General relativistic Boltzmann equation

HAL Id: hal-00771861

https://hal.archives-ouvertes.fr/hal-00771861

Submitted on 2 Jul 2013

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Ismaël Bailleul, Fabrice Debbasch

To cite this version:

I. Bailleul

1

and F.Debbas h

2

1

Statisti alLaboratory,CentreforMathemati alS ien es,Wilberfor eRoad, Cambridge,CB30WB,UK

2

UPMC,ERGA-LERMA,UMR8112,3rueGalilée,94200Ivry,Fran e

E-mail: i.bailleulstatslab. am.a .uk ,fabri e. debb as h gm ail. om

Abstra t. Anewprobalisti approa hto generalrelativisti kineti theoryis proposed.Thegeneralrelativisti BoltzmannequationislinkedtoanewMarkov pro essina ompletelyintrinsi way. Thistreatmentisthenusedtoprovethe ausal hara teroftherelativisti Boltzmannmodel.

Keywords: Generalrelativity,Boltzmann equation,probability,Markovpro ess

AMS lassi ations hemenumbers: Primary: 83C75,60H10;Se ondary: 60H30

1. Introdu tion

Relativisti transport arises in a large variety of ontexts; these in lude not only

astrophysi s[1℄and osmology[2℄,butalsoplasmaphysi s[3℄andheavyion ollisions

[4℄,andeven ondensedmatterphysi s[5,6℄wheretransport,allbeitnonrelativisti ,

o urs at bounded speed [5℄. There are three main types of models for relativisti

transport: the purely ma ros opi ,so- alled hydrodynami almodels [7℄, themodels

basedkineti theory [8℄, and thesto hasti models [9, 10℄. The purely ma ros opi

models havebeen developpedsin e the1940's [11℄, but have serious limitations. In

parti ular,traditionalLandau-E kharttheorieshavebeenproventobenon- ausal[12℄

and strong arguments[13℄, bothmathemati al and physi al, exist against themore

re entma ros opi theoriesbasedonextended thermodynami s[14℄.

As explained below, the main tool of relativisti kineti theory is arelativisti

generalization of Boltzmann famous transport equation. Various relativisti

Boltzmannequationshavebeenproposedsin e1940. Asystemati treatmenthasbeen

proposed onlyre entlyin [15,16℄ (seealso[17℄for arelativisti generalizationofthe

Vlasovequation). Thetreatmentproposedin[17℄is ovariant,butnotmanifestlyso.

Ontheotherhand,[15,16℄oersseveralequivalent,manifestly ovariantapproa hes,

but fails to oer a purely intrinsi presentation. The relativisti Boltzmann and

Boltzmann-Vlasov models havelong be assumed [13℄ to be ausal

‡

, but there is, to thebestofourknowledge,noformalproofofthisassertionintheexistinglitterature.

Physi ally realisti sto hasti models have been developed sin e 1997 [18℄ and

relativisti sto hasti pro esses now onstitute a rapidly expanding eld in both

mathemati sandphysi s. Re entreferen esaree.g.[19,20,21,22℄and[23,24,25,26℄,

wherethereaderwillndintrinsi , ovariantandmanifestly ovariantapproa hesto

relativisti diusions. All relativisti sto hasti pro essesstudiedsofarare ausal.

Theaimof thepresentarti leisthreefold.

(i) proposea lear,intrinsi presentationoftherelativisti Boltzmannequation,

(ii) usethisintrinsi presentationtoestablish,forthersttime,a learlinkbetween

the two main bran hes of relativisti transport models i.e. relativisti kineti

theoryandrelativisti sto hasti pro esses,

(iii) usethatlinkto oerasimpleproofthat therelativisti Boltzmannis ausal.

Allresultsarepresentedinanarbitraryorientedandtime-orientedspa e-time.

The material is organized as follows. Se tion 2. oers a presentation of the

physi al aspe ts of relativisti kineti theory. Se tion 3 sets up the geometri al

tools while Se tion 3 reviews the denition and properties of the relativisti

one-parti ledistribution. Se tion4presentsanintrinsi probabilisti interpretationofthe

relativisti Boltzmannequation;theproofthatthiseqautionis ausalisalsooutlined

inthisse tion,allte hni aldetailsbeingrelegatedtotheAppendix.

2. Physi al aspe ts ofrelativisti kineti theory

ThetraditionalBoltzmannequation[27,28℄aimsatdes ribingtheout-of-equilibrium

dynami s of a dilute gas of non relativisti and non quantum point parti les. This

equationistodaybestderivedfromtheso- alledBBGKYhierar hy[29,28℄. Suppose

thereare

N

parti lesinthegaz;a ordingto lassi alme hani s,theevolutionofthe gazisthen ompletely determinedby asystemof

6N

ordinarydierentialequations xingthedynami softheparti lepositionsandvelo ities. Theprobabilityofnding,

ata ertaintime

t

,any

k

≤ N

parti lesata ertainpointofthe

k

-dimensional phase-spa ethenadmitsadensitywithrespe ttotheLebesguemeasureinthisphase-spa e

andthisdensityisobviouslytheprodu t ofDira distributions.

