Existence, uniqueness and convergence of a particle approximation for the Adaptive Biasing Force process

0.61 MB·English

by  Benjamin Jourdain

Existence, uniqueness and convergence of a particle
approximation for the Adaptive Biasing Force process
0
1
0
2 Benjamin Jourdain1, Tony Leli`evre2, Rapha¨el Roux2

n
October 2, 2009
a
J

6
1 1
Universit´e Paris-Est, CERMICS, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vall´ee Cedex 2,

France
]
R 2 Universit´e Paris-Est, CERMICS, Project-Team MICMAC ENPC-INRIA, 6 et 8 avenue Blaise
P Pascal, 77455 Marne-La-Vall´eeCedex 2, France
.
h
Abstract
t
a
Westudyafreeenergycomputationprocedure,introducedin[5,8],whichreliesonthelong-time
m
behaviorof anonlinear stochastic differentialequation. This nonlinearity comesfrom aconditional
[
expectation computed with respect to one coordinate of the solution. The long-time convergence

2 of the solutions to this equation has been proved in [11], under some existence and regularity
v assumptions.
8 In this paper, we prove existence and uniqueness under suitable conditions for the nonlinear
1 equation,andwestudyaparticleapproximation techniquebased on aNadaraya-Watsonestimator
5 of the conditional expectation. The particle system converges to the solution of the nonlinear
4 equationifthenumberofparticlesgoestoinfinityandthenthekernelusedintheNadaraya-Watson
.
3 approximation tendsto a Dirac mass.
0 Wederivea rate for this convergence, and illustrate it bynumerical examples on a toy model.
9
0
:
v
i
X
r
a
This work was supported by the french National Research Agency (ANR) under the programs ANR-08-BLAN-0218-03
BigMCandANR-09-BLANMEGAS.
1
Introduction
Free energy computations are an important problem in the field of molecular simulation (see [4]).
Thedifficultyofthosecomputationsliesinthefactthatmostdynamicsinmolecularsimulationsare
highlymetastable: manyfreeenergybarrierspreventagood sampling. Westudyheretheadaptive
biasing force (ABF) method, which was introduced in [5, 8] to get rid of those metastabilities.
Thetypicalproblemsonecanthinkaboutarethestudyofastructuralangleintheconformation
of a protein, or the measure of the evolution of a chemical reaction. Mathematically, each configu-
ration of the system is modelized by an element of a high-dimensional state space , typically an
D
opensubsetofRd,whichisendowedwithaprobabilitymeasure,calledthecanonicalmeasure. This
measure is given by ( e−βV(x)dx)−1e−βV(x)dx, where V denotes the potential energy undergone
D
bythe physicalsystem, and β is proportional tothe inverseof thetemperature of thesystem.
R
Forsomexinthestatespace,oneisinterestedinaparticularquantity,denotedbyξ(x),ξ being
assumed to be a smooth function from to the one-dimensional torus T. The quantity ξ(x) has
D
to be understood as a coarse-grained information on the system, which is the relevant information
for the practitioner. In the examples above, ξ(x) would be a structural angle in a protein with
conformation x,or a numbermeasuring theevolution of a chemical system in state x.
We call free energy the effective energy associated to the quantity ξ(x), that is, the function A(z)
suchthate−βA(z)dz istheimagemeasureofthecanonicalmeasurebythefunctionξ. Ourobjective
is to compute numerically the function A. When =Rd, a naive method to do so is to simulate,
for a given random variable X and an independeDnt Rd valued Brownian motion W, the process
0
−
definedby the(overdamped) Langevin dynamics
dX = V(X )dt+ 2β−1dW , (0.1)
t t t
−∇
p
which, under some regularity assumptions on the potential, is ergodic and admits the canonical
measure as unique invariant measure. This approach appears to be untractable in practice, since
the convergence to equilibrium is very slow, due to multiple metastabilities appearing in most
problems: typically, a molecule moves microscopically within times of order 10−15 seconds, while
thetypical time scale of themacroscopic movesis of order 10−9 seconds.
