Two mathematical tools to analyze metastable stochastic processes

0.69 MB·English

by  Tony Lelièvre

Two mathematical tools to analyze metastable
stochastic processes
TonyLelie`vre
2
1
0
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n
a
J AbstractWepresenthowentropyestimatesandlogarithmicSobolevinequalitieson

8 theonehand,andthenotionofquasi-stationarydistributionontheotherhand,are
1 usefultoolstoanalyzemetastableoverdampedLangevindynamics,inparticularto

quantifythedegreeofmetastability.Wediscusstheinterestoftheseapproachesto
]
A estimatetheefficiencyofsomeclassicalalgorithmsusedtospeedupthesampling,
N and to evaluate the errorintroduced by some coarse-graining procedures. This pa-
per is a summary of a plenary talk given by the author at the ENUMATH 2011
.
h conference.
t
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m
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1 Introductionandmotivation

1
v
5 The aim of this paper is to present two mathematical viewpoints on metastability.
7 Roughlyspeaking,adynamicsissaidtobemetastableifitspendsalotoftimein
7 a region (called a metastable state) before hopping to another region. To be more
3 specific,wewillfocusinthefollowingontheoverdampedLangevindynamics:
.
1
(cid:112)
0 dX =−∇V(X)dt+ 2β−1dW (1)
t t t
2
1
which is used for example in molecular dynamics to describe the evolution of a
v: molecular system. In this context, the configuration of the system Xt ∈Rn is the
i coordinatesoftheparticles(thinkoftheatomsofalargemolecule),V :Rn→Ris
X
thepotentialenergy,whichtoaconfigurationx∈RnassociatesitsenergyV(x),and
r
a β−1 =kBT is proportional to the temperature (kB being the Boltzmann constant).
The stochastic processW is a standard n-dimensional Brownian motion. For such
t
TonyLelie`vre
CERMICS, Ecole des Ponts Paristech, Universit Paris-Est, 6 et 8 avenue Blaise Pascal, 77455
Marne-la-Valle´e,FranceandINRIARocquencourt,MicMacprojectteam,DomainedeVoluceau,
B.P.105,78153LeChesnayCedex,France.e-mail:[email protected]
1
2 TonyLelie`vre
a dynamics, metastability typically originates from two mechanisms. First, in the
small temperature regime, the dynamics (1) can be seen as a perturbation of the
simple gradient dynamics y˙=−∇V(y) for which, from any initial condition, the
solutionconvergestoalocalminimumofV.Havingthisinmind,thedynamics(1)
ismetastablebecauseittakesalotoftimetoleavethevicinityofalocalminimum
before jumping to the neighborhood of another local minimum. This is due to the
energybarrierswhichhavetobeovercome.Suchbarriersandthezerotemperature
limit can be analyzed in particular with large deviation techniques [19]. Second,
metastabilitymaycomefromentropiceffects.Imaginethattheconfigurationspace
is made of two boxes linked by a narrow corridor (the potentialV is zero on this
configurationspace,andinfiniteoutside).Then,thedynamics(1)ismetastablebe-
causeittakesalotoftimetofindthesmallcorridortogofromoneboxtotheother.
Metastabilityishereduetoentropicbarriers.Thiscanbequantifiedusingtheno-
tion of free energy, see [36, 29]. In practice, metastability thus originates from a
combinationofbothenergeticandentropiceffects,therelativeimportanceofeach
dependingonthesystemunderconsideration,andonthetemperature.SeeFigure1
foranumericalillustration.Generallyspeaking,metastabilityisrelatedtothemul-
timodalityofthemeasureµ,namelythefactthatsomehighprobabilityregionsare
separatedbylowprobabilityregions.
