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ON THE TREND TO EQUILIBRIUM FOR THE
FOKKER-PLANCK EQUATION : AN INTERPLAY
BETWEEN PHYSICS AND FUNCTIONAL ANALYSIS
⁄ ⁄⁄P. A. MARKOWICH AND C. VILLANI
Abstract. Wepresentconnectionsbetweentheproblemoftrend
toequilibriumfortheFokker-Planckequationofstatisticalphysics,
and several inequalities from functional analysis, like logarithmic
Sobolev or Poincar´e inequalities, together with some inequalities
arising in the context of concentration of measures, introduced by
Talagrand, or in the study of Gaussian isoperimetry.
Contents
1. The Fokker-Planck equation 2
2. Trend to equilibrium 3
3. Entropy dissipation 4
4. Logarithmic Sobolev inequalities 6
5. The Bakry-Emery reversed point of view 7
6. Log Sobolev) Poincar´e 8
7. The nonuniformly convex case 9
8. Generalizations to other physical systems 10
9. An example : generalized porous medium equations 12
10. Gaussian isoperimetry 14
11. Talagrand inequalities and concentration of the Gauss
measure 15
12. Log Sobolev) Talagrand) Poincar´e 16
13. Related PDE’s 18
13.1. The Monge-Amp`ere equation 18
13.2. The Hamilton-Jacobi 19
13.3. The sticky particles system 19
14. HWI inequalities 19
15. Displacement convexity 20
References 22
Acknowledgement: The authors acknowledge support by the EU-funded TMR-
network ‘Asymptotic Methods in Kinetic Theory’ (Contract # ERB FMRX CT97
0157) and from the Erwin-Schr odinger-Institute in Vienna.
12 P. A. MARKOWICH AND C. VILLANI
1. The Fokker-Planck equation
The Fokker-Planck equation is basic in many areas of physics. It
reads
¡ ¢@‰ n(1) =r¢ D(r‰+‰rV) ; t‚0; x2R ;
@t
where D = D(x) is a symmetric, locally uniformly positive deﬁnite
(diﬀusion) matrix, and V = V(x) a conﬁning potential. Here the un-
known ‰ = ‰(t;x) stands for the density of an ensemble of particles,
and without loss of generality can be assumed to be a probability dis-
ntribution onR since the equation (1) conserves nonnegativity and the
nintegral of the solution over R . The phase space can be a space of
position vectors, but also a space of velocities v; in the latter case the
2potential V is usually the kinetic energyjvj =2.
We refer to [32] for a phenomenological derivation, and a lot of basic
references. TheFokker-Planckequationcanbesetonanydiﬀerentiable
structure, in particular on a Riemannian manifold M, rather than on
nEuclidean spaceR . It can also be considered in a bounded open set,
with (say) a vanishing out-ﬂux condition at the boundary.
Far from aiming at a systematic study of equation (1), our intention
here is to focus on some tight links between this equation, and several
functional inequalities which have gained interest over the last decade,
and especially in the last years. In order to simplify the presentation,
we restrict to the case when the diﬀusion matrix is the identity – but
in order to keep some generality in (1), we allow any underlying Rie-
mannian structure. Thus we shall study
@‰
n(2) =r¢(r‰+‰rV); t‚0; x2R or M:
@t
Moreover, wedonotaddressregularityissues, andshallalwaysassume
2that V is smooth enough, say C , perform formal calculations and do
not deal with their rigorous justiﬁcations in this paper.
As dictated by physical intuition, we mention that the stochastic
diﬀerential equation underlying (2) is
dX =dW ¡rV(X )dt;t t t
with W a standard Wiener process (or Brownian motion). Thus thet
Fokker-Planck equation models a set of particles experiencing both
diﬀusionand drift. The interplaybetween these twoprocesses is at the
basis of most of its properties.THE FOKKER-PLANCK EQUATION 3
2. Trend to equilibrium
Let us begin an elementary study of the Fokker-Planck equation.
¡VFrom (2) we see that there is an obvious stationary state : ‰ = e
¡V(adding a constant to V if necessary, one can always assume that e
is a probability distribution). It is then natural to change variables by
¡Vsetting ‰=he . Then we obtain for (2) the equivalent formulation
@h n(3) =Δh¡rV ¢rh; t‚0; x2R or M:
@t
¡VThe operator L=Δ¡rV ¢r is self-adjoint w.r.t. the measure e .
More precisely,
(4) hLh;gi ¡V =¡hrh;rgi ¡V:e e
2(weusetheobviousnotationforweightedL -scalarproductsandnorms).