This des riptionof thegaz isof ourse of littleuse be ause

N

is verylarge. It isalsooflittlephysi alinterest,be ausethepositionsandvelo itiesoftheindividual

gaz parti les annor be measured. What one observes are rather smoothed out or

averagedquantities. Oneususallysupposesthattheaveragedprobabilityofndingat

a ertaintimeany

k

≤ N

parti lesata ertainpointofthe

k

-dimensionalphasespa e, still admits,for all

k

, adensitywith respe t tothe Lebesgue measure in this phase-spa e. Itisthenstraightforwardtodedu efrom theequationsof lassi alme hani s,

asystemofequations obeyed exa tlybyallthese densities. This system onstitutes

whatone allstheBBGKYhierar hy.

Ifonlypair-intera tionsaretakenintoa ount,theequation

k

ofthehierar hyis anintegro-dierentialtransportequationxingthedynami softhe

k

-parti ledensity in termsof the

k

+ 1

-parti ledensity. TheBoltzmannequation is dedu edfrom the

k

= 1

-equation of the BBGKY hierar hy by taking into a ount the high dilution ofthegazandpostulatingthat thetwoparti ledensityis ompletely determinedby

the one parti le density, and by supposing all intera tions between parti les to be

loserangeintera tions whi h anthus be assimilatedtopoint ollisions. Notethat

thisrestri tiondoesnotpre ludeintera tionsof theparti les withan`exterior'eld,

independentofthegasdynami s.

N

intera ting parti les involves,notonly the parti ledegrees offreedom (positions, velo ities), but also the intera tion eld degrees of freedom. In other words, the

`me hani al' equations are, in the relativisti regime, a set of dierential equations,

notfor

6N

, but for an innitenumber of degreesof freedom. Extending theabove approa h to therelativisti realm would thus ne essitateintrodu ingdensities in an

innitedimensionalspa e. Writingarelativisti equivalentoftheBBGKYhierar hy

thus seems arather formidable task, and dealingwith su h ageneralized hierar hy

appearsevenmoredaunting.

The lassi al Boltzmann equation however, taken by itself, an be generalized

to the relativisti regime; this is possible be ause the Botzman equation onsiders

onlyoneparti ledensities, whi h admitani erelativisti formulation,andtreatsall

parti leintera tionsaspoint ollisions.

Therelativisti Boltzmanequationisthenbuiltintwosteps. Step1: itispossible

tointrodu eanaturalgeometri alobje twhi hgeneralizestotherelativisti realmthe

standardnotionofone-parti ledensity. Thiswillbe alledtherelativisti one-parti le

density. Step2: Consideragazof

N

nonquantumbutrelativisti parti lesimmersed in an`exterior'gravitationaleld,independentof thedynami sof thegaz. Suppose

also that allgaz parti lesintera t only through ollisions and that the gazparti les

arefreebetweentheir ollisionsi.e. followgeodesi softheexteriorgravitationaleld.

Itis thenpossibletowrite down fomallyarelativisti Boltzmanequationobeyedby

therelativisti distributionfun tion. Theleft-handsideof thistransport equationis

simplythea tionofthegeodesi owontherelativisti one-parti ledensityandthe

left-handsideisa ollisionterm,thegeneralformofwhi hisindependentofthedetail

oftheparti leintera tions.

This arti le makes lear the probabilisti ontent of the relativisti Boltzmann

equation by introdu ing a well hosen random dynami s and by showing that the

distributionfun tionobeysBoltzmannequationif,andonlyif,itisthedensityofthe

invariantmeasureoftherandompro ess. Theprobabilisti pointofviewalsomakes

the ausal hara teroftherelativisti Boltzmannequationvery lear.

The useof probabilisti methods to investigate the lassi alhomogeneous

non-relativisti Boltzmann equation is not newand dates ba kto Ka 's suggestion[30℄,

followed by M Kean'swork [31℄. The rst breakthrough ame from Tanaka'swork

[32℄ whoproved someexponentialrateof onvergen e toequilibrium forMaxwellian

gasesbyusing probabilisti tools. Theenormousindustrywhi hfollowed ([33℄, [34℄,

[35℄, [36℄ et .) ulminated re entlywith theresultsby Fournieret al. [37℄, [38℄ who

obtainedsomeuniqueness resultsfor somesingular ollisionkernels byprobabilisti

methods based on ouplings. It is however, to the best of our knowledge, the rst

timethat aprobabilisti viewis givenontherelativisti Boltzmann equation. More

generally,ithasbe omemoreandmore learthatthestudyofrandompro esseswith

values in Lorentzian manifolds an bring interesting insights on dierent questions

ranging from the irreversibility problem in relativisti statisti al me hani s [39℄ to

plasmaphysi s[40℄, [26℄,tothestudyofspa etimesingularities[22℄.

3. Geometri alsetting

Let

(M, g)

be aLorentzian manifold, orientedand time-oriented. Denote by

T M

its tangentbundle,withgeneri point

ϕ

= (m, ˙

m)

, andby

T

1

_M

theunit future-oriented

DenotebyVol

M

thevolumeformon

M

asso iatedwiththeLorentzianmetri

g

(forwhi hVol

M

(e

0

, . . . , e

3

) = 1

,if

(e

0

, . . . , e

3

)

isanorthonormalbasisof

M

atsome point). Identify the volume form and the volume measure Vol

M

(dm)

. Thetangent bundle

T M

inheritsfromtheLorentzianstru tureof

M

avolumemeasureVol

T M

(dϕ)

whi histhesemi-dire tprodu tofVol

M

bytheLebesguemeasureLeb

m

(d ˙

m)

inea h ber

T

m

M

,normalizedtoassignmeasure

1

toanyhyper ubeof

T

m

M

onstru tedon anorthonormalbasis:

Vol

T M

(dϕ) =

Leb

m

(d ˙

m) ⊗

Vol

M

(dm),

ϕ

= (m, ˙

m).