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oordinate x coordinat0
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0.0 5000 10000 15000 20000
x coordinate Time
Fig.1 Above:(left)anexampleofa2dimensionalpotential,withenergeticbarriers(thereare
actually two possible saddle points to go from left to right), with (right) the x-component as a
functionoftimeoftheassociatedstochasticprocesssolutionto(1).Below(left)anexampleofa
2dimensionalpotential(whichiszeroinsidetheclosedsolid-lineshape,andinfiniteoutside)for
whichanentropicbarrier(right)isobserved.
Twomathematicaltoolstoanalyzemetastablestochasticprocesses 3
An important concept related to metastability is ergodicity. Under adequate as-
sumptions onV, the dynamics (1) can be shown to be ergodic with respect to the
canonicalmeasure:
µ(dx)=Z−1exp(−βV(x))dx, (2)
(cid:82)
whereZ= exp(−βV(x))dxisassumedtobefinite.Ergodicityactuallyrefersto
Rn
twodifferentproperties:(i)anaveragealongatrajectoryconvergesinthelong-time
limittowardsanaveragewithrespecttoµ:foranytestfunctionsϕ :Rn→R,
1 (cid:90) T (cid:90)
lim ϕ(X)dt= ϕdµ, (3)
t
T→∞T 0 Rn
and(ii)thelawoftheprocessX attimet convergestoµ inthelong-timelimit:for
t
anytestfunctionsϕ :Rn→R,
(cid:90)
lim E(ϕ(X ))= ϕdµ. (4)
T
T→∞ Rn
IfX ismetastable,bothlimits(3)and(4)aretypicallyverydifficulttoreach,sinceT
t
shouldbesufficientlylargetovisitallthemetastablestates.Fromanumericalview-
point,metastabilityraisesthussamplingissues,bothtocomputecanonicalaverages
(namelyaverageswithrespectto µ)andtocomputeaveragesoverpathswhichre-
quirestogenerateefficientlymetastabledynamics,thelatterbeingofcourseamore
complicatedtaskthantheformer.
Inthefollowing,wewouldliketointroducetwomathematicaltoolstomeasure
the“degreeofmetastability”ofthedynamics(1).InSection2,wediscussthenotion
ofLogarithmicSobolevinequality,whichisawaytoquantifytheergodicfeaturesof
aprocess,andmoreprecisely,howfasttheconvergence(4)happens.Inthiscontext,
the slower the convergence, the more metastable the process is. In Section 3, we
introduce the notion of quasi-stationary distribution, and identify the typical time
it takes, for a given region of the state space, to reach a quasi equilibrium in this
regionbeforeleavingit.Metastabilityinthiscaseisrelatedtothefactthatthistime
issmallcomparedtothetypicaltimeittakestoleavetheregion.Inbothcases,we
willexplainhowthesetoolscanbeusedto(i)analyzenumericalmethodswhichare
usedinmoleculardynamicsto“accelerate”metastabledynamics(seeSections2.3
and3.2)and(ii)obtaincoarse-graineddescriptionsofthemetastabledynamics(see
Sections2.4and3.3).
We would like to emphasize the importance of quantifying metastability for
practical aspects. Indeed, there exists in the literature many asymptotic analysis
in some limiting regimes (zero temperature limit, time-scale separation limit en-
forcedthroughanexplicitsmallparameterintroducedinthedynamics,etc...)where
it is shown how a metastable dynamics converges to some effective Markovian
dynamics. In practice, the parameters which are considered to go to zero in the
asymptoticregimesmayindeedbesmall,butarecertainlynotzero.Anaturalques-
tionistoquantifytheerrorintroducedbyassumingthattheseparametersarezero,
an assumption which is behind many numerical methods, and coarse-graining ap-
4 TonyLelie`vre
proaches. This requires in turn quantifying the metastable features of the original
dynamics.Thisispreciselytheaimofbothapproachespresentedbelow.
Let us finally clearly state that the aim of this paper is not to provide proofs of
the announced results and sometimes even not to state them very precisely math-
ematically, but to gather in a new and hopefully enlightening way various recent
studies,inparticular[35,29,28].Allthestatementsbelowcanbereformulatedas
precise mathematical claims, with rigorous proofs, except the discussion in Sec-
tion3.3whichismoreprospective.