In particular,
2hLh;hi ¡V =¡krhk ;2 ¡Ve L (e )
so that L is a nonpositive operator, whose kernel consists of constants
¡V(since e is a positive function). This shows that the only acceptable
¡Vequilibria for (2) are constant multiples of e – the constant being
1 ¡Vdetermined by the norm of h in L (e ), which is preserved.
Now, consider the Cauchy problem for the Fokker-Planck equation,
which is (2) supplemented with an initial condition
Z
‰(t=0;¢)=‰ ; ‰ ‚0; ‰ =1:0 0 0
We expect the solution of the Cauchy problem to converge to the equi-
¡Vlibrium state e , and would like to estimate the rate of convergence
in terms of the initial datum. Let us work with the equivalent formu-
Vlation (3), with the initial datum h = ‰ e . Since L is a nonpositive0 0
self-adjoint operator, we would expect h(t;¢) to converge exponentially
fastto1ifLhasaspectralgap‚>0. Thiseasilyfollowsbyelementary
spectral analysis, or by noting that the existence of a spectral gap of
¡Vsize ‚ for L is equivalent to the statement that e satisﬁes a Poincar´e
inequality with constant ‚, i.e
(5) •Z Z Z ‚
2 ¡V ¡V 2 ¡V 2 ¡V8g2L (e ); ge dx=0=) jrgj e ‚‚ g e :
Indeed, knowing (5), and using (3), one can perform the computation
Z Z Z
d 2 ¡V 2 ¡V 2 ¡V(6) ¡ (h¡1) e =2 jrhj e ‚2‚ (h¡1) e ;
dt4 P. A. MARKOWICH AND C. VILLANI
which entails
Z Z
2 ¡V ¡2‚t 2 ¡V(h¡1) e •e (h ¡1) e :0
Thus, if h solves (3) with initial datum h ,0
2 ¡V ¡‚th 2L (e ))kh(t;¢)¡1k 2 ¡V •e kh ¡1k 2 ¡V :0 L (e ) 0 L (e )
Equivalently, if ‰ solves (2) with initial datum ‰ ,0
2 V ¡V ¡‚t ¡V(7) ‰ 2L (e ))k‰(t;¢)¡e k 2 V •e k‰ ¡e k 2 V :0 0L (e ) L (e )
Thisapproachisfastandeﬀective,buthasseveraldrawbacks, which
are best understood when one asks whether the method may be gen-
eralized :
1) Note that the functional space which is natural at the level of (3)
2 ¡V(h2 L (e )) is not at all so when translated to the level of (2) (‰2
2 VL (e )). Formathematicalandphysicalpurposes,itwouldbedesirable
1to be as close as possible to the space ‰ 2 L (which corresponds to
the assumption of ﬁnite mass).
2)Thephysicalcontentoftheestimate(7)isquiteunclear. Strongly
based on the theory of linear operators, this estimate turns out to
be very diﬃcult, if not impossible, to generalize to nonlinear diﬀusion
equations(likeporousmediumequations,ortheFokker-Planck-Landau
equations) which arise in many areas of physics, cf. Section 8.
3)Also,itisoftenquitediﬃculttoﬁndexplicitvaluesofthespectral
gap of a given linear operator. Many criteria are known, which give
existence of a spectral gap, but without estimate on its magnitude the
results obtained in this manner are of limited value only.
3. Entropy dissipation
2 ¡VInsteadofinvestigatingthedecayinL (e )normforh,wecouldas
well consider a variety of functionals controlling the distance between
hand1. Actually,whenever`isaconvexfunctiononR,onecancheck
that
Z Z ‡ ·
‰¡V ¡V(8) `(h)e dx= ` e dx
¡Ve
deﬁnes a Lyapunov functional for (3), or equivalently for (2). Indeed,
Z Z
d ¡V 00 2 ¡V(9) `(h)e dx=¡ ` (h)jrhj e dx:
dtTHE FOKKER-PLANCK EQUATION 5
2Our previous computation in the L norm corresponds of course to
2thechoice`(h)=(h¡1) ;buttoinvestigatethedecaytowardsequilib-
rium, we could also decide to consider any strictly convex, nonnegative
function ` such that `(1)=0.