The idea of the ABF method is to prevent the process X from staying in metastable states by
t
introducing a biasing force which repel X from the states where it stayed for too long a time. To
t
do this, we use the following representation of A, that can be deduced from the co-area formula
(see [11]):
A′(z)=E[F(X)ξ(X)=z], (0.2)
|
whereX isarandomvariabledistributedaccordingtothecanonicalmeasure,andF isthefunction
definedby
ξ V 1 ξ
F(x)= ∇ ·∇ div ∇ . (0.3)
ξ2 − β ξ2
|∇ | (cid:18)|∇ | (cid:19)
ThefunctionA′ iscalledthemeanforce. Actually,(0.2)alsoholdswhenX isdistributedaccording
tothe measure
−1
e−β(V(x)+W◦ξ(x))dx e−β(V(x)+W◦ξ(x))dx,
(cid:18)ZD (cid:19)
which is the canonical measure associated with the biased potential V +W ξ where W is any
◦
smooth function.
Equation(0.2)leadsustoconsiderthefollowingdynamics,whichshouldgetridofmetastabilities
forawellchosenξsinceit“flattens”theenergylandscapeintheξdirection(see[11]andLemma1.2
2
below for more precise statements):
dX = V A ξ β−1ln( ξ−2) (X ) ξ−2(X )dt+ 2β−1 ξ−1(X )dW ,
t −∇ − t◦ − |∇ | t |∇ | t |∇ | t t (0.4)
(A′t(z) =E[F(cid:0)(Xt)|ξ(Xt)=z]. (cid:1) p
The second equality in (0.4) shows that if X is distributed according to the canonical measure
t
associated with the potential V A ξ, then the biasing force A′ is actually the derivative A′ of
− ◦ t
the free energy, and the first equation in (0.4) consists in a Langevin dynamics associated to the
potential V A ξ. Consequently, the dynamics (0.4) admits a stationary point: A′ = A′ and
Law(X )=(− e−◦V+A◦ξdx)−1e−V+A◦ξdx. t
t D
If we actually have convergence to this stationary state, we have a method, that should be
R
efficient (i.e. that should not see the metastabilities), to sample the canonical measure up to a
known perturbation eA◦ξ. This algorithm has thus two applications: it allows the computation of
thefreeenergyA,anditcanbeusedasanadaptativeimportancesamplingmethodforthecanonical
measure.
The long time behavior of Equation (0.4) has been studied in [11], where it has been proven
that for a sufficiently regular solution, one has, in some sense, an exponential convergence to the
stationary state, with a rate that is better (for a well chosen ξ) than the rate of convergence to
equilibrium for (0.1).
The practical difficulty in simulating (0.4) is to compute the conditional expectation, which is
a highly nonlinear term. Stochastic differential equations involving conditional expectations have
already been studied, in a case where the conditional expectation is computed with respect to
a random initial condition (see [16, 18]) or where the variable whose conditional expectation is
computed is fixed (see [7]). Our situation is much more complex since both the conditioning and
theconditionedvariableschangewithtimeandareaffectedbythepreviousconditionalexpectations.
The same difficulty arises in Lagrangian stochastic models which are commonly used in the
simulation of turbulent flows (see [2]). The main difference between the system studied in [2]
and(0.4)isthattheauthorsconsidersaLangevindynamicswithnoiseonlyonthevelocity. Thelack
of ellipticity then leads toadditional difficulties. Inoursetting we are ableto deriveaquantitative
error estimate for theparticle discretization while thisseems more difficult for Langevin dynamics.
In this paper, we prove that existence and uniqueness hold for Equation (0.4) under suitable
conditions,andwestudyanapproximationofX byaninteractingparticlesystem(seeTheorems1.3
t
and 1.4 below).
The paperis organized as follows. In Section 1 we state ourmain results.