Remark1.There are other techniques to quantify metastability, that we do not re-
view here. We would like to mention in particular spectral approaches [24, 44],
potentialtheoreticapproaches[3,4]andapproachesbasedondriftconditions[26].
Drawingconnectionsbetweenthesevarioustechniquesisaninterestingsubject,see
forexample[7]forconnectionsbetweenapproachesbasedondriftconditions,and
functionalinequalitiessuchasLSI.
Remark2.Inthispaper,weconcentrateontheoverdampedLangevindynamics(1)
eventhoughthisisnotthemostwidelyuseddynamicsinmoleculardynamics.All
thealgorithmswepresentbelowgeneralizeto(andareusedwith)thephase-space
Langevindynamics,whichismuchmorepopular.However,generalizingthemath-
ematicalapproachesoutlinedbelowtotheLangevindynamicsisnotaneasytask,
due to the lack of ellipticity of the associated infinitesimal generator, see [23, 47]
forexamplesofstudiesinthatdirection.
2 LogarithmicSobolevinequality
Asexplainedabove,wequantifyinthissectionthemetastabilityof(1)byconsid-
ering the rate of convergence of the limit (4). We thus consider the law at time t
of X, which has a density ψ(t,x). The probability density function ψ satisfies the
t
Fokker-Planckequation:
∂ψ =div(∇Vψ+β−1∇ψ). (5)
t
Notice that the density of µ with respect to the Lebesgue measure, denoted by
ψ (x)=Z−1exp(−βV(x)),isobviouslyastationarysolutionto(5).
∞
2.1 Definition
LetusintroducethenotionoflogarithmicSobolevinequality(seeforexample[1]).
Definition1.Theprobabilitymeasureµ issaidtosatisfyalogarithmicSobolevin-
equalitywithconstantR(inshortLSI(R))ifandonlyif,foranyprobabilitymeasure
Twomathematicaltoolstoanalyzemetastablestochasticprocesses 5
ν suchthatν isabsolutelycontinuouswithrespecttoµ,
1
H(ν|µ)≤ I(ν|µ)
2R
(cid:16) (cid:17)
whereH(ν|µ)=(cid:82) ln dν dν istherelativeentropyofν withrespecttoµ,and
Rn dµ
(cid:12) (cid:16) (cid:17)(cid:12)2
I(ν|µ)=(cid:82) (cid:12)∇ln dν (cid:12) dν istheFisherinformationofν withrespecttoµ.
Rn(cid:12) dµ (cid:12)
When both µ and ν admit densities (respectively ψ and φ) with respect to the
Lebesgue measure, we shall also use the notation H(ψ|φ) for H(µ|ν). A crucial
propertyisthefollowing:
Proposition1.Themeasureµ definedby(2)satisfiesaLSI(R)ifandonlyif,forall
probabilitydensityfunctionsψ ,foralltimet≥0,
0
H(ψ(t,·)|ψ )≤H(ψ |ψ )exp(−2β−1Rt) (6)
∞ 0 ∞
whereψ isthesolutionto(5)withinitialconditionψ(0,·)=ψ .
0
Thisisasimpleconsequenceofthestandardcomputation:ifψ satisfies(5),then
d
H(ψ(t,·)|ψ )=−β−1I(ψ(t,·)|ψ ).
dt ∞ ∞
AnaturalwaytoquantifymetastabilityifthustorelateittoR:
ThesmallerR,themoremetastablethedynamics(1)is. (7)
Before we proceed, let us make two remarks. First, by combining the classical
Bakry-Emery criteria with the perturbation result of Holley and Stroock, the mea-
sure µ actuallysatisfiesaLSIunderverymildassumptionsonV:basically,ifV is
smooth,andisα-convexatinfinity,thenaLSIfor µ holds.Whatismorecompli-
catedistogettheoptimalconstantR.Second,inthesimplecaseofadoublewell
potential in dimension 1, it is standard to show that the average time spent by the
processX inawellbeforehoppingtoanotheroneincreasesasthetemperaturegoes
t
tozero,usingforexamplelargedeviationtechniques.Itcanalsobecheckedinthis
simplesettingthattheconstantRscaleslikeexp(−βH)inthelimitoflargeβ (small
temperature),whereHistheheightofthebarriertoovercometoleaveagivenwell.