Atany point

m

∈ M

, the metri

g

m

on

T

m

M

indu es on ea h hyperboloid

T

1

m

M

a Riemannian metri ; denoteby Vol

1

m

(d ˙

m)

its asso iatedvolume measure,where

d

m

˙

isunderstoodhereasasurfa eelementin

T

1

m

M

. Thevolumemeasure Vol

T

1

M

is Vol

T

1

M

(dϕ) =

Vol

1

m

(d ˙

m) ⊗

Vol

M

(dm),

ϕ

= (m, ˙

m).

As is well-known, geodesi motion indu es a dynami s in

T M

whi h leaves the bundle

T

1

_M

stable: freely falling parti les have a velo ity of onstant norm. Let

denote by

H

0

the ve tor eld on

T M

generating the geodesi motion. Given a lo al oordinate system

x

: U ⊂ M 7→ R

4

, any tangent ve tor

m

˙

of

T

m

M

an uniquelybewritten

P

3

i=0

m

˙

i

∂

x

i

,withtheusualnotations. Themap

(m, ˙

m) ∈ T M 7→

(x

i

₎

06i63

,

( ˙

m

i

)

06i63

denesalo al oordinatesystemon

T M

. Inthese oordinates, thegeodesi ve toreld

H

0

reads

3

X

i=0



˙

m

i

∂

∂x

i

− Γ

k

ij

m

˙

i

m

˙

j

∂

∂

m

˙

k



,

wherethe

Γ

'saretheChristoelsymbolsof

g

. Givenaspa elikehypersurfa e

V

, write

T

1

_V

for



(m, ˙

m) ∈ T

1

_M

_{; m ∈ V}

. This

bundleinheritsfromtheLorentzianmetri anaturalvolumemeasureVol

T

1

V

whi his

thesemi-dire tprodu toftheRiemannianvolumemeasureon

V

andtheRiemannian measureinea hhyperboloid

T

1

m

M

,for

m

∈ V

. Notethat

T

1

_V

isnottheunittangent

bundleto

(V, g)

.

4. One parti ledistributionfun tion

Follow the random motion of atypi alparti le of a relativisti gas, in a spa etime

(M, g)

; itdes ribesarandom path

ψ

s

= (m

s

,

m

˙

s

)

in

T

1

_M

, where

§ s

is amultiple of thepropertimeofthetimelikepath

(m

s

)

. Withoutlossofgenerality,we anrestri t ourselvestothe asewhere

m

˙

s

hasunitnormand

s

isthepropertimeoftheparti le. The random path an be thought of as a su ession of (potentially innitesimal)

geodesi segmentsseparatedbypointswhererandom ollisions hangethevelo ityof

theparti le.

Statisti alphysi s[41,13, 42,15℄ suggeststhefollowingassumptions aboutthis

pro ess.

•

One anasso iatetoanyspa elikesubmanifold

V

ameasure

µ

T

1

V

(dϕ)

ontheunit tangentbundle

T

1

_V

over

V

,tobeunderstoodasthedistributionofatypi algas parti lehitting

V

. Themeasure

µ

T

1

V

(dϕ)

hasadensity

f

V

(ϕ)

withrespe ttothe naturalvolumemeasureVol

T

1

V

(dϕ)

on

T

1

_V

.

•

Given any point

ϕ

= (m, ˙

m) ∈ T

M

, dene

V

ϕ

as the set of spa elike hypersurfa es

V

of

M

ontaining

m

and orthogonal to

m

˙

at

m

. The value at

ϕ

of the density

f

V

does not depend on the arbitrary hoi e of hypersurfa e

V

_{∈ V}

_ϕ

,so

f

V

(ϕ)

isawell-deneds alar;denoteitby

f

(ϕ) = f (m, ˙

m)

.

•

Atanypoint

m

inspa etime,theve toreld

j(m) =

R

˙

m∈T

_m

1

M

mf

˙

(m, ˙

m)

Vol

1

m

(d ˙

m)

representstheparti le urrentatpoint

m

. Inparti ular,givenanyspa elike sub-manifold

V

with future unit normal

̟

V

(m)

at

m

∈ V

, the omponent of

j(m)

normalto

V

atpoint

m

represents,forsomeonewho onsiders

V

as3Dspa e,the 3Dorspatial parti le density

n(m)

, dened withrespe t to thenaturalvolume measureVol

V

(dm)

indu edbytheLorentzianmetri of

V

. Su hanobserverwill also onsider

T

1

_V

as6D phase-spa e and will thus view

g

m, ̟

˙

V

(m)

f

(m, ˙

m)

asthe phase-spa e parti le density, dened with respe t to the naturalvolume

measureon

T

1

_V

.

The fun tion

f

is usually alled the parti le density in

T

1

_V

, or one parti le

distribution fun tionofthe gas. Thisdensity ompletelydetermines,throughits

rst moment

j

, the parti le ontent of the spa etime. The stress-energy ontent is determinedthroughthese ondmomentsof

f

. Notethatthezerothmomentof

f

has nophysi alinterpretation.