Section 2 is devoted to some uniqueness and regularity results. More precisely, we prove that
thetime marginals of a solution to Equation (0.4) satisfy some partial differential equation. Then,
under an integrability condition on the initial condition, we prove uniqueness for the solutions to
thisequation,sothatthenonlinearterm in(0.4)isreducedtoaboundeddriftcoefficient. Wethus
provepathwise uniquenessand uniquenessin distribution for thesolutions of (0.4).
Section 3 is devoted to existence results. More precisely, we introduce a regularization of the
dynamics(0.4) involvingtwoparameters αand ε,which isanothernonlinear stochastic differential
equation whose nonlinearity is less singular. We prove that strong existence, pathwise uniqueness
and uniqueness in distribution hold for this equation and then we show that the solutions to this
stochastic differential equation converge to some process which satisfies (0.4) in the limit (α,ε)
→
(0,0), yielding strong existence. Wealso prove that this convergenceholds with rate (α+√ε).
O
InSection 4weintroduceaninteractingparticlesystem toapproximatetheregularized dynam-
ics, and we prove a propagation-of-chaos result for this particle system. We also derive a rate of
convergencefor this propagation of chaos.
In Section 5, we illustrate the efficiency of the particle approximation of the ABF method and
therate of those convergences with some numerical examples in small dimension.
3
Acknowledgement: WewarmlythankMathiasRoussetforhiscarefulreadingofanearlyversion
of themanuscript.
Notation
We denote by T = R/Z the one dimensional torus, and for x R, we denote by x T its
projection on T. In the following, we will work in two different d∈omains : T Rd−1{or}T∈d. The
D ×
case = T Rd−1 will be called the non compact case, and the case = Td will be called the
D × D
compactcase. Forx Rd,dependingonthecaseconsidered,wewillalsodenoteby x theelement
of T Rd−1 (resp. Td∈) defined by x =( x1 ,x2,...,xd) (resp. x =( x1 ,..., {xd})).
× { } { } { } { } { }
In the following, we will call “function defined on T” (resp. on T Rd−1, resp. on Td), a
×
Z periodical(resp. Z periodicalinthefirstcoordinate, resp. Zd-periodical)functiondefinedonR
(r−esp on Rd). Integral−s on T, T Rd−1 or Td mean integrals on [0,1), [0,1) Rd−1 or [0,1)d.
× ×
We denote by L2(Td) the space of functions on Td whose square is integrable on Td, and by
H1(Td) the space of functions in L2(Td) whose weak gradient is square integrable on Td. We use
similar notations on T Rd−1 and T.
×
Fortwofunctionsf andgdefinedonT Rd−1orTd,wedenotef gtheconvolutionwithrespect
× ∗
to the first coordinate, that is,
f g(x)= f(x1 y1,x2...d)g(y1,x2...d)dy1.
∗ −
ZT
Iff is definedon T, we also usethe notation f g to denote
∗
f g(x)= f(x1 y1)g(y1,x2...d)dy1.
∗ −
ZT
Whenf andg aredefinedon =T Rd−1 orTd,theconvolutioninallthecoordinatesisdenoted
D ×
f⋆g:
f⋆g(x)= f(x1 y1,x2...d y2...d)g(y1,y2...d)dy1dy2...d.
− −
ZD
In the following, we call “probability measure on T” (resp. on T Rd−1, Td) a nonnegative
×
Z-periodical (resp. Z-periodical with respect to thefirst coordinate, Zd-periodical) measure µ such
that µ([0,1))=1 (resp µ([0,1) Rd−1)=1, µ([0,1)d)=1).
When X isarandomvaria×bletakingvaluesinT(resp. inT Rd−1,Td),wecall“distribution
{ } ×
of X ” or “law of X ” theprobability measure µ on T (resp. on T Rd−1, Td) such that
{ } { } ×
E[f( X )]= f(x)µ(dx).
{ }
Z
For a given probability measure µ on T Rd−1 (resp. a probability density u) and a given
×
bounded function g, we denote µg (resp. ug(x1)dx1) the marginal on T of the measure g.µ (resp.
g(x)u(x)dx). Namely:
µg(A)= gdµ
ZA×Rd−1
and
ug(x1)= g(x1,x2...d)u(x1,x2...d)dx2...d.
