PROBABILITY METRICS AND UNIQUENESS OF THE SOLUTION TO THE BOLTZMANN EQUATION FOR A

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PROBABILITY METRICS AND UNIQUENESS OF THE SOLUTION TO THE BOLTZMANN EQUATION FOR A MAXWELL GAS G. TOSCANI AND C. VILLANI Abstract. We consider a metric for probability densities with finite variance on Rd, and compare it with other metrics. We use it for several applications, including a uniqueness result for the solution of the spatially homogeneous Boltzmann equation for a gas of true Maxwell molecules. Key-words : spatially homogeneous Boltzmann equation, probability metrics, Maxwellian molecules. 1. Introduction Denote by Ps(Rd), s > 0, the class of all probability distributions F on Rd, d ≥ 1, such that ∫ R d |v|s dF (v) < ∞. We introduce a metric on Ps(Rd) by (1) ds(F,G) = sup ??Rd |f?(?)? g?(?)| |?|s where f? is the Fourier transform of F , f?(?) = ∫ R d e?i?·v dF (v). Let us write s = m + ?, where m is an integer and 0 ≤ ? < 1. In order that ds(F,G) be finite, it suffices that F and G have the same moments up to order m. The norm (1) has been introduced in [6] to investigate the trend to equilibrium of the solutions to the Boltzmann equation for Maxwell molecules.