For several reasons, a very interesting choice is
(10) `(h)=hlogh¡h+1:
R
¡VIndeed, in this case, taking into account the identity (h¡1)e =0,
we ﬁnd
Z Z Z
‰¡V(11) `(h)e = ‰log = ‰(log‰+V):
¡Ve
This functional is well-known. In kinetic theory it is often called the
free energy, while in information theory it is known as the (Kullback)
¡Vrelative entropy of ‰ w.r.t. e (strictly speaking, of the measure ‰dx
¡Vw.r.t. the measure e dx).
As we evoke in section 8, the occurrence of the relative entropy is
rather universal in convection-diﬀusion problems, linear or nonlinear.
Soanassumptionofboundednessoftheentropy(fortheinitialdatum)
issatisfactorybothfromthephysicalandfromthemathematicalpoint
¡Vof view. We shall denote the relative entropy (11) by H(‰je ), which
is reminiscent of the standard notation of Boltzmann’s entropy.
The relative entropy is an acceptable candidate for controlling the
distance between two probability distributions, in view of the elemen-
tary inequality
1 2(12) H(‰j‰e)‚ k‰¡‰ek :1L2
Inequality (12) is known as Csisz´ar-Kullback inequality by (many) an-
alysts, and Pinsker inequality by probabilists (cf. [2] for a detailed
account).
By (9), if ‰ is a solution of the Fokker-Planck equation (2), then
Z ﬂ ‡ ·ﬂ2d ‰ﬂ ﬂ¡V ¡V(13) H(‰je )=¡ ‰ﬂr log ﬂ dx·¡I(‰je ):
¡Vdt e
ThefunctionalI isknownininformationtheoryasthe(relative)Fisher
information, and in the theory of large particle systems as the “Dirich-pR
2 ¡Vlet form” (since it can be rewritten as 4 jr hj e ). In a kinetic
context, it is simply the entropy dissipation functional associated to
the Fokker-Planck equation.
Now, let us see how the computation (13) can help us investigating
the trend to equilibrium for (2).6 P. A. MARKOWICH AND C. VILLANI
4. Logarithmic Sobolev inequalities
2¡n=2 ¡jxj =2 nLet ?(x) = (2…) e denote the standard Gaussian on R .
The Stam-Gross logarithmic Sobolev inequality [23, 33] asserts that
for any probability distribution ‰ (absolutely continuous w.r.t. ?),
1
(14) H(‰j?)• I(‰j?):
2
By a simple rescaling, if ? denotes the centered Gaussian with vari-?
2¡n=2 ¡jxj =(2?)ance ?, (2…?) e , then
?
H(‰j? )• I(‰j? ):? ?
2
In the study of the trend to equilibrium for (2), this inequality plays
2jxj
precisely the role of (5). It implies that (by (13) with V(x) = ), if
2?
‰ is a solution of
‡ ·@‰ x
(15) =r¢ r‰+‰ ;
@t ?
then ‰ satisﬁes an estimate of exponential decay in relative entropy,
¡2t= (16) H(‰(t;¢)j? )•H(‰ j? )e :? 0 ?
Why is (14) called a logarithmic Sobolev inequality ? because it can
be rewritten
? ¶ ? ¶Z Z Z Z
2 2 2 2 2u logu d ¡ u d log u d •2 jruj d :
1 2 2ThisassertstheembeddingH (d )‰L logL (d ),whichisaninﬁnite-
1 n 2⁄ n˙dimensionalversionoftheusualSobolevembeddingH (R )‰L (R ).
Now, of course, (16) is a quite limited result, because it concerns
a very peculiar case (quadratic conﬁnement potential). All the more
that the solution of (15) is explicitly computable ! So it is desirable to
understand how all of this can be extended to a more general setting.
¡VBy deﬁnition, we shall say that the probability measure e sat-
isﬁes a logarithmic Sobolev inequality with constant ‚ > 0 if for all
probability measures ‰,
1
¡V ¡V(17) H(‰je )• I(‰je ):
2‚
Byacomputationcompletelysimilartothepreviousone,weseethatas
¡Vsoon as e satisﬁes a logarithmic Sobolev inequality with constant ‚,
then the solution of the Fokker-Planck equation (with V as conﬁning
¡2‚tpotential) goes to equilibrium in relative entropy, with a rate e
at least. So the question is now : which probability measures satisfy
logarithmic Sobolev inequalities ?THE FOKKER-PLANCK EQUATION 7
5. The Bakry-Emery reversed point of view
In1985,BakryandEmery[3]provedthebasicfollowingresult,which
goes a long way towards the solution of the preceding question.