ZRd−1
In particular, µ1 is the first coordinate marginal of µ. When we do not specify the measure in an
integral, it is theLebesgue measure.
4
Wewill need the weighted spaces
1/p
Lp(w)=(ψ∈Lp(T×Rd−1) s.t. kψkLp(w) d=ef(cid:18)ZT×Rd−1|ψ|pw(cid:19) <∞),
for 1 p< , and
≤ ∞
1/2
H1(w)= ψ H1(T Rd−1) s.t. ψ d=ef ψ2+ ψ2 w <
( ∈ × k kH1(w) (cid:18)ZT×Rd−1 | | |∇ | (cid:19) ∞)
(cid:0) (cid:1)
with w(x) = (1+ x2...d 2)λ, for some λ > (d 1)/2. Notice that w does not depend on the first
coordinate x1, and|that|there is a positive cons−tant K such that
d
x T Rd−1, w(x) 2λ(1+ x2...d 2)λ−1 xi Kw(x). (0.5)
∀ ∈ × |∇ |≤ | | | |≤
i=2
X
Wewill use several times the following statement:
Lemma 0.1. For a bounded function g, and u L2(w) one has, for some constant K,
∈
ug K g u .
k kL2(T) ≤ k kL∞(T×Rd−1)k kL2(w)
If moreover, g has bounded derivatives and u H1(w), then
∈
ug K g u .
k kH1(T) ≤ k kW1,∞(T×Rd−1)k kH1(w)
The same inequalities hold with the non weighted norms in the right-hand side, for u respectively in
L2(Td) and H1(Td).
Proof. Recallthatweassumedλ d−1,sothat 1 isintegrableonRd: 1dx< . Consequently,
≥ 2 w Rd w ∞
we havetheestimation
R
2
ug 2 = gu
k kL2(T)
ZT(cid:12)ZRd−1 (cid:12)
(cid:12) (cid:12)
≤kgk(cid:12)(cid:12)2L∞(T×Rd−(cid:12)(cid:12)1) |u|2w w1
ZT(cid:18)ZRd−1 ZRd−1 (cid:19)
K g 2 u 2 .
≤ k kL∞(T×Rd−1)k kL2(w)
Theproof is similar in the space H1(w).
Inthefollowing, K willdenotesomepositiveconstant,whosevaluecanchangefromlinetoline.
1 Assumptions and statement of the main results
In this paper, we consider a particular case of Equation (0.4) to simplify the argumentation: we
assumeβ =1(thiscanberealizedbyachangeofvariable), =T Rd−1or =Td. Weconsideras
D × D
reactioncoordinatethefirstcoordinatefunctionξ: Rdefinedbyξ(x)=ξ(x1,x2,...,xd)=x1.
D→
This should not change the theoretical results, but will simplify the proofs. The definition (0.3) of
F is then reduced to
F =∂ V,
1
where V is defined on Td or T Rd−1.
×
5
The two settings = Td and = T Rd−1 will be respectively called the compact and the
D D ×
non-compactcase. Ourresultsholdinbothsettings, andtheproofsaremostlyidentical,withsome
slight additional difficulties in the non compact case. Thus, in those situations, we only give the
proofs in thenon-compact case.
With those assumptions, Equation (0.4) rewrites
dX = V(X )+E ∂ V(X ) X1 e dt+√2dW , (1.1)
t −∇ t 1 t |{ t} 1 t
e1 denoting the first vector (cid:0)in the canonical (cid:2)basis of Rd. We(cid:3)wi(cid:1)ll call solution to Equation (1.1) a
process X whereX satisfies(1.1). Theinitialconditionof (1.1)isarandomvariabledenotedX ,
t t 0
{ }
and is supposed to be independent of the Brownian motion W. Wedenote by P the law of X ,
0 0
{ }
which is a probability measure on .