• cluding d2


• then


• called mc


• called csiszar-kullback inequality


• probability metrics


• csiszar


• supn ∫


• see also


• also apply


• boltzmann equation

Sujets

• See also
• Boltzmann equation

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 46
• Langue: English

Extrait

PROBABILITY METRICS AND UNIQUENESS OF THE
SOLUTION TO THE BOLTZMANN EQUATION FOR A
MAXWELL GAS
G. TOSCANI AND C. VILLANI
Abstract. We consider a metric for probability densities with
dfinite variance onR , and compare it with other metrics. We use
it for several applications, including a uniqueness result for the
solution of the spatially homogeneous Boltzmann equation for a
gas of true Maxwell molecules.
Key-words: spatiallyhomogeneousBoltzmannequation,probability
metrics, Maxwellian molecules.
1. Introduction
dDenote by P (R ), s>0, the class of all probability distributions Fs
donR , d‚1, such that
Z
sjvj dF(v)<1:
d
dWe introduce a metric on P (R ) bys
bjf(»)¡gb(»)j
(1) d (F;G)= sups s
d j»j»2
bwhere f is the Fourier transform of F,
Z
¡i»¢vbf(»)= e dF(v):
d
Let us write s = m+fi, where m is an integer and 0 • fi < 1. In
order that d (F;G) be finite, it suffices that F and G have the sames
moments up to order m.
The norm (1) has been introduced in [6] to investigate the trend to
equilibrium of the solutions to the Boltzmann equation for Maxwell
molecules. There, the case s = 2+fi, fi > 0, was considered. Further
applicationsofd , withs=4, werestudiedin[3], whilethecases s=2s
and s = 2+fi, fi > 0, have been considered in [4] in connection with
the so-called Mc Kean graphs [8].
1
RRR2 G. TOSCANI AND C. VILLANI
In this paper, we shall be interested with the case s = 2. To under-
stand why this case separates in a natural way from the other ones, let
dus briefly introduce and discuss other well-known metrics on P (R ).s
dLet F, G in P (R ), and let Π(F;G) be the set of all probability dis-s
d dtributions L in P (R £R ) having F and G as marginal distributions.s
Let
Z
s(2) T (F;G)= inf jv¡wj dL(v;w):s
L2Π(F;G)
1=s dThen¿ =T metrizestheweak-*topology TW onP (R ). Wenotess ⁄ s
thatT istheKantorovich-Vasershteindistanceof F andG[7,18]. For1
adetaileddiscussion,andapplicationofthesedistancestostatisticsand
informationtheory,seeVajda[17]. Seealso[10]forarecentapplication
to kinetic theory.
The case s = 2 was introduced and studied independently by Tana-
ka [15] who, in the one-dimensional case d=1, applied T to the study2
of Kac’s equation. Subsequently, the properties of T were studied in2
the multidimensional case by Murata and Tanaka [9]. Applications
to the kinetic theory of rarefied gases were finally studied by Tanaka
in[16];severaloftheseapplicationsweregivenasimplifiedproofin[12].
TheimportanceofTanaka’sdistance¿ mainlyreliesonitsconvexity2
and superadditivity with respect to rescaled convolutions. We recall
this property, that is at the basis of most of the applications of T .2
LetfX ;Y g,fX ;Y gbetwoindependentpairsofrandomvariables,0 0 1 1
andletF (resp. G )betheprobabilitydistributionofX (resp. Y ),i=i i i i
0;1. For 0 < ‚ < 1, let F (resp. G ) be the probability distribution‚ ‚p pp p
of ‚X + 1¡‚X (resp. ‚Y + 1¡‚Y ), i.e.0 1 0 1
? ¶ ? ¶
1 ¢ 1 ¢
(3) F = F p ⁄ F p :‚ 0 1d=2 d=2‚ (1¡‚)‚ 1¡‚
Then,
(4) T (F ;G )•‚T (F ;G )+(1¡‚)T (F ;G ):2 ‚ ‚ 2 0 0 2 1 1
Superadditivity is also known for convex functionals (relative en-
tropies), like Boltzmann’s relative entropy
Z
f(v)f(5) H(fjM )= f(v)log dv;
f
d M (v)
fwhere f is a probability density and M is the Gaussian density with
the same mean vector and variance as those of f. This means that
the property (4) holds with T replaced by H and fG ;G g replaced2 0 1
f f0 1by fM ;M g. This is a consequence of Shannon’s entropy power
RPROBABILITY METRICS AND UNIQUENESS 3
inequality (Cf [13, 1]). The same property holds for the relative Fisher
information,
Z
fl fl2f ffl fl(6) I(fjM )= rlogf(v)¡rlogM (v) f(v)dv
d
(see again [13, 1]). As discussed by Csiszar [5], by means of the rela-
tive entropy H, one can define the so-called H-neighbourhoods. Even
ifthosedonotdefineatopologicalspace,intheusualsense,theirtopo-
logical structure is finer than the metric topology defined by the total
