Logarithmic Sobolev inequalities for some nonlinear PDE's F. Malrieu

Laboratoire de Statistique et Probabilites, Universite Paul Sabatier, 118 route de Narbonne, 31064 Toulouse Cedex, France

Abstract The aim of this paper is to study the behavior of solutions of some nonlinear partial dierential equations of Mac Kean-Vlasov type. The main tools used are, on one hand, the logarithmic Sobolev inequality and its connections with the concentration of measure and the transportation inequality with quadratic cost; on the other hand, the propagation of chaos for particle systems in mean eld interaction. Key words: Interacting particle system, logarithmic Sobolev inequality, propagation of chaos, relative entropy, concentration of measure

1 Introduction A probability measure  on Rn satis es a logarithmic Sobolev inequality with constant C if Z   (1) Ent f  C jrf j d 2

2

for all smooth enough functions f where Z Z   Z Ent f = f log f d ? f d log f d : 2

2

2

2

2

Let us recall two consequences of this property for . First, for every r  0, and every Lipschitz function f on Rn (equiped with the Euclidean topology) with kf k = sup jf (xjx) ?? fyj(y)j  1 ; x6 y one has (jf ? E  (f )j  r)  2e?r =C : This is the so-called concentration phenomenon, see Ledoux (1997). Lip

=

2

Preprint submitted to Elsevier Preprint

22 December 2000

Secondly, denote by W2 (;  ) the Wasserstein distance with quadratic cost i.e. s ZZ  1 W2 (;  ) = inf 2 jx ? yj d(x; y) 2

where the in mum is running over all probability measure  on Rn  Rn with respective marginals  and  . By the Monge-Kantorovitch representation (see Rachev (1984)), we have Z Z  W2 (;  ) = sup gd ? fd 2

where the supremum is running over all the bounded functions f and g such that g(x)  f (y) + 21 jx ? yj for every x; y 2 Rn. Following Otto and Villani (2000), if  is absolutely continuous and satis es (1), then, for every probability measure  absolutely continuous with respect to , 2

W2(;  )  C2 H( j ) where H is the relative entropy: 8 Z d Z > < log d = d log d d if    d d d H( j ) = > : +1 otherwise: 2

(2)

An other approach of this theorem has been done by Bobkov et al. (2001). Consider now a diusion process with in nitesimal generator L and semigroup (Pt)t. As usually, we associate to L its carre du champ ? de ned by and the operator ?2

?(f; g) = 21 (L(fg) ? f Lg ? gLf ) ?2 (f ) = 21 (L(?f ) ? 2?(f; Lf )) :

Then, following Bakry (1997), for any  2 R, the following properties are equivalent

 For every f , ?q2(f )  ?(f ) p  For every f , ?(Ptf )  e?tPt ?f 2

 The probability measures (Pt(
)(x))x satis es a logarithmic Sobolev inequality with constant

  Ct = 2 1 ? e?t :

Besides, if  is positive, the last property remains true for the invariant measure  with constant 2=. This provides an exponential rate for the ergodicity of the semigroup: H(Pt j )  Ke? t: 2

Let us now introduce the rst nonlinear PDE (on Rd) we want to study: 8 @u > < = div [ru + (urU + urW  u)] ; @t (3) > : u(0;
) = u ; 0

where W is convex, even, with polynomial growth and U is uniformly convex (i.e. Hess U (x)  I for some > 0). The solution of (3) can be interpreted as the law of the stochastic process (X t)t solution of 8 p > > dX = 2 dBt ? rU (X t) dt ? rW  ut(X t) dt ; t > < (4) L(X t) = ut(x) dx ; > > > : u smooth 0

0

where L(X t) is the law of X t. Remark 1.1. Let us motivate in few words the study of (3). Benedetto et al. (1998) give a physical interpretation when

d = 1 ; U (x) = x2 and W (x) = jxj : 2

3

Equation (3) is the homogeneous version of a transport equation in a thermal bath with temperature 1= that can be derived from inelastic collisions:

(@t + v@x)u(t; x; v) = @vv + @v [( v + W 0 v u(t; x; v))u(t; x;v)] 2

where v stands for the convolution with respect to the velocity v.