D
To ensurethe integrability of ∂ V(X ),we makethefollowing assumption :
1 t
Assumptioni. V isatwicecontinuouslydifferentiablefunction,whichhasboundedfirstandsecond
order partial derivatives.
Notice that Assumption i yields boundedness of the drift coefficient in (1.1). In the compact
case, assumption i is satisfied as soon as V is a twice differentiable function.
We have to make some assumptions on the initial condition X . What is needed to prove our
0
results will depend on whether we consider the compact or the non compact case. In the compact
case, we consider thefollowing assumption:
Assumption ii. The probability measure P has a density p lying in L2(Td) and whose first
0 0
coordinate marginal p1 is bounded from below by a positive constant. (Notice that p1 isa probability
0 0
density on T.)
In the non compact case, we will need a stronger assumption: we have to control the decay of
theinitialconditionatinfinity,soweworkintheweightedspaceL2(w). Wewilluse,inadditionto
Assumption ii,the following one:
Assumption iii. The density p of P lies in both L1(w) and L2(w).
0 0
NoticethatAssumptionsiiiimpliesthat X hasfinitemomentsoforderlessthan2λ,andthat
0
{ }
Assumptionithenyieldsacontrolonthecorrespondingmomentsofanysolutionto(1.1),uniformly
in t R :
∈
Lemma 1.1. Under Assumptions i and iii, on any bounded time interval [0,T], the moments of
order less than 2λ of any solution X of (1.1) are bounded:
sup E[X 2λ]< .
t
| | ∞
0≤t≤T
Proof. Thiscomesfrom theboundednessofthedrift coeficient b (x)= V(x)+E[∂ V(X)X1 =
s 1
x1], which holds in regard of Assumption i. Indeed, we have E[X 2λ−] ∇= E[X + tb (X|)ds+
| t| | 0 0 s s
√2W 2λ] K E[X 2λ]+t2λ+tλ , which is boundedon [0,T].
t| ≤ | 0| R
Accordingt(cid:0)othefollowingfunda(cid:1)mentallemma,thesolutionto(1.1)isgoingtosampleefficiently
thecoordinate reaction state space T.
Lemma 1.2. Denote by P the law of X , where X is a solution to Equation (1.1). Then, P1
t { t} t t
has a density p1, such that p1 satisfies the heat equation on T with initial condition p1. Thus, p1 is
t 0
uniquely defined on T [0, ), and smooth on T (0, ).
× ∞ × ∞
Proof of Lemma 1.2. Let f bea smooth function on T. One has, byIt¯o’s formula
∂ E f(X1) = E f′(X1)∂ V(X ) +E f′(X1)E ∂ V(X ) X1 +E f′′(X1) .
t t − t 1 t t 1 t |{ t} t
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:2) (cid:3)(cid:3) (cid:2) (cid:3)
6
But,f beingafunctiononT,f′(X1)onlydependson X1 ,sothatthetwofirsttermsintheright
t { t}
hand side cancel. Then, it holds
∂ E f(X1) =E f′′(X1) ,
t t t
which is exactly theheat equation in th(cid:2)eweak(cid:3)sense(cid:2)for t (cid:3)p1, p1 being thedistribution of X1 .
7→ t t { t}
Foruniquenessand regularity of this solution, see [6, Chapter XIV].
Lemma 1.2 allows us to rewrite equation (1.1) using the distribution of X1 . Indeed,since P1
has a density, the measure given for A [0,1) by P∂1V(A) = E ∂ V(X{)1t(} X1 ) also has ta
⊂ t 1 t A { t}
density p∂1V. Wecan thuswrite
t (cid:2) (cid:3)
dX = V(X )+ p∂t1V(Xt1)e dt+√2dW ,
 t (cid:18)−∇ t p1t(Xt1) 1(cid:19) t (1.2)
Pt =distribution of Xt .