ThisprototypicalsituationthusshowsthattheLSIconstantindeedallowstoquan-
tifyingtheintuitivedefinitionofmetastabilitywegaveintheintroduction.
2.2 Metastabilityalongareactioncoordinate
Manyalgorithmsandmodellingdiscussionsarebasedontheintroductionofaso-
calledreactioncoordinateξ,namelyasmoothlow-dimensionalfunctionwhichtyp-
icallyindicestransitionfromonemetastablestatetoanother.Forsimplicity,letus
6 TonyLelie`vre
assume that ξ :Rn →T has values in the one-dimensional torus (think of an an-
gleinamolecule,whichcharacterizesitsconformation).Then,onemayintroduce
probabilitymeasuresassociatedtoµ andξ:
• the image ξ∗µ of the measure µ by ξ, which is a probability measure on the
torus T, and which is also written as ξ∗µ(dz)=exp(−βF(z))dz, F :T→R
being then the so-called free energy associated to µ and ξ. Using the co-area
formula,aformulaforF isgivenby
(cid:90)
F(z)=−β−1ln Z−1exp(−βV(x))δ (dx) (8)
ξ(x)−z
Σ(z)
whereΣ(z)={x∈Rn,ξ(x)=z}andδ (dx)isameasuresupportedbyΣ(z)
ξ(x)−z
suchthatdx=δ (dx)dz.
ξ(x)−z
• thefamilyofconditionalprobabilitymeasuresµ(·|ξ(x)=z)withsupportΣ(z),
whichareindexedbyz∈Tanddefinedas
exp(−βV(x))δ (dx)
ξ(x)−z
µ(dx|ξ(x)=z)= .
exp(−βF(z))
These two measures are completely defined through the conditioning formula: for
anytestfunctions:ϕ :Rn→Randψ :T→R,
(cid:90) (cid:90) (cid:90)
ϕ(ξ(x))ψ(x)µ(dx)= ϕ(z) ψ(x)µ(dx|ξ(x)=z)exp(−βF(z))dz.
Rn T Σ(z)
Forproofs,wereferforexampleto[36].
Letusassumethatthemeasuresµ(·|ξ(x)=z)satisfyaLSI(ρ)(withρ uniform
inz),andthatthemeasureξ∗µ=Z−1exp(−βF(z))dzsatifiesaLSI(r).Inthespirit
oftheprevioussubsection,wewillsaythat(thiswillbeassumption(H1)below):
“themetastabilityoftheprocessX isalongξ”ifandonlyifρ (cid:29)r. (9)
t
AtypicalexampleofsuchasituationisgivenonFigure1,forwhichξ(x,y)=x.In
suchatwo-dimensionalsetting,noticethatF(x)=−β−1ln(cid:82) Z−1exp(−βV(x,y))dy
R
exp(−βV(x,y))dy
andµ(dy|ξ(x,y)=x)= .
exp(−βF(x))
ItispossibletorelatetheLSIconstantofthemeasure µ (namelyR)totheLSI
constantsofthemeasuresµ(·|ξ(x)=z)(namelyρ)andξ∗µ (namelyr),see[32].
Roughlyspeaking,
Ifrissmallandρ issmall,thenRissmall. (10)
Moreover,ifRissmallandξ iswellchosen,thenrissmallbutρmaybeverylarge.