The probabilistic approach of this type of equations is to substitute to the nonlinear process a particle system in mean eld interaction i.e. to replace, in Equation (4), u by the empirical measure of the particle system. This leads to 3

the following SDE on Rd : 8 N     p X > > < dXti; = 2 dBti ? rU Xti; dt ? N1 rW Xti; ? Xtj; dt j > > L(X ) = u dx for i = 1; ::; N : N

N

N

N

N

(5)

=1

N

0

0

N

As it is expected, the particle system is a \good approximation" of the solution of Equation (4) in a precise way described below. In Section 2, we establish the propagation of chaos property, uniform in time.

Theorem 1.2. There exists K such that, for every N  1,   K ; sup E Xt ? X t  N t2R

1

1 2

N

Then we show that the laws at time t of the particle system and of the nonlinear process satisfy a logarithmic Sobolev inequality from which we derive the exponential rate for the convergence to equilibrium for the nonlinear PDE. Theorem 1.3. There exists a constant K such that for every positive t,

kut ? uk  K e? t=

2

1

where u is the unique solution of

u(x) = Z1 exp (?U (x) ? W  u(x))

Z

with Z = exp (?U (x) ? W  u(x)) dx :

At last we obtain con dence intervals for the convergence of the empirical measure at time t to u: for every r  0, 1 0 N s # " Z X K N r 1 i; ? tA @  2 exp ? 2 : P N i f (Xt ) ? f (x)u(x) dx  r + N + Ke 2

N

=1

Section 3 is dedicated to the study of the case U = 0 which has been investigated by Benachour et al. (1998a,b). Section 4 deals with the general case of McKean-Vlasov type equations. 4

2 Dissipative equations 2.1 The model

Consider the following nonlinear dierential equation on Rd: 8 @u > < = div [ru + (urU + urW  u)] ; @t > : u(0;
) = u ;

(6)

0

where W is convex, even, with polynomial growth and U is uniformly convex (i.e. Hess U (x)  I for some > 0). The real will be the greater real (nonnegative) satisfying Hess W (x)  I. The solution of (6) can be interpreted as the law of the stochastic process (X t)t solution of 8 p > > dX = 2 dBt ? rU (X t) dt ? rW  ut(X t) dt ; t > < (7) > L(X t) = ut(x) dx ; > : u smooth. 0

0

x

Remark 2.1. In order to prove existence and unicity for Equation (7), it is possible to use the technics of Benachour et al. (1998a). In particular, notice that it can be rewritten

8 p > > = 2 dBt ? rU (X t) dt ? rWt(X t) dt ; dX t > < h i r W ( x ) = E r W ( x ? X ) ; t t > > > : u smooth.

(8)

0

One can prove rst existence of ut by a xed point argument using the Wasserstein distance. Then, once ut is known, strong uniqueness for (X t) holds.

At last, the associated particle system is the solution of 8 N     p X > > < dXti; = 2 dBti ? rU Xti; dt ? 1 rW Xti; ? Xtj; dt Nj > > : L(X ) = u dx for i = 1; ::; N N

N

N

=1

N

0

0

N

(9)

N

Remark 2.2. In what follows, (X i:) will stand for the solution of (7) where

the Brownian motion (B: ) is replaced by (B:i ).

5

2.2 Uniform propagation of chaos

Theorem 2.3. There exists K such that, for every N  1,   sup E Xt ; ? X t  K N 1

t2R

(10)

1 2

N

and

; ! T E sup Xt ? X t  K : N t Remark 2.4. The constant K depends on , and on the p-th moment of u for a power p that control the growth of U and W . 1

0

1 2

N

T

0

Proof of Theorem 2.3. The proof is quite similar to the one in Benachour et al. (1998a). Nevertheless the term \rU "provides a better estimate in time. For i = 1; :::; N ,

Z t  Xti; ? X it = Xsi; ? X is ? rU (Xri; ) ? rU (X ir ) dr s Z N   t X ? N1 s rW (Xri; ? Xrj; ) ? rW  ur (X ir ) dr: j N

N

N

N

N

=1

By It^o's formula, N N X Xti; ? X i = X X i; ? X i s t s i i N Z t   X ?2 s (Xri; ? X ir )
rU (Xri; ) ? rU (X ir ) dr i N Z t X ? N2 ij (r) dr (11) s i;j 2

N

=1

2

N

=1

N

N

=1

(1)