5. Evolutionequation for the oneparti le distributionfun tion

Letusasso iatetotheoneparti uledistributionfun tionofthegasaMarkovpro ess

performing geodesi motion in between sho k times where it is hit by parti les of

the gas, resulting in a hange of its speed. Parametrize this pro ess by the proper

time of its traje tories in

M

. The rate at whi h the sho ks happen is supposed to depend only on the one parti le distribution fun tion and on the hosen model for

the ollisionme hanismof pairsof parti les. Giventwoparti lesatlo ation

m

∈ M

, withvelo ity

m

˙

and

m

˙

′

,denote by

p

and

p

′

_{∈ T}

1

_M

theout omeof the ollisionof

thetwoparti les orrespondingtothes attering angle

θ

∈ S

2

. Wedenote lassi ally

by

W

(m ; ˙

m,

m

˙

′

; θ)

the ollisionkernel,whi hrepresentstherateatwhi htheabove ollisionholds;ithasthesymmetryproperty

W

(m ; ˙

m,

m

˙

′

; θ) = W (m ; p, p

′

; θ),

(1)

forall

m

∈ M

and

m,

˙

m

˙

′

_{∈ T}

m

M

,usuallyreferedtoasthemi ros opi reversibility. See e.g. thebook[43℄ofCer ignaniandKremerforpre isemodelsof ollisionme hanisms

and ollisionkernels;these pra ti allyimportantdetails areirrelevant forus in this

work. WedeneourMarkovpro essbyitsgenerator:

Gh

(m, ˙

m) = H

0

h

(m, ˙

m)

+

Z

T

m

M×S

2



h(m, p) − h(m, ˙

m)

W

(m ; ˙

m,

m

˙

′

; θ) dθ f (m, ˙

m

′

)

Vol

1

m

(d ˙

m

′

).

(2)

Re all that the out ome

p

of the ollisionis afun tion of in oming momenta

m,

˙

m

˙

′

andthes atteringangle

θ

. Notethatasthetotalrateof ollision

mightbeinnite,onewouldreally needthesophisti atedtoolsofsto hasti al ulus

tojustifytheaboveintuitivepi tureofthemotionasgeodesi traje toriesinbetween

sho ktimes,asthese sho kstimeswouldnotbedis retein asetheaboveintegralis

innite. Thisisnotourpurposehere,though.

5.1. The relativisti Boltzmann equation

One anasso iatetoany

ϕ

= (m, ˙

m) ∈ T

1

_M

,theset

V

ϕ

ofspa elikehypersurfa es

V

of

M

ontaining

m

andorthogonalto

m

˙

at

m

. Supposenowthehittingdistributionby thepro essofanyspa elikehypersurfa e

V

hasadensity

g

V

withrespe ttoVol

T

1

V

. The value of the density

g

V

at point

ϕ

will not depend on the arbitrary hoi e of hypersurfa e

V

∈ V

ϕ

. Indeed,

g

V

(ϕ)

isthelimit ofthe ratioof themean numberof parti leshittinganeighbourhood

U

of

ϕ

in

T

1

_V

bythevolumeofthatneighbourhood,

asitde reasesto

{ϕ}

. Givenanother

V

′

in

V

ϕ

,we anmap

U

toaneighbourhood

U

′

of

ϕ

in

T

1

_V

′

byadieomorphismarbitrarily losetotheidentitysin e

V

and

V

′

havethe

sametangentspa eat

m

,provided

U

issmallenough. Thetwolimitratios

g

V

(ϕ)

and

g

V

′

(ϕ)

willthushavethesamevalue. So

g

V

(ϕ) = g(ϕ)

isanaturalfun tion(s alar)on theunit tangentbundle

T

1

_M

, named one parti le distribution fun tion of the

Markov pro ess. The fun tion

g

enjoys the following ru ial analyti al property, provedin Appendix.

Proposition1. Wehave:

G

∗

g

= 0

. So the measure

g(ϕ)

Vol

T

1

M

(dϕ)

is invariant for the random dynami s, and the equation

G

∗

g

= 0

isadetailledbalan eequation. Inamore on reteway,proposition 1meansthattheintegral

Z

T

1

M

g(ϕ)

n

(H

0

h)(ϕ)+

Z

T

m

M×S

2

˘h(m, p)−h(m, ˙

m)¯W (m ; ˙

m,

m

˙

′

; θ) dθ f (m, ˙

m

′

)

Vol

m

(d ˙

m

′

)

o

Vol

T

1

M

(dϕ)

(3)

isnullforanysmoothfun tions

h

with ompa tsupport. Wewritehere

ϕ

= (m, ˙

m)

for ageneri element

ϕ

∈ T

1

_M

. The symmetry property (1) of the ollision kernel

andanintegrationbyparts

k

enabletore-write(3)undertheform

Z

T

1

M

−H

0

g

+ C(f, g)

(ϕ)h(ϕ)

Vol

T

1

M

(dϕ) = 0,

where

C f, g

(ϕ) =

Z

T

1

M

Z

S

2



g(m, p)f (m, p

′

)−g(m, ˙

m)f (m, ˙

m

′

)

W

(m ; ˙

m,

m

˙

′

; θ) dθ

Vol

m

(d ˙

m

′

),

that is

H

0

g

= C(f, g)

. Boltzmann's fundamental haos hypothesis is equivalent to saying that the one parti le distribution fun tion of the gas and the one parti le

distributionfun tion oftheMarkovpro ess oin ide:

g

= f

. Equation

H

0

f

= C(f, f )

is the usual form of the relativisti Boltzmann equation. Consult [44℄ for a totally

dierentandaxiomati presentationofthegeneralrelativisti Boltzmann equation.

k

Theve toreld

H

0

hasan

L

2

`

Vol

T

1

M

´

-dualequalto

−H

0

asitpreservesLiouvillemeasureon

T

1

_M

5.2. Causal hara terof therelativisti Boltzmannequation

We show in this se tion howthe introdu tion of the above random dynami s leads

toa learunderstandingofthe ausal hara terofthegeneralrelativisti Boltzmann

equation, through proposition 1. We referthe reader to the works [45, 46, 47℄ and

[48℄of DudynskiandEkiel-Jezewskaformathemati alworksonthat questionin the

spe ialrelativisti ase.

