towards the analytic continuation of the solution, although we cannot prove this. The advantage of considering free diffusion equations is that we can study with the same techniques equations involving more than one variable, indeed for models such as the AB model where

P(A,B)= αA4 + βB4 + A2 + B2 + γ(AB+ BA) one can prove that for |γ | < 1 and for negative values of α and β close to zero the solution starting from 0 again exists and remains bounded over all times, suggesting that an analytic continuation of the model exists for such α and β. Of course this argument works for much more general models, but we do not have any rigorous argument for the convergence of the solution towards some limit distribution.

7.2. The case of the free Ornstein–Uhlenbeck process

On the other hand in the case where f(x)is a linear function, then we know that the distribution of Xt always converges to the semicircular distribution. We even have an exponential rate of convergence for the relative free entropy, thanks to the free analogue of the log Sobolev inequality due to Voiculescu. Let F(x) = λx2 for a positive λ>0. Then we have f(x)= 2λx and the diffusion equation dXt = dSt − λXt dt has the solution

t −λt −λ(t−τ) Xt = e X0 + e dSτ . 0

−λt The distribution of Xt is given by the free convolution of the distribution of e X0 with 2λt a semi-circle of variance (1 − e )/(2λ). In particular, for t →∞the distribution of Xt converges towards a semi-circle of variance 1 , i.e., towards the distribution of √1 S, 2λ 2λ where S is a semi-circular of variance 1. In this case we have 2 2 (X∞) − (X)= χ(X∞) − λϕ X∞ − χ(X)+ λτ X 1 1 = χ(S)− log(2χ)− − χ(X)+ λτ X2 2 2 and I(X)= 0(X) − 4λ + 4λ2τ X2 , where χ and 0 are the free entropy and the free Fisher information, respectively. We claim now that we have a corresponding free logarithmic Sobolev-inequality for ρ = 2λ, i.e., 1 I(X) (X∞) − (X) 2ρ P. BIANE, R. SPEICHER / Ann. I. H. Poincaré – PR 37 (2001) 581–606 605 or 0(X) 4λχ(S) − 2λ log(2λ) + 2λ − 4λχ(X). To prove this it sufﬁces to have this inequality for that value of λ which maximizes the right hand side. This value of λ is determined by

log(2λ) = 2χ(S)− 2χ(X), and the free logarithmic Sobolev inequality for the Ornstein–Uhlenbeck process is thus equivalent to the statement

2πe 2χ(X) 2χ(S)− log 0(X) = log . 0(X)

But this inequality was proved by Voiculescu in Proposition 7.9 of [21].

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