=1

where

  h i ij (r) = Xri; ? X ir
rW (Xri; ? Xrj; ) ? rW  ur (X ir )  i  h = Xri; ? X ir
rW (Xri; ? Xrj; ) ? rW (X ir ? X jr)  i  h + Xri; ? X ir
rW (X ir ? X jr ) ? rW  ur (X ir ) = ij (r) + ij (r): The vector eld rW is odd and satis es (rW (x) ? rW (y))
(x ? y)  0 (1)

N

N

N

N

N

N

N

(2)

(3)

6

then, by de nition of ij (r), (2)

ij = ij (r) + ji (r) ih i h = Xri; ? Xrj; ? (X ir ? X jr )
rW (Xri; ? Xrj; ) ? rW (X ir ? X jr )  0: (4)

(2)

(2)

N

N

N

N

It has been shown that N X i;j =1

ij (r) =

X

(2)

(4)

i 0. Then, we still have the propagation of chaos for blocks of size o(N ). Proposition 3.3. For k  N , we have s k;

k sup kut ? ut k  K k : N (

N)

t0

1

4 The general case This part is dedicated to the study of the generic model of nonlinear diusions that can be approximated by particles in mean eld interaction. The beginning consists in a straightforward adjustment of the classical method presented by Sznitman (1991). 4.1 The model

Let us start with Borel-measurable functions bi(x; y), ij (x; y), 1  i; j  d, from Rd  R to Rd. Let (Bt)t be a d-dimensional Brownian motion. We study 0

17

the following equation: 8 p > > dX = 2(X t;   u(t; X t)) dBt + b(X t;   u(t; X t)) dt t > < > L(X t) = u(t; dy) > : Xt = X : =0

(20)

0

The density of a solution at time t is known to be a weak solution of: d @ u(s; x)= X @ [a (x;   u(s; x))u(s; x)] @t @xi@xj ij i;j d @ X ? @x [bi(x;   u(s; x))u(s; x)] i i 2

=1

=1

where a is equal to . Theorem 4.1. If  and b are bounded and globally Lipschitz functions, strong existence and uniqueness hold for equation (20). 4.2 Convergence of the associated particle system

Let (B i)i2N be independent Brownian motions on Rd. The interacting particle system associated to (20) is the solution of 8 p i; > i; i; i > dX t = 2 (Xs ;   s (Xs )) dBs > < (21) +b(Xsi; ;   s (Xsi; )) ds for i = 1; :::; N > > > X i; = X i : N

N

N

N

N

N

N

N

0

0

where t is the empirical measure of the system at time t i.e. it is equal to n X t = N1 X : k N

N

k;N

t

=1

How it is expected, the propagation of chaos holds. Indeed we have the following estimate: Theorem 4.2. If  and b are bounded and globally Lipschitz, for all T 2 R , there exists K 2 R such that for all i and N in N, " # K i i; E sup Xt ? X t  (22) N: tT +

T

2

N

18

T

Remark 4.3. It can be shown rst that for some K 0 , T

   = K 0 i i; p : sup E Xt ? X t N tT 2

N

1 2

(23)

T

Once again, K 0 is of the order of exp(KT ) like the Log-Sobolev constant C of the particle system at time T (see section 4.3). T

T

4.2.1 An improvement in a special case

In this section, we will suppose that  is equal to I . Then, it is still true that the nonlinear process has a bounded-below (in R) curvature. The its law at time t satis es a logarithmic Sobolev inequality. This ensures that, for k  N , we have s sup kutk; ? u t k k  K Nk : (

N)

1

tT

T

In fact, it is possible to get a best estimate (at the level of processes). Following a suggestion of Professor Del Moral we use a method developed by Ben Arous and Zeitouni (1999) in an abstract framework. Nevertheless our case is simpler because we already have established Theorem 4.2. Let us start with few notations:    P stands for the law of (Xt ) tT = (Xt ; ; :::; Xt ; ) tT the particle system until time T ,    Pk; denotes the law of (Xt ; ; :::; Xtk; ) tT . By exchangeabilty, the kmarginals of P do not depend on the choice of coordinates,  let P be the law until time T of the nonlinear process solution of (20).   The key point is that we are able to show that the relative entropy H P j P

is bounded in N . Proposition 4.4. There exists a constant K such that for all N ,   H P j P  K : N