variation distance,
Z
”(f;g)= jf(v)¡g(v)jdv;
in the sense that
p
(7) ”(f;g)• 2H(fjg);
which is the so-called Csiszar-Kullback inequality.
It turns out that these properties of superadditivity and convexity
also hold for d , the proofs being in fact much more simple. As an2
illustration of the interest of these properties, we shall give a version
of the central limit theorem and a very simple proof of Kac’s theorem.
We shall also apply d to the study of the Boltzmann equation with2
Maxwellian molecules,
@f
(8) (t;v)=Q(f;f)(t;v)
@t
Z ? ¶
u¢n 0 0= ? [f(v)f(w)¡f(v)f(w)] dwdn;
3 2 juj£S
where u = v ¡ w is the relative velocity of colliding particles with
velocity v and w, and
v+w juj v+w juj0 0v = + n; w = ¡ n
2 2 2 2
are the postcollisional velocities. ?(”) is a nonnegative function which
for true Maxwell molecules has a nonintegrable singularity of the form
¡5=4(1¡”) as ”!1. Usually, one truncates ? in some way, so that it
become integrable (cut-off assumption).
We shall prove that d shares a remarkable property with Tanaka’s2
distance : it is nonexpanding with time along trajectories of the Boltz-
mann equation; that is, if f and g are two such solutions,
(9) d (f(t);g(t))•d (f(0);g(0)):2 2
This holds even if ? is singular. As an immediate corollary, we obtain
thatthesolutiontotheCauchyproblemfortheBoltzmannequationis
RR4 G. TOSCANI AND C. VILLANI
unique. Up to our knowledge, this is so far the only uniqueness result
available for long-range interactions.
The organization of the paper is as follows. First, in section 2, we
dinvestigate the connections between several distances on P (R ), in-2
cluding d and ¿ . In section 3, we establish the basic properties of2 2
superadditivity for d and give applications. Then, in section 4, we2
apply this metric to the study of the Boltzmann equation.
d2. Metrics on P (R )2
In order that d be well-defined, we need to restrict it to some space2
of probability densities with the same mean vector. For simplicity, we
shall restrict to probability measures with zero mean vector, and we
shall work on
‰ ?Z Z
d 2(10) D = F 2P (R ); v dF(v)=0; jvj dF(v)=d¢?? 2 i
where ? is some positive real number. We begin with two elementary
lemmas.
dLemma 1. Let (F ) in D , F *F in P (R ). Thenn ? n 0
Z
2(11) F 2D () lim sup jvj 1 dF (v)=0:? jvj‚K n
K!1n!1
R
2Proof. It is clear that if lim sup jvj 1 dF (v) = 0, thenK!1 jvj‚K nn!1
F 2 D . On the other hand, if this condition is not satisfied, then? R
2there exists ">0 and (K )!1 with jvj 1 dF (v)•d¢?¡".n njvj•KnR
2 2 2Sincejvj 1 *jvj , this implies that jvj dF(v)•d ¡", hencejvj•Kn
F 2= D . ⁄?
Lemma 2. Let (F ) and F in D , such that F * F. Then, theren ? n
exists a nonnegative function ` such that `(r)=r ¡! 1 as r ! 1,
which can be chosen smooth and convex, and a constant M > 0, such
that
Z
2(12) sup `(jvj )dF (v)•M:n
n
Proof. Using Lemma 1, copy the construction that was done in [6] forR
2one single function : there exists k such that sup jvj dF (v)•1 nn jvj‚k1R
21=2; then there exists k ‚ k such that sup jvj dF (v)• 1=4;2 1 nn jvj‚k2
and so on... for all p‚2, there exists k ‚k such thatp p¡1
Z
12sup jvj dF (v)• :n p2n jvj‚kpPROBABILITY METRICS AND UNIQUENESS 5
2 2Thenchoose`(jvj )=pjvj ifjvj2 [k ;k ); smooththisfunctionandp p+1
slow its growth if necessary, as in [6]. ⁄
In the sequel of the paper, for M >0, we shall denote by
‰ ?Z
M d 2+fi(13) P (R )= F 2D ; jvj dF(v)•M ;?2+fi
‰ Z ?
M d 2(14) P (R )= F 2D ; `(jvj )dF(v)•M?`
for ` nonnegative, `(r)=r ¡!1 as r ! 1. Lemma 2 enables us to
Mrestrict to spaces P in all the cases when one is interested with weak`
convergence in D .?
In addition to the metrics d and ¿ = ¿ which were introduced in2 2
the last section, we consider
d† Prokhorov’s distance ‰(F;G) : for –‚0 and U ‰R , we define
' “ ' “
– d –] dU = v2R ; d(v;U)<– ; U = v2R ; d(v;U)•– ;
where d(v;U)=inffkv¡wk;w2Ug. Let
' “
" d?(F;G)=inf ">0=F(A)•G(A )+" for all closed A‰R ;
we set
(15) ‰(F;G)=maxf?(F;G);?(G;F)g:
m ⁄ ⁄ m d† the (C ) distancekF¡Gk : form‚1, letC (R )bethesetm
of m-times continuously differentiable functions, endowed with
its natural normk¢k . Then, letm
fl fl‰Z ?
fl fl
⁄ mfl fl(16) kF ¡Gk =sup ’dF(v) ;’2C ;k’k &#