{ }
Moreover,underAssumptioniithedensityp1t satisfies0<infTp10≤p1t,uniformlyintime,thanksto
themaximum principle. This assumption will consequently prevent thedenominator in thesecond
term of (1.2) from vanishing.
In view of Equation (1.2), a natural particle approximation of X is then obtained using the
t
Nadaraya-Watsonestimatorofaconditionalexpectation(see[19]),given,forsomeparameterηand
for a positive integer N, bythe system of N stochastic differential equations
N ϕ (Xη,1 Xη,1 )∂ V(Xη )
dXη = V(Xη )+ m=1 η t,n,N − t,m,N 1 t,m,N e dt+√2dWn, 1 n N
t,n,N −∇ t,n,N P Nm=1ϕη(Xtη,,n1,N −Xtη,,m1,N) 1! t ≤ ≤
(1.3)
P
where (Wn) is a sequence of independent Brownian motions, and ϕ is a smooth approximation
t η
for the Dirac measure at the origin on T. For the initial condition, we work with the following
assumption
Assumption iv. The initial condition of Equation (1.3) is (Xη ) =(X ) , where
0,n,N 0≤n≤N 0,n 0≤n≤N
(X0,n)n∈N is a sequence of i.i.drandom variables with density p0, and independent of the Brownian
motions (Wn) .
t t≥0
We also need an assumption on the shape of ϕ . The parameter η = (α,ε) will be chosen
η
in (0, )2, and ϕ will havetheform
η
∞
ϕ (x)=α+ψ (x), (1.4)
η ε
where ψ is a sequence of mollifiers on T as ε 0. Namely, assuming ε < 1/2, ψ is a smooth
ε ε
→
non-negativeZ-periodical function, such that ψ 0 on [ 1/2,1/2] [ ε,ε] and such that
ε
≡ − \ −
1/2
ψ =1.
ε
Z−1/2
Asimple waytoconstructsuchasequenceistoconsiderasmooth non-negativefunctionψ defined
on R, with support in [ 1,1] such that ψ = 1, and then consider the Z-periodization ψ of
− R ε
ψ = 1ψ(.)(ψ iswelldefinedforε<1/2). Thisexamplemakesthefollowing assumptionnatural:
ε ε ε ε R
Assumption v. The function ψ satisfies
ε
K K
kψεkL∞(T) ≤ ε , and kψε′kL∞(T) ≤ ε2.
7
The reason for adding a positive constant α to the mollifier is to avoid singularities at the
denominatorintheright-handsideof (1.3). Noticethat(1.4)yieldsstrongexistenceanduniqueness
for (1.3), since thedrift is globally Lipschitz continuous.
Weare going to provethefollowing two results:
Theorem 1.3. [Existence and uniqueness of the solution] In both the compact and non compact
cases, under Assumption i, weak existence holds for Equation (1.1). If P denotes the distribution
of a solution, then for all s > 0 the time marginals P of P admits a density p , such that for all
s s
0<t<T,
p L∞((t,T),L2( )) L2((t,T),H1( )). (1.5)
∈ D D
Moreover, underbothAssumptionsiandiiforthec\ompactcase, andunderAssumptionsi,iiandiii
for the non compact case, strong existence, pathwise uniqueness and uniqueness in distribution also
hold, and one can take t=0 in (1.5).
Theorem 1.4. [Particle approximation of the process X ] In the compact case, under Assump-
t
tions i, ii, iv, and v or in the non-compact case under the additional Assumption iii, define the
processes X by (1.3). Then, it holds that, for any positive T, and for α and ε small enough,
t,n,N
EZ0T(cid:13)(cid:13)PNn=1∂1VNn=(1Xϕtη,ηn(,.N−)ϕXη(tη,.,n1−,NX)tη,,n1,N) −A′t(cid:13)(cid:13)L∞(T)dt=O(cid:18)α+√ε+ √1NeαKε2(cid:19).
 (cid:13)(cid:13) P (cid:13)(cid:13) 
Theorem (cid:13)1.3 is a consequence of Theorem 2.8 and(cid:13)Corollary 3.10 below, and Theorem 1.4 is a
consequenceof Theorems 3.11 and 4.1 below.