N

1

N

1

0

T

N

N N

0

N

N

T

0

N T

T

N

N T

T

T

N

N T

T

T

Proof. By Girsanov theorem, P has a density with respect to P given by N

N T

dP (Y ) = exp(H ) dP

N T

N

T

N

where 19

3 2 N N Z T X X 1 4b(Ysi; (Ysi ? Ysj )) ? b(Ysi;   us(Ysi))5 dBsi H = N j i 3 2 N Z T N X X 1 1 i j i i i 5 4b(Ys ; ?2 N (Ys ? Ys )) ? b(Ys ;   us(Ys )) ds: N

T

0

=1

=1

2

i=1

0

j =1

Under the measure P , ((Bti) i ) t is a N -dimensional Brownian motion and (Bit) i t de ned by 3 2 Z N T X B it = Bti ? 4b(Ysi; N1 (Ysi ? Ysj )) ? b(Ysi;   us(Ysi))5 ds j N

1

1

N

0

N

0

T

T

0

=1

is a N -dimensional Brownian motion under P . Then it follows from (2) N T

H



P jP

N

N T



!# d P = E P log

dP Z N  i X Th = 12 E b(Xsi; ;   s (Xsi; )) ? b Xsi; ;   us (Xsi; ) ds i Z T  N  X K i; i;  2 E   s (Xs ) ?   us (Xs ) ds i N Z T   X K i; j; i;  2N E (Xs ? Xs ) ?   us (Xs ) ds: i;j "

N T

N T

N

N

=1

N

N

N

N

2

0

N

N

2

N

0

=1

N

=1

N

N

2

0

We write

(Xsi; ? Xsj; ) ?   us(Xsi; ) = (Xsi; ? Xsj; ) ? (Xsi; ? X js) +(Xsi; ? X js) ? (X is ? X js ) +(X is ? X js) ?   us(X is ) +  us(X is) ?   us (Xsi; ) N

N

N

N

N

N

N

N

and get, by Theorem 4.2,   K i; j; i; E (Xs ? Xs ) ?   us (Xs )  N which achieves the proof. N

N

N

2

T

Now we just have to use (16) and the Csiszar and Kullback inequality (15) to establish a strong convergence estimation. 20

Theorem 4.5. For all T and N , there exists a constant K such that, for all

k  N,

T

s

k;

k

P ? P T V  K Nk : N

T

T

T

This implies a strong version of propagation of chaos for blocks of size o(N ). 4.3 Concentration of the empirical measure

We now want to establish that the particle system (in the generic case) has a bounded-below curvature. This is much more complicated than in the case where  is equal to I . Indeed, the second order part of the in nitesimal generator equips Rd with a Riemannian metric which changes the notion of gradient. N

4.3.1 Control of the Ricci curvature

For simplicity's sake, we will suppose, in this section, d = 1. The case d  2 can be treated by the same way but with complicated notations. In Remark 4.9, we sketch the changes that occur when d is greater than 2. We treat here the case of Equation (20). An advanced study of Riemannian geometry can be found in Gallot et al. (1990) but let us try to describe the setting in few words. Consider the dierential operator L on R de ned by N

L=

N X

N X @ @: g (x) @x @x + hi(x) @x i j i i 2

ij

i;j =1

=1

Its second order part equips R with an intrinsic metric g which depends on x 2 R and which is characterized by the matrix (gij ) (the inverse of (gij )). The geometry of (R ; g) is quite dierent from the usual one and it appears a curvature. We now show how to control Ricci curvature of the dierential operators L associated to (21) and de ned by N

N

N

N

L = N

where

N X i=1

N @ +X @ gi(x) @x bi(x) @x 2

2

i

i=1

i

1 0 N X 1 g (x) = a@xi; N (xi ? xj )A , gi(x) = gi (1x) i

j =1

21

(24)

(25)

and

1 0 N X 1 bi(x) = b@xi; N (xi ? xj )A: j =1

Now let us recall the general de nition of the Ricci curvature.