Fix an open spa elike hypersurfa e

V

and denote by

D

+

_(V)

its future domain

of dependen e: it is the set of points

m

of

M

su h that any past-dire ted tiemlike pathstarted from

m

hits

V

. This set is known to be globallyhyperboli , [49℄. The nextproposition holdsforallglobally hyperboli spa etimesalthoughwestateitfor

D

+

(V)

.

Proposition 2. One an asso iate to any point

m

of

D

+

_(V)

a positive onstant

T

(m)

su h thatanypastdire tedtimelikepathstarted from

m

, parametrizedbyits propertime,hits

V

before time

T

(m)

.

Proof Itsu estotakefor

T

(m)

thelengthofafuture-dire tedmaximalgeodesi from

V

to

m

,whoseexisten eisguaranteedbytheglobalhyperboli ityof

D

+

_(V)

seee.g. prop. 2.33inSenovilla'sreview[50℄,or onsult[51℄.



Considerthe

T

1

_M

-valuedMarkovpro ess

(ψ

s

)

s>0

= (m

s

,

m

˙

s

)

s>0

withgenerator

G

∗

h

= −H

0

h

+ C(f, h);

it has past-dire ted timelike paths. Start it from a point

(m, ˙

m) ∈ T

1

_M

with

m

∈ D

+

_(V)

. Sin e

f

is

G

∗

-harmoni (by proposition 1), the random pro ess

f

(ψ

s

)

s>0

is a non-negativemartingale. Denote by

H

the hitting time of

T

1

_V

by

(ψ

s

)

s>0

; itisalmost-surelyboundedaboveby

T

(m)

,byproposition2. One anthus applytheoptionalstoppingtheorem andget

f

(m, ˙

m) = E

(m, ˙

m)



f

(ψ

H

)



.

This identity proves the rst part of the following statement. Write

T

1

_D

+

_(V)

for



(m, ˙

m) ∈ T

1

_M

_{; m ∈ D}

+

_(V)

.

Theorem3. Let

(M, g)

beanyLorentzianmanifoldand

V

beaspa elikehypersurfa e. The one parti le distribution fun tion of a gas is a ausal fun tion: its values on

T

1

_D

+

_(V)

are determined by its values on

T

1

_V

. The restri tion of

f

to

T

1

_V

is the

minimalsetofdataneededtodetermine

f

on

T

1

_D

+

_(V)

.

Proof The se ond part of the statement dire tly omes from the fa t that the

distribution of

ψ

H

has support in the whole of

T

1

_I

−

_(m

0

) ∩ V

for a pro ess

startedfrom thepoint

ψ

0

= (m

0

,

m

˙

0

)

.



Appendix

The result of proposition 1 omes from Kolmogorov's forward equation for the

transitionsemi-groupofageneralMarkovpro ess

X

;were allithere. Denoteby

x

ageneri elementofthestatespa eofthepro essandwrite

P

t

(x, h) = E

x



h(X

t

)

for the expe tation of

h(X

t

)

for a pro ess started from

x

; write

P

t

(x, dy)

for the asso iated kernel on the state spa e. Write, as above,

G

for the generator of the pro ess. Kolmogorov'sforward equation omes from thesemi-group propertyof the

kernels

P

t

(x, ·)

, en odedin theChapman-Kolmogorovequation

P

t+s

(x, h) =

Z

P

t

(y, h)P

s

(x, dy),

∀ s, t > 0, x

in thestatespa e

,

andreads(seee.g. Chap. 1of[52℄)

d

dt

P

t

(x, h) = P

t

(x, Gh).

(A.1)

Ina ontext wherethekernels

P

t

(x, ·)

aregivenbyadensity

p

t

(x, y)

withrespe t to somereferen emeasure

dy

,equation(A.1)re-writes

d

dt

p

t

(x, y) = G

∗

p

t

(x, y),

(A.2) where

G

∗

a tson

y

andisthedualof

G

in

L

2

_(dy)

. Notehoweverthatthereisnoneed

ofdensitiestomakesenseofequation(A.1).

Theresultofproposition1islo alin

M

;itwill omeasanappli ationofequation (A.2) by reparametrizinglo ally the traje tories of the pro ess by a time fun tion

denedlo ally on

M

. Thefollowinglo al onstru tionwillbeusedtothatend. a)Normalvariationofaspa elikehypersurfa e. Let

V

bearelatively ompa t spa elikehypersurfa e of

M

. For

m

∈ V

and

ε

∈ R

small enough, dene

Φ

ε

(m)

as theposition attime

ε

of thegeodesi started from

m

,leaving

V

orthogonallyin the future dire tion with aunit speed. Then there exists, as a onsequen eof thelo al

inversiontheorem, apositive onstant

η

andan openset

U ⊂ M

su h that the map

Φ : (−η, η) × V → U

,

(ε, m) 7→ Φ

ε

(m)

, isadieomorphism. Letusfurther suppose

η

and

V

smallenoughfor

U

tobestrongly ausal. Writing

V

ε

for

Φ

ε

(V)

,themap

Φ

0

is theidentityon

V

,and

∂

ε

Φ

ε

(m) ∈ T

1

Φ

ε

(m)

M

isorthogonalto

T

Φ

ε

(m)

V

ε

. Thefamilyof spa elikehypersurfa es

{V

ε

}

ε∈(−η,η)

is alledthenormalvariationof

V

. Thefollowing relatednotationwill beuseful.