Insuchacase,themetastabilityoftheprocessX isessentiallyencodedinthelow-
t
dimensional observable ξ(X). It is then possible to use numerical and analytical
t
techniques to accelerate the long-time convergence and go around the difficulties
associated to metastability, as explained in the two next subsections. This could
Twomathematicaltoolstoanalyzemetastablestochasticprocesses 7
alsobeusedtoyieldadefinitionofwhatagoodreactioncoordinateis:itisa(low-
dimensional)functionξ suchthatρ/risaslargeaspossible.Designinganumerical
method which would look for the best ξ in this respect would be very interesting
forpracticalapplications.
Beforeweproceed,weprovideaformulathatwillbeusefulbelow.Byusingthe
co-area formula, starting from (8), it is possible to check that the derivative of F
writes:
(cid:90) ∇V·∇ξ (cid:18) ∇ξ (cid:19)
F(cid:48)(z)= f(x)µ(dx|ξ(x)=z)with f = −β−1div . (11)
|∇ξ|2 |∇ξ|2
Inthesimplecaseξ(x,y)=xmentionedabove,thefunction f issimply f =∂ V.
x
2.3 Afirstexample:theadaptivebiasingforcetechnique
Asexplainedabove,onedifficultyrelatedtometastabilityisthattheconvergence(4)
isveryslow.Inparticular,thisimpliesthatitwillbedifficulttosamplethecanonical
measure µ fromatrajectoryX.Thefactthatitisdifficulttosampleamultimodal
t
measureisawellknownproblemsharedwithotherfieldsthanmoleculardynamics.
Instatisticsforexample,MarkovChainMonteCarlomethodsareverypopularand
similarsamplingproblemsoccurforBayesianinference[10].
Underassumption(9),anaturalimportancesamplingideaisthefollowing.Ifthe
metastabilityofX isalongξ,itissensibletotrytoremovethesemetastablefeatures
t
bychangingthepotentialV toV−F◦ξ,whereF isthefreeenergy(8)(andF◦ξ
denotesthecompositionofF withξ).Indeed,ifwedenote
µ (dx)=Z−1exp(−β(V−F◦ξ)(x))dx
F F
the associated tilted measure, we clearly have ξ∗µF =1T(x)dx and µF(·|ξ(x)=
z)= µ(·|ξ(x)=z): the conditional measures remain the same, but the marginal
along ξ is now a uniform measure, namely a very gentle measure, without any
multimodality, and thus easy to sample. In other words, following (10), we may
hope that the LSI constant of the tilted measure µ is much smaller than the one
F
of the original measure µ. As a numerical illustration of this fact, we present on
Figure 2 the equivalent of the trajectories presented on Figure 1, using the biased
potentialV−F◦ξ.Weclearlyobservemuchmoretransitionsfromlefttorightwith
thebiasedpotential,bothinthecaseofanenergeticbarrierandanentropicbarrier.
ThedifficultyisofcoursethatcomputingthefreeenergyF isachallengeinit-
self(itcorrespondstoasamplingproblemof,apriori,asimilarcomplexityasthe
sampling of µ). The free energy is actually a quantity of great interest for practi-
tionersinitself[9],andmanymethodshavebeendesignedtocomputeit[36].Thus,
biasing the measure using F does not seem to be such a good idea. The principle
ofadaptivebiasingtechniqueistoactuallyuse,atagiventimet,anapproximation
F ofthefreeenergyF (inviewoftheconfigurationsvisitedsofar)inordertobias
t
8 TonyLelie`vre
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0.