De nition 4.6. Let us de ne

(1) Christophel symbols for i; j; k = 1; :::; N :

! N X @g @g @g 1 pi kp ki ip ? = 2 g @x + @x ? @x ; k i p p   (2) the Riemann curvature tensor Riqkl where j ki

=1

N N X @ ?i @ ?i X Riqkl = @xqk ? @xql + ?ipl?pqk ? ?ipk ?pql ; l k p p =1

=1

(3) the Ricci curvature tensor (Rql ) where

Rql =

N X i=1

Riqil;

(4) the drift tensor (mij ) where N X 1 @b j Mij = 2 (ribj + rj bi) ou ribj = @x + ?jkibk : i k =1

In our context, the matrix (Gij ) is diagonal and then many terms in the previous de nitions are equal to 0. For example, ?jki is equal to 0 as soon as the three index are dierent. By a straightforward computation we have the explicit expression of the Ricci and drift tensors. Lemma 4.7. The entries of Ricci tensor are

! ! X @ 1 g :::g 1 @ @ Rlq = 2 @x @x log g g + 4 log gi @x log gi l q l q q i6 q;l @xl gq @ log g :::g ? 1 @ log gl @ log g :::g ? 41 @ log @xl @xq glgq 4 @xq @xl gl gq 2

1

N

=

1

1

N

N

and

2 !3 i @g X X X @ @ log g g :::g 1 @g 1 @ g q i q 5 Rqq = 2 4 @x log g + gi @x + @x @x + 2 @x q i i q i q i i6 q i6 q ! ! X i @ log gq X i @ log gq @ log(g :::g ) ! 1 1 + 8 gq g @x : ? 4 gq g @x @xi i i i i6 q 2

2

1

2

2

N

=

2

2

=

1

=

22

N

Moreover, the entries of the drift tensor have the following form: N N @b ? 1 X 0 (x ? x ) @b + 1 X @ log gi Mii = ribi = @x  k i N @u 2 @x k=1

et

k

k=1

!   log gi   @b @b gj b : 1 j i bi + 1 ? gj gi @ log Mij = 2 @x + @x + 1 ? gigj @ @x @xi j i j j

We have now to control the spectrum of the matrix T , equal to R ? M . Proposition 4.8. If there exist c such that 1=c  a(x; y)  c and if the rst (resp the rst and second) derivatives of b (resp a) are bounded by c, the spectrum of T is uniformly bounded (above and below) in N 2 N by a real number . Proof. By Gershgorin-Haddamard Theorem (see Grifone (1990)), if  is an eigenvalue of T , 0 1

@X jTij jA: jj  max iN N

1

j =1

We are going to show that, under the assumptions of proposition 4.8, Rqq et Mqq are uniformly bounded in N whereas Rql and Mql are of order 1=N . In order to deal with simple notations,  will be a common bound greater than 1 for  and its derivatives. Then we have @ log g :::g  X @ log gi @xl@xq glgq i6 l;q @xl@xq X 0 @ log a 1  (xi ? xl)0(xi ? xq ) @y N i6 l;q X 1 N 2 c i6 l;q 2

1

2

N

=

2

2

2 2

=

2 4

2

=

and

! X 0 X @ @ 1 @ log a 0 j (xi ? xl) (xi ? xq )j @y log gi @x log gi  N q i6 q;l i6 q;l @xl X  N1 c: i6 q;l 2

=

2

=

2 4

2

=

Then a short computation gives

jRlq j  74Nc : 2 4

23

2

By the same way, it can be shown that 8 > > > < jRqq j  7 c > jMii j  3c > : jMij j  5c =N: 2 6 2

5

Then we have obtained the bound we were looking for. All the operators L have a curvature bounded below by  = 17 c .

N

2 6

Of course, a more precise result could be obtained by specifying the bound of each derivative but this work is not very essential from a theoretic point of view. Remark 4.9. When d  2, gi is a d  d matrix and it appears a \local curvature". 4.3.2 Concentration of measure

Theorem 4.10. For all Lipschitz function with kf k  1, we have, for all

r  0,

Lip

0 s 1 ! Z 1 X i; K Nr sup P@ N f (Xt ) ? f (y)u(t; y) dy  r + N A  2 exp ? C : t i 2

N

T

N

T

where

=1

T

  C = 2 1 ? e?  ; 2

T

T

and u is solution of (21). Remark 4.11. Explicit forms of the constants  and K can be given. With the notations we have introduce in section 4.3.1, T

8 > <  = 17 c > : K = 6 c exp (12 c T ): 2 6

T

4 2

2 2

Acknowledgments This work is a part of my PhD thesis. I would want to thank my advisor Professor D. Bakry for his useful advices and encouragements. 24

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25