Notations. Wedeneave toreld

̟

on

U

asfollows. Givenapoint

m

∈ V

ε

,denote by

̟(m)

the futureunit timelike ve tororthogonal to

T

m

V

ε

;set

̟(ϕ) := ̟(m)

,for

ϕ

= (m, ˙

m)

.

• γ = γ(ϕ) := g ̟(ϕ), ˙

m

will be afun tion of

ϕ

= (m, ˙

m)

inthe tangentbundle of

U

.

•

The

∗TV

ε

-operationwillstandfortaking the

L

2

₍

Vol

TV

_ε

)

-dualandthe

∗

-operation

for takingthe

L

2

₍

Vol

T

1

M

)

-dual.

•

For larity, all obje ts dened on

V

or

TV

will have a hat on them:

ϕ, b

b

G, b

ψ

ε

...

denedbelow.

•

Last,

H

ε

willdenote the hittingpropertimeof

TV

ε

⊂ T

1

_M

.

b) Reparametrization of the traje tories of the pro ess. Given a point

ϕ

of

T

1

_M

take a relatively ompa t spa elike hypersurfa e

V

ontaining

m

and do the pre eding onstru tion. To provethat

G

∗

use the parameter

ε

as a time parameter rather than using the proper time of the randomtraje tories. That is, onsider there-parametrized pro ess

{ψ

H

_ε

}

ε∈(−η,η)

; it hasgenerator

γ

−1

_G

. De omposethis operatorasfollows

∀ ϕ = Φ

ε

( b

ϕ) ∈ V

ε

,

Gf

γ

(ϕ) = (̟f )(ϕ) + b

G(f ◦ φ

ε

) ( b

ϕ) = (̟f )(ϕ) + Gf

(ϕ),

(A.3)

where

G

b

isanoperatoron

TV

,andwhere,asa onsequen e,

G

a tsonlyon

TV

ε

. Now, dene the

TV

-valuedpro ess

 bψ

ε

ε∈(−η,η)

:=



Φ

−1

ε

(ψ

H

ε

)

ε∈(−η,η)

and denote by

ℓ

b

ε

itstime-dependentgenerator.

)Proofofproposition1. Withoutlossofgenerality,one anassumethat

ρ

b

0

hasa smoothdensitywithrespe ttoVol

TV

anddenoteby

b

ρ

ε

thedensityofthedistribution of

ψ

b

ε

with respe t to Vol

TV

. By Kolmogorov's forward equation, it satises the equation

∂

ε

ρ

b

ε

= b

ℓ

∗TV

ε

ρ

b

ε

,

forall

ε

∈ (−η, η)

. Theoperator

b

ℓ

∗TV

ε

standshereforthe

L

2

₍

Vol

TV

)

-dual of

ℓ

b

ε

. Let usnowdenoteby Vol

(ε)

TV

thepull-ba kon

TV

by

φ

ε

of themeasureVol

TV

_ε

on

TV

ε

, anddenoteby

G

ε

itsdensitywithrespe ttoVol

TV

. Then

ϕ

b

ε

hasadensity

b

µ

ε

=

b

ρ

ε

G

ε

withrespe ttoVol

(ε)

TV

;itsatisestheequation

∂

ε

µ

b

ε

+

∂

ε

G

ε

G

ε

b

µ

ε

= b

ℓ

∗TV

; ε

ε

µ

b

ε

.

(A.4)

We have written here

b

ℓ

∗TV; ε

ε

g

for

b

ℓ

∗TV

ε

(G

ε

g)

G

ε

. Denote by

µ

ε

the density of

ψ

H

ε

with

respe t to Vol

TV

ε

, and onsider

µ

and

G

as fun tions of

ε

and

ϕ

∈ TV

ε

, that is, onsiderthemasfun tionsdenedonthetangentbundleof

U

. Byitsverydenition, the fun tion

µ

and the one parti le distribution fun tion of the pro ess are linked throughtherelation

µ(ϕ) = γ g(ϕ),

(A.5)

dis ussedinthethird pointofse tion4. Equation(A.4) anbewrittenin termsof

µ

as

̟µ

+

̟ G

G

µ

= G

∗TV

ε

µ.

(A.6)

Theoperator

G

hasbeen introdu ed in equation (A.3). It is usefulat that stage to remarkthatwehave

G

∗TV

ε

= G

∗

as a onsequen e of the hange of variable formula, and sin e we have a normal

variationof

V

. Thefollowinglemmaisneededtomakethenalstep. Lemma. Wehaveforanysmoothfun tion

f

Proof Asabove,thisis onsequen eofthe hangeofvariableformulaandthefa t

that we have a normal variation of

V

. Write

T

1

_U

for the future unit tangent

bundleover

U

andtake

h

asmoothfun tionover

T

1

_U

with ompa tsupport.