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Fig.2 SeeFigure1forcomparison.Thereactioncoordinateisξ(x,y)=x.Above:Intheenergetic
barriercase,(left)the2dimensionalpotentialminusthefreeenergyand(right)thex-component
of the associated stochastic process. Below: In the entropic barrier case, (left) the free energy
and(right)thex-componentofthestochasticprocesssimulatedagainwiththefreeenergybiased
potential.
thedynamics.Inotherwords,insteadofusingthebiaseddynamics(whichassumes
thatF isalreadyknown)
(cid:40) (cid:112)
dX =−∇(V−F◦ξ)(X)dt+ 2β−1dW,
t t t
(12)
F(cid:48)(z)=E (f(X)|ξ(X)=z),
µ
wheretheformulaforF(cid:48)(z)isexactly(11)(hereandinthefollowing,E denoting
µ
anexpectationtakenwithrespecttothemeasureµ),oneratherconsiders
(cid:40) (cid:112)
dX =−∇(V−F ◦ξ)(X)dt+ 2β−1dW,
t t t t
(13)
F(cid:48)(z)=E(f(X)|ξ(X)=z).
t t t
ThebottomlineisthatifX solutionto(13)wasatequilibriuminstantaneouslywith
t
respect to µ , then F(cid:48) would be exactly F(cid:48). Of course, this instantaneous equilib-
Ft t
riumassumptioniswrong,butthehopeisthatF(cid:48) learnsonthefly(andeventually
t
convergesto)F(cid:48).
Thedynamics(13)isthesocalledadaptivebiasingforce(ABF)process,andis
oneofthemostefficientmethodstocomputefreeenergydifferences,see[13,25]
for the original idea. One way to understand such a method is that already visited
states are penalized (the potential is flattened in these regions, or equivalently, the
Twomathematicaltoolstoanalyzemetastablestochasticprocesses 9
probabilitytovisitthemisincreased)inordertoforcethestochasticprocesstovisit
newregions.Therearemanyothertechniquesalongtheselines,see[34].Thehope
isthat,byforcingthesystemtovisitallthepossiblevaluesofξ,themetastability
oftheoriginaldynamicsiscompletelyovercome.Thisissomewhatinthespiritof
simulated annealing or parallel tempering, where the temperature (which in some
senseplaystheroleofξ)ofthesystemischangedinordertovisitnewregions.
Whatcanbeshownisthatundertheassumptions:
• (H1)themetastabilityoftheprocessX isalongξ (see(9)),
t
• (H2) the cross derivative ∇ f is bounded (where ∇ denotes the gradient
Σ(z) Σ(z)
projectedontothetangentspacetoΣ(z)),
then the convergence of the ABF process (12) to equilibrium is much faster (ba-
sically exponential with rate β−1ρ) compared to the convergence of the original
process(1)toequilibrium(whichisexponentialwithrateβ−1R,see(6)).Inpartic-
ular,F convergestothefreeenergyF veryfastifξ issuchthatρ islarge(thisis
t
assumption(H1)).Wereferto[35]foraprecisemathematicalstatement.Theproof
isbasedonentropytechniques[2],andtheideaoftwo-scaleanalysisforlogarithmic
Sobolevinequalities[22,32].
Wealsoreferto[33]forsomerefinementsinthecaseswhenρ isonlylargefor
someofthefamilyofconditionalprobabilitymeasures µ(·|ξ(x)=z)indexedbyz
(theso-calledbi-channelsituation).Forpracticalaspects(discretizationtechniques
andnumericalillustrations),wereferto[34,8,27]andto[10]foranapplicationof
suchtechniquesinthecontextofBayesianinferenceinstatistics.
Remark3.Inthelong-timelimit,F isobtained,andthemeasuresampledby(13)is
not µ but µ .Therearebasicallytwowaystorecoveraverageswithrespectto µ.
F
First,asinstandardimportancesamplingapproaches,reweightingcanbeused:
E (ϕ(X)exp(−βF◦ξ(X))
E (ϕ(X))= µF .