TheconvergencerateinTheorem1.4isaprettybadone,sinceforagivenN itexplodesasεgoes
to 0. Consequently, the size ε of the window has to be chosen carefully depending on the number
N of particles. This is discussed more precisely in Section 5.
2 Notion of solution, regularity and uniqueness results
Inthissection weconsidertheFokker-Planckequationassociated tothenonlinearstochasticdiffer-
ential equation (1.1) and prove that uniqueness holds for weak solutions of this partial differential
equation. From this uniqueness result, the study of Equation (1.1) can be reduced to the study of
a linear stochastic differential equation. Wecan thusproveuniquenessfor Equation (1.1).
LetusderivetheFokker-Planckequationassociated toEquation(1.1). Letψ beatwicecontin-
uously differentiable function. Applying It¯o’s formula and taking the expectation, we obtain that
thelaw P of a weak solution X to equation (1.1) satisfies
t t
{ }
T T
ψ(x)dP (x)= ψ(x)dP (x) ψ(x) V(x)dP (x)dt+ ∆ψ(x)dP (x)dt
T 0 t t
− ∇ ·∇
ZD ZD Z0 ZD Z0 ZD
(2.1)
T p∂1V
+ ∂ ψ(x) t (x1) dP (x)dt,
1 p1 t
Z0 ZD (cid:18) t (cid:19)
which is a weak formulation of thefollowing partial differential equation
p∂1V
∂ P =div(P V + P ) ∂ P t , (2.2)
t t t∇ ∇ t − 1 t p1
(cid:18) t (cid:19)
withinitialconditionP .Usingintegrationbyparts,weintroduceastrongerdefinitionforsolutions
0
to(2.2) which will allow us toprove existenceand uniqueness.
8
Definition 2.1. In the compact case, a function u is said to be a solution to (2.2) if, for any
positive T,
u belongs to L∞((0,T),L2(Td)) L2((0,T),H1(Td)) ;
•
for any function ψ H1(Td), we have:
T
• ∈
u∂1V
∂ u ψ= u V ψ u ψ+ u t ∂ ψ, (2.3)
t t − t∇ ·∇ − ∇ t·∇ t u1 1
ZD ZD ZD ZD t
in the sense of distributions in time ;
u =p .
0 0
•
In the non compact case, u is said to be a solution to (2.2), if, for any positive T,
u belongs to L∞((0,T),L2(w)) L2((0,T),H1(w)) ;
•
for any ψ H1(w)
T
• ∈
u∂1V
∂ u ψw= u V (w ψ+ψ w) u (w ψ+ψ w)+ u t (∂ ψ)w, (2.4)
t t − t∇ · ∇ ∇ − ∇ t· ∇ ∇ t u1 1
ZD ZD ZD ZD t
holds in the sense of distributions in time ;
u =p .
0 0
•
Notice that (2.3) is a variational formulation of (2.2) in the space L2(Td) and that (2.4) is a
variational formulation of (2.2) in thespace L2(w).
These conditions make sense. Indeed, in both cases, the conditions on u and ψ are such that
the variational formulations (2.3) and (2.4) are well defined (notice that one has w Kw).
|∇ | ≤
Moreover, for the compact case, if u lies in L2((0,T),H1(Td)), and satisfies (2.3) then ∂ u lies in
t
L2((0,T),H−1(Td)), so that (see [13, page 23]) u lies in ([0,T],L2(Td)), allowing us to define the
C
valueof u at time t=0. The same argument holds for the non compact case.