¡V nTheorem 1. Let e be a probability measure on R (resp. a Rie-
2 2mannian manifold M), such that D V ‚‚I (resp. D V +Ric‚‚I ),n n
where I is the identity matrix of dimension n (and Ric the Ricci cur-n
¡Vvature tensor on the manifold M). Then, e satisﬁes a logarithmic
Sobolev inequality with constant ‚.
2(IntheRiemanniancase,D V standsofcoursefortheHessianofV.)
Moreover, there is room for perturbation in this theorem, as can be
seen from the standard Holley-Stroock perturbation lemma [24]: If V
1 ¡V0is of the form V +v, where v 2 L and e satisﬁes a logarithmic0
¡VSobolevinequalitywithconstant‚,thenalsoea
¡osc(v)Sobolevy, witht ‚e , with osc (v) = supv ¡
infv. By combining the Bakry-Emery theorem with the Holley-Strock
lemma, one can generate a lot of probability measures satisfying a
logarithmic Sobolev inequality. We also refer to [2] for more general
statements with a non-constant diﬀusion matrix D.
But what is striking above all in the Bakry-Emery theorem, is that
its proof is obtained by a complete inversion of the point of view (with
respect to our approach). Indeed, while our primary goal was to estab-
lish the inequality (17) in order to study the equation (2), they used
theequation(2)toestablishtheinequality(17)! Hereishowtheargu-
ment works, or rather how we can understand it from a physical point
of view, developed in [2] (the original paper of Bakry and Emery takes
a rather abstract point of view, based on the so-called Γ , or carr´e du2
champ it´er´e).
1)Recognizetheentropydissipation(inthiscase, therelativeFisher
information)astherelevantobject, andanalyseitstimeevolution. For
this purpose, compute (under suitable regularity conditions)
d
¡V ¡V(18) J(‰je )·¡ I(‰je );
dt
2) Prove that under the assumptions of Theorem 1, the following
functional inequality holds
1¡V ¡V(19) I(‰je )• J(‰je );
2‚
so that the entropy dissipation goes to 0 exponentially fast,
¡V ¡2‚t ¡V(20) I(‰(t;¢)je )•e I(‰ je ):08 P. A. MARKOWICH AND C. VILLANI
3) Integrate the identity (20) in time, from 0 to +1. Noting thatR R+1 +1¡2‚t ¡V ¡Ve dt=1=(2‚) and I(‰je )dt=H(‰ je ), recover00 0
1¡V ¡VH(‰ je )• I(‰ je );0 0
2‚
which was our goal.
4) Perform a density argument to establish the logarithmic Sobolev
inequality for all probability densities ‰ with bounded entropy dissi-0
pation, getting rid of regularity constraints occured in performing the
steps 1)–3).
Thereadermayfeelthatthediﬃcultyhassimplybeenshifted: why
should the functional inequality (19) be simpler to prove than (17) ?
It turns out that (19) is quite trivial once the calculation of J has been
rearranged in a proper way :
Z ? ¶‡ · ‡ ·T‰ ‰¡V 2 2(21) J(‰je )=2 ‰tr D log D log
¡V ¡Ve e
Z D ‡ · ‡ ·E‰ ‰
2+2 ‰ (D V +Ric)rlog ;rlog :
¡V ¡Ve e
The ﬁrst term in (21) is obviously nonnegative without any assump-R
¡V 2tions,whilethesecondoneisboundedbelowby2‚ ‰jrlog(‰=e )j =
¡V 22‚I(‰je ), if ‚ is a lower bound for D V +Ric. Of course, the diﬃ-
cultyistoestablish(21)! Asweshallseelater, therearesimpleformal
waystowardsit. LetusonlymentionatthisstagethattheRiccitensor
comes naturally through the Bochner formula,
¡ ¢1 2 2 T 2¡ru¢r4u+4 jruj = tr (D u) D u +hRic¢ru;rui:
2
6. Log Sobolev ) Poincare
The reader may wonder what price we had to pay for leaving the
2 VL (e ) theory in favor of the more general (and physically more nat-
ural) framework of data with ﬁnite entropy. It turns out that we lost
nothing : this is the content of the following simple theorem, due to
Rothaus and Simon :
¡VTheorem 2. Assume that e satisﬁes the logarithmic Sobolev in-
¡Vequality (17) with constant ‚. Then e also satisﬁes the Poincar´e
inequality (5) with c ‚.