µ E (exp(−βF◦ξ(X))
µF
For this idea to be efficient, the weights should not be too widespread (otherwise
thevariancemaybelarge),whichmeansthatsupF−infF shouldnotbetoolarge,
see [10] for a discussion of these aspects. Another idea is to use a conditioning
approach:
(cid:90)
E (ϕ(X)|ξ(X)=z)exp(−βF(z))dz
µ
Eµ(ϕ(X))= T (cid:90) .
exp(−βF(z))dz
T
The conditional probabilities E (ϕ(X)|ξ(X)=z) can then be computed using ei-
µ
ther the fact that E (ϕ(X)|ξ(X) = z) = E (ϕ(X)|ξ(X) = z) (so that the ABF
µ µF
process (13) can be used to compute them) or using dedicated techniques to sam-
pletheconditionalprobabilitymeasure µ(·|ξ(x)=z),whichshouldbeeasyunder
assumption (H1). Indeed, if ρ is large, in virtue of Proposition 1, the overdamped
Langevin dynamics associated tothe measure µ(·|ξ(x)=z)should converge very
10 TonyLelie`vre
fasttoequilibrium.Suchadynamicsisroughlyspeakingaprojectionoftheoriginal
gradient dynamics (1) to the submanifold Σ(z). We refer to [11, 36, 37] for more
informationonsuchconstrainedsamplingtechniques.
2.4 Asecondexample:obtaininganeffectivedynamicsonξ(X )
t
Ifweareinthesituation(9)wherethemetastabilityofX isalongξ,anotherideais
t
totrytoderiveaneffectivedynamicsforξ(X),whichwouldthenbeeasytosimu-
t
latesinceitislow-dimensional,andhopefullyassociatedwithasmallercharacter-
istic timescale than the original dynamics (1). The idea is that if the metastability
is along ξ, then ξ(X) should move much more slowly than the components of X
t t
whicharealong“directionsorthogonaltoξ”,sothatsomeaveragingshouldbepos-
siblealongthosedirections.Thisisverymuchinthespiritofprojectionoperatoror
Mori-Zwanzig techniques [20]. Below, we first derive an effective Markovian dy-
namicsforξ(X)andthenassessthequalityofthiseffectivedynamicsbyderiving
t
quantitativebounds.Thesequantitativeboundsareagainobtainedusinglogarithmic
Sobolevinequalitiesandentropycomputations.
Theideatoderiveaneffectivedynamicsonξ(X)startsfromasimpleItoˆ calcu-
t
lus.If(X) satisfies(1),then
t t≥0
dξ(X)=(−∇V·∇ξ+β−1∆ξ)(X)dt+(cid:112)2β−1|∇ξ(X)| ∇ξ(Xt) ·dW.
t t t t
|∇ξ(X)|
t
Ofcourse,thisisnotaclosedequationfortheevolutionofξ(X).Itisnotdifficult
t
tocheckthatifwetheconsider
(cid:112)
dz˜ =b˜(t,z˜)dt+ 2β−1σ˜(t,z˜)dB
t t t t
with
(cid:16) (cid:12) (cid:17)
b˜(t,z˜)=E (−∇V·∇ξ+β−1∆ξ)(X)(cid:12)ξ(X)=z˜
t (cid:12) t
and
(cid:16) (cid:12) (cid:17)
σ˜2(t,z˜)=E |∇ξ|2(X)(cid:12)ξ(X)=z˜ ,
t (cid:12) t
then, for all time t ≥0, the law of the random variable ξ(X) is equal to the law
t
of the random variable z˜. The difficulty of course is that b˜ and σ˜ are intractable
t
numerically,sincetheyarefunctionsdependingontime.Anaturalidea(following
theintuitiongivenatthebeginningofthissection)isthatonecouldreplacethecon-
ditionalexpectationsdefiningb˜ andσ˜2 byconditionalexpectationsatequilibrium,
namely:
(cid:112)
dz =b(z)dt+ 2β−1σ(z)dB (14)
t t t t
with
(cid:16) (cid:12) (cid:17)
b(z)=E (−∇V·∇ξ+β−1∆ξ)(X)(cid:12)ξ(X)=z
µ (cid:12)