Z

T

1

U

(̟

∗

f

) (ϕ) h(ϕ)

Vol

T

1

M

(dϕ) =

Z

T

1

U

f

(ϕ) (̟ h)(ϕ)

Vol

T

1

M

(dϕ)

=

Z

(−η,η)

Z

TV

f

(ε, b

ϕ) (∂

ε

h

)(ε, b

ϕ) G

ε

( b

ϕ)

Vol

TV

(d b

ϕ) dε

= −

Z

(−η,η)×TV

(∂

ε

f

)(ε, b

ϕ

) h(ε, b

ϕ) G

ε

( b

ϕ)

Vol

TV

(d b

ϕ) dε

−

Z

(−η,η)×TV

(f h)(ε, b

ϕ) ∂

ε

G

ε

( b

ϕ)

Vol

TV

(d b

ϕ) dε

= −

Z

T

1

U



̟f

+

̟ G

G



(ϕ) h(ϕ)

Vol

T

1

M

(dϕ).

(A.7)



Asa onsequen eofthislemmawe anusethede ompositiongiveninequation(A.3)

towriteequation(A.6) as

G

∗

µ

γ

= 0

,thatis

G

∗

_g

_{= 0}

,byequation(A.5).

[1℄ A.W.Guthmann,M.Georanopoulos,A.Mar owith, andK.Manolakou, editors. Relativisti FlowsinAstrophysi s,Berlin,2002.Springer.

[2℄ J. Bernstein. Kineti theory in the expanding universe. Cambridge Monographs on Mathemati alPhysi s.CambridgeUniversityPress,Cambridge,1988.

[3℄ J.Freidberg. PlasmaPhysi sandFusionEnergy. Cambridge,2007.

[4℄ J.Dunkeland P.Hänggi. One-dimensionalnonrelativisti andrelativisti Brownianmotions: Ami ros opi ollisionmodel. Phys.A,374:559572,2007.

[5℄ C. Chevalier, F.Debbas h, and J.P.Rivet. Areview of nite speed transport models. In Pro eedings of the Se ond International Forum on Heat Transfer (IFHT08), Sept. 17-19, Tokyo,Japan.HeatTransferSo ietyofJapan,2008.

[6℄ A.Pototskyetal. Relativisti brownianmotiononagraphene hip. arXiv:1103.0945,2011. [7℄ L.D.LandauandE.M.Lifshitz. FluidMe hani s. PergamonPress,Oxford,1987.

[8℄ S.R.deGroot,W.A.vanLeeuwen,andC.G.vanWeert. Relativisti Kineti Theory. North-Holland,Amsterdam,1980.

[9℄ F.Debbas handC.Chevalier. Relativisti sto hasti pro esses: areview. InO.Des alzi,O.A. Rosso,andH.A.Larrondo,editors,Pro eedingsofMedynol2006,NonequilibriumStatisti al Me hani sandNonlinearPhysi s,XVConferen eonNonequilibriumStatisti alMe hani s and Nonlinear Physi s, Mar del Plata, Argentina, De . 4-8 2006'., volume 913 of A.I.P. Conferen ePro eedings,Melville,NY,2007.Ameri anInstituteofPhysi s.

[10℄ J.DunkelandP.Haenggi. Relativisti brownianmotion. Physi sReports,471(1):174,2009. [11℄ C.E kart. Phys.Rev.,58:919,1940.

[12℄ W.A.His o kandL.Lindblom. Phys.Rev.D,31:725,1985.

[13℄ W.Israel. Covariantuidme hani sandthermodynami s: Anintrodu tion. InA.Anileand Y.Choquet-Bruhat, editors,Relativisti Fluid Dynami s,volume1385 ofLe ture Notesin Mathemati s,Berlin,1987.Springer-Verlag.

[14℄ I.MüllerandT.Ruggeri. ExtendedThermodynami s,volume37ofSpringerTra tsinNatural Philosophy. Springer-Verlag,New-York,1993.

[15℄ F. Debbas h and W. van Leeuwen. General relativisti Boltzmann equation i. ovariant treatment. Physi aA,388(7):10791104,2009.

[16℄ F. Debbas h and W. van Leeuwen. General relativisti Boltzmann equation ii. manifestly ovarianttreatment. Physi aA,388(9):18181834,2009.

[17℄ I.Y.DodinandN.J.Fis h. Vlasovequationand ollisionlesshydrodynami sadaptedto urved spa etime. Phys.Plasmas,17:112118,2010.

[19℄ J.Fran hiand Y.LeJan. Relativisti diusions andS hwarzs hild geometry. Comm. Pure Appl.Math.,60(2):187251,2007.

[20℄ J.AngstandJ.Fran hi. Centrallimittheoremfora lassofrelativisti diusions. J.Math. Phys.,48(8):083101,20,2007.

[21℄ I.Bailleul. Poissonboundaryofarelativisti diusion. ProbabilityTheoryandRelatedFields, 141(1):283330,2008.

[22℄ I.Bailleul. Aprobabilisti viewonsingularities. J.Math.Phys.,52:023520,2011.

[23℄ F.Debbas h. Adiusionpro essin urvedspa e-time. J.Math.Phys.,45(7):27442760,2004. [24℄ M. Rigotti and F.Debbas h. An

H

-theorem for the general relativisti Ornstein-Uhlenbe k

pro ess. J.Math.Phys.,46(10):103303,11,2005.