2.1 Existence of regular densities for solutions to the nonlinear
equation
Inthissection,weconsiderasolutionX toEquation(1.1)andwedenotebyP thelawof X . We
t t
{ }
showthatP hasadensityp ,andthatpisasolutiontoEquation(2.2),inthesenseofDefinition2.1.
t t
Lemma 2.2. Consider both the compact and the non compact cases. Under Assumption i, for
any t 0, P admits a density p with respect to the Lebesgue measure satisfying the followingmild
t t
≥
representation
t t p∂1V
p =G ⋆P + G ⋆( Vp )ds ∂ G ⋆ s p ds, (2.5)
t t 0 ∇ t−s ∇ s − 1 t−s p1 s
Z0 Z0 (cid:18) s (cid:19)
where G is the density of √2 times the Brownian motion on , namely
t
D
Gt(x)= (4πt1)d/2 e−|x−4kte1|2
Xk∈Z
for the non-compact case, and
Gt(x)= (4πt1)d/2 e−|x−4tk|2
kX∈Zd
for the compact case.
9
Proof. Let χ be a smooth function with compact support on T Rd−1 and T > 0. Then, for
×
t [0,T], thefunction ψ defined by
∈
ψ =G ⋆χ,
s t−s
is theuniquesmooth solution tothe following problem
∂ ψ = ∆ψ on (0,t) T Rd−1,
s − × × (2.6)
(ψt =χ on T Rd−1.
×
Computing ψ (X ) by It¯o’s formula and using (2.6) we get
s s
t t
ψ dP = ψ dP ∆ψ dP ds+ ∆ψ dP ds
t t 0 0 s s s s
−
ZT×Rd−1 ZT×Rd−1 Z0 ZT×Rd−1 Z0 ZT×Rd−1
t t p∂1V
ψ VdP ds+ ∂ ψ s dP ds
− ∇ s·∇ s 1 s p1 s
Z0 ZT×Rd−1 Z0 ZT×Rd−1 s
t t p∂1V
= ψ dP ψ VdP ds+ ∂ ψ s dP ds.
0 0− ∇ s·∇ s 1 s p1 s
ZT×Rd−1 Z0 ZT×Rd−1 Z0 ZT×Rd−1 s
Usingthe expression of ψ and Fubini’s Theorem, we have:
t
t
χdP = χ(G ⋆P )+ χ G ⋆(P V)ds
t t 0 t−s s
∇ ∇
ZT×Rd−1 ZT×Rd−1 ZT×Rd−1 Z0
t p∂1V
χ ∂ G ⋆ s P ds.
− 1 t−s p1 s
ZT×Rd−1 Z0 (cid:18) s (cid:19)
This last equation being true for any smooth function χ with compact support, then P is given
t
by the right-hand side of (2.5), which is an integrable function, so that for any positive t, P has a
t
density p satisfying (2.5).
t
In regard of the following lemma, p necessarily satisfies some integrability conditions.
Lemma 2.3. In both the compact and the non compact case, under Assumptions i and ii, p lies in
L∞((0,T),L2( )) for any T > 0, and we have p C, where C is some constant
D k kL∞((0,T),L2(D)) ≤
only depending on P , V and T.
0
∇
In the non compact case, under Assumptions i, ii and iii, p lies in L∞((0,T),L2(w)) for any
T > 0, and we have a bound p C, where C is some constant only depending on
k kL∞((0,T),L2(w)) ≤
P , V and T.
0
∇
WeonlygivetheproofofLemma2.3inthenoncompactcase,theoneinthecompactcasebeing
similar.
Proof. The mild formulation (2.5) will allow us to prove that u L∞((0,T),L2(w)). Since p lies
0
inbothL1(w)andL2(w),itliesinLq(w),forany1 q 2.We∈firstprovethatwehaveauniform
≤ ≤
in time estimate in Lq(w),1 q 2, for p .
t
≤ ≤
From equation (2.5), it follows
t p∂1V
kptkLq(w) ≤kp0kLq(w)+ k∇Gt−s⋆(∇Vps)kLq(w)+ ∂1Gt−s⋆ sp1 ps ds. (2.7)
Z0 (cid:13) (cid:18) s (cid:19)(cid:13)Lq(w)
(cid:13) (cid:13)
(cid:13) (cid:13)
It holds, from Jensen’s inequality, (cid:13) (cid:13)
10