Actually the Poincar´e inequality is a linearized version of the loga-
rithmic Sobolev inequality : to see this, it suﬃces to notice that if g isTHE FOKKER-PLANCK EQUATION 9
R
¡Vsmooth and satisﬁes ge dx=0, then as "!0,
2¡ ¢ "¡V ¡V 2H (1+"g)e je ’ kgk 2 ¡VL (e )2
¡ ¢
¡V ¡V 2 2I (1+"g)e je ’" krgk :2 ¡VL (e )
In [2] a more general study is undertaken. Actually one can deﬁne a
wholefamilyofrelativeentropyfunctionals,oftheform(8),whose“ex-
2tremals”aregivenby`(h)=hlogh¡h+1atoneend,`(h)=(h¡1) =2
at the other end. For each of these entropies one can perform a Bakry-
Emery-type argument to prove logarithmic-Sobolev-type inequalities;
and they are all the stronger as the nonlinearity in the relative entropy
is weaker (the strongest one corresponding to the hlogh nonlinearity).
CorrespondingvariantsoftheHolley-Stroockperturbationlemma,and
of the Csisz´ar-Kullback-Pinsker inequality are also established in great
generality in [2].
7. The nonuniformly convex case
SofarwesawthatthecombinationoftheBakry-Emerytheoremand
the Holley-Stroock perturbation lemma is enough to treat the case of
1conﬁning potentials which are uniformly convex (+L -perturbations).
ﬁWhathappensifV(x)behavesatinﬁnitylike,say,jxj with0<ﬁ< 2?
If 1• ﬁ < 2, then there is no logarithmic Sobolev inequality, while a
Poincar´e inequality still holds. This seems to indicate that the linear
approach is better for such situations. This is not the case : a sim-
ple way to overcome the absence of logarithmic Sobolev inequality is
exposed in [37]. There, modiﬁed Sobolev inequalities are
established, in which the degeneracy of the convexity is compensated
by the use of moments to “localize” the distribution function. For
instance, in the situation we are considering,
¡V ¡V 1¡– –H(‰je )•CI(‰je ) M (‰) ;s
where M (‰) is the moment of order s of ‰,s
Z
2 s=2M (‰)= ‰(x)(1+jxj ) dx (s>2)s
N
and
2¡ﬁ
– = 2(0;1=2):
2(2¡ﬁ)+(s¡2)
Combining this estimate with a separate study of the time-behavior
¡1ofmoments,onecanproveconvergencetoequilibriumwithrateO(t )
¡•(this means O(t ) for all •) if the initial datum is rapidly decreasing.
A striking feature of the argument is that it is not at all necessary to
R10 P. A. MARKOWICH AND C. VILLANI
prove that the moments stay uniformly bounded : it only suﬃces to
show that their growth is slow enough.
8. Generalizations to other physical systems
Letusnowgiveashortreviewofsomephysicalmodelsforwhichthe
methodologyofentropydissipationestimateshasenabledasatisfactory
solution of the problem of trend to equilibrium :
1)nonlinearlycoupledFokker-Planckequation,likethedrift-diﬀusion-
Poissonmodel (see[2], the references therein and [1], [7], [8]). This is a
Fokker-Planck equation @ ‰=r¢(r‰+‰rV), in which the conﬁningt
potential is equal to the sum of an external potential (say, quadratic),
and a self-consistent potential obtained through Poisson coupling. In
other words,
2jxj
V(x)= +W(x); ¡ΔW =‰:
2
1Once uniform in time L bounds on V are established, the use of log-
arithmic Sobolev inequalities leads to the conclusion that solutions of
this model converge exponentially fast to equilibrium. For the corre-
sponding bipolar problem we refer to [7].
2) nonlinear diﬀusion equations of porous-medium or fast diﬀusion
type, with a conﬁning potential, like
@‰ ﬁ n(22) =Δ‰ +r¢(‰x); t‚0; x2R ;
@t
where the exponent ﬁ satisﬁes
1
ﬁ‚1¡ :
n
For this model one can establish logarithmic Sobolev-type inequalities
of the same type as for the linear Fokker-Planck equation, see [19, 20,
28]. More generally, if one considers the equation
@‰ n(23) =r¢(rP(‰)+‰rV(x)); t‚0;x2R
@t
under the assumptions
P(‰)2D V ‚‚I ; ‚>0; nondecreasing; P(‰) increasing;n 1¡1=n‰
one can prove [17] exponential decay to equilibrium with rate at least
¡2‚t 1e . We shall elaborate on this example in the next section .
1Note added in proof : equations like (22) naturally arise as rescaled versions
ﬁof porous-medium equations, @‰=@t = Δ‰ . The rescaling method is robust even
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