[25℄ JörnDunkelandPeterHänggi. Relativisti Brownianmotion. Phys.Rep.,471(1):173,2009. [26℄ Z.Haba. Relativisti diusionof elementaryparti les withspin. J.Phys.A,42(44):445401,

17,2009.

[27℄ L.Boltzmann.VorlesungenüberGastheorie. ErweiterterNa hru kder1896-1898beiAmbrosius BarthinLeipzigers hienenAusgabe,1981. Akademis heDru ku.Verlagsanstalt,Graz. [28℄ K.Huang. Statisti alMa hani s. JohnWiley&Sons,NewYork,2ndedition,1987.

[29℄ N.N.Bogoliubov. Problemsofadynami altheoryinstatisti alphysi s. InG.E.Uhlenbe kJ.de Boer,editor,StudiesinStatisti alMe hani s,volume1,Amsterdam,1962.North-Holland. [30℄ M.Ka . Foundations ofkineti theory. InPro eedings of theThird BerkeleySymposium on

Mathemati al Statisti sand Probability, 19541955, vol.III, pages 171197,Berkeley and LosAngeles,1956.UniversityofCaliforniaPress.

[31℄ H.P.M Kean,Jr. A lassofMarkovpro essesasso iatedwithnonlinearparaboli equations. Pro .Nat.A ad.S i.U.S.A.,56:19071911,1966.

[32℄ H. Tanaka. Probabilisti treatmentof the Boltzmann equation ofMaxwellianmole ules. Z. Wahrs h.Verw.Gebiete,46(1):67105,1978/79.

[33℄ K.U hiyama. Onderivation ofthe Boltzmannequationfromadeterministi motionofmany parti les. InProbabilisti methods in mathemati al physi s(Katata/Kyoto, 1985), pages 421441.A ademi Press,Boston,MA,1987.

[34℄ C. Graham and S. Méléard. Sto hasti parti le approximations for generalized Boltzmann modelsand onvergen eestimates. Ann.Probab.,25(1):115132,1997.

[35℄ G.Tos aniandC.Villani. Probabilitymetri sanduniquenessofthesolutiontotheBoltzmann equationforaMaxwellgas. J.Statist.Phys.,94(3-4):619637,1999.

[36℄ F.RezakhanlouandC.Villani. EntropymethodsfortheBoltzmannequation,volume1916of Le tureNotesin Mathemati s. Springer, Berlin, 2008. Le tures froma Spe ialSemester onHydrodynami LimitsheldattheUniversitédeParisVI,Paris,2001,EditedbyFrançois GolseandStefanoOlla.

[37℄ N. Fournier and H. Guérin. On the uniqueness for the spatially homogeneous Boltzmann equationwithastrongangularsingularity. J.Stat.Phys.,131(4):749781,2008.

[38℄ N.Fournierand Cl.Mouhot. Onthewell-posednessofthespatiallyhomogeneousBoltzmann equationwithamoderateangularsingularity. Comm.Math.Phys.,289(3):803824,2009. [39℄ F.Debbas h,K.Malli k,andJ.P.Rivet. Relativisti Ornstein-Uhlenbe kpro ess. J.Statist.

Phys.,88(3-4):945966,1997.

[40℄ Z. Haba. Relativisti diusion with fri tion on a pseudo-Riemannian manifold. Classi al QuantumGravity,27(9):095021,15,2010.

[41℄ J.Ehlers. InPro eedings of theVarennaSummer S hoolon Relativsiti Astrophysi s, New-York,1971.A ademi Press.

[42℄ F. Debbas h, J.P. Rivet, and W.A.van Leeuwen. Invarian e of the relativisti one-parti le distributionfun tion. Physi aA,301:181195,2001.

[43℄ C.Cer ignaniandG.M.Kremer. Therelativisti Boltzmannequation: theoryandappli ations, volume22ofProgressinMathemati alPhysi s. BirkhäuserVerlag,Basel,2002.

[44℄ C. Marle. Sur l'établissement des équations de l'hydrodynamique des uides relativistes dissipatifs.I.L'équation deBoltzmann relativiste. Ann. Inst.H.Poin aré Se t.A(N.S.), 10:67126,1969.

[45℄ M.Dudy«skiandM.L.Ekiel-Je»ewska. Onthe linearizedrelativisti Boltzmann equation.I. Existen eofsolutions. Comm.Math.Phys.,115(4):607629,1988.

[46℄ M. Dudy«skiandM. L.Ekiel-Je»ewska. Causalityprobleminthe relativisti kineti theory. In Re ent developments in nonequilibrium thermodynami s: uids and related topi s (Bar elona, 1985),volume 253ofLe tureNotesinPhys., pages357360.Springer, Berlin, 1986.

[48℄ M. Dudy«ski and M. L. Ekiel-Je»ewska. Causality of the linearized relativisti Boltzmann equation. Phys.Rev.Lett.,55(26):28312834,1985.

[49℄ S.W.HawkingandG.F.R.Ellis.Thelarges alestru tureofspa e-time. CambridgeUniversity Press,London,1973. CambridgeMonographsonMathemati alPhysi s,No.1.

[50℄ J. Senovilla. Singularity theorems and their onsequen es. Gen. Relativity Gravitation, 30(5):701848,1998.

[51℄ J. K.Beem, P. E. Ehrli h, and K.L. Easley. Global Lorentzian geometry, volume 202 of Monographs and Textbooks in Pure and Applied Mathemati s. Mar el Dekker In ., New York,se ondedition,1996.