From a stochastic Becker-Döring model to the Lifschitz-Slyozov equation with boundary value

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From a stochastic Becker-Döring model to the
Lifschitz-Slyozov equation with boundary value
Romain Yvinec, Julien Deschamps, Erwan Hingant
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Romain Yvinec, Julien Deschamps, Erwan Hingant. From a stochastic Becker-Döring model to the
Lifschitz-Slyozov equation with boundary value. 2014. ￿hal-01123221￿
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From a stochastic Becker-D¨oring model
to the Lifschitz-Slyozov equation with
boundary value∗
Julien Deschamps
DIMA, Universita`degli Studi di Genova, Italy.
e-mail: [email protected]
and
Erwan Hingant
CI2MA, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile.
e-mail: [email protected]
and
Romain Yvinec
BIOS group, INRA, UMR85, Unit Physiologie de la Reproduction et des Comportements,
F-37380 Nouzilly, France.
e-mail: [email protected]
Abstract: Weinvestigatetheconnection betweentwoclassicalmodelsof
nt phase transition phenomena, the (discrete size) stochastic Becker-Do¨ring
ri equationsandthe(continuoussize)deterministicLifshitz-Slyozovequation.
p
e For general coefficients and initial data, we introduce a scaling parameter
r and show that the empirical measure associated to the stochastic Becker-
p
D¨oringsystemconvergesinlawtotheweaksolutionoftheLifshitz-Slyozov
n
o equation when the parameter goes to 0. Contrary to previous studies, we
si useaweaktopologythatincludestheboundaryofthestatespaceallowing
er us torigorouslyderiveaboundary value forthe Lifshitz-Slyozovmodel in
V thecaseofincomingcharacteristics.Itisthemainnoveltyofthisworkand
it answers to a question that has been conjectured or suggested by both
mathematiciansandphysicists.Weemphasizethattheboundaryvaluede-
pendsonaparticularscaling(asopposedtoamodelingchoice)andisthe
result of a separation of time scale and an averaging of fast (fluctuating)
variables.
MSC2010subjectclassifications:60H30;60B12;35F31;35L50;82C26.
Keywords and phrases: Limit theorem, Averaging, Stochastic Becker-
D¨oring, Lifschitz-Slyozov equation, Boundary value, Measure-valued solu-
tion.
∗This work has been supported by ANR-14-CE25-0003 (JD), FONDECYT Grant no.
3130318(EH)andINRA(RY)
1
Comment citer ce document :
Yvinec, R., Deschamps, J., Hingant, E. (2014). From a stochastic Becker-Döring model to the
Lifschitz-Slyozov equation with boundary value. arXiv preprint arXiv:1412.5025 [math.PR], 1-69.
J. Deschamps, et al./Stochastic Becker-D¨oringto Lifschitz-Slyozov 2
Contents
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 The stochastic Becker-Do¨ringprocess . . . . . . . . . . . . . . . . . . 6
3 Scaling law and main results . . . . . . . . . . . . . . . . . . . . . . . 8
3.1 The measure-valued stochastic Becker-Do¨ringprocess . . . . . . 8
3.2 Definition of the scaling and the associate process. . . . . . . . . 10
3.3 Convergence towards the Lifschitz-Slyozov equation. . . . . . . . 11
4 Equations and martingale properties . . . . . . . . . . . . . . . . . . . 15
4.1 The original process . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 The rescaled process . . . . . . . . . . . . . . . . . . . . . . . . . 18
5 Estimations and technical results . . . . . . . . . . . . . . . . . . . . . 21
5.1 Moment estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2 Tightness of the rescaled process . . . . . . . . . . . . . . . . . . 24
5.3 Tightness of the boundary term . . . . . . . . . . . . . . . . . . . 27
6 Convergence and limit problem . . . . . . . . . . . . . . . . . . . . . . 32
6.1 The weak limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.2 Identification of the occupation measure . . . . . . . . . . . . . . 37
7 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7.1 Boundary Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.3 Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.4 Deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
t
n
8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
ri
p A Topology and Compactness . . . . . . . . . . . . . . . . . . . . . . . . 49
e
r A.1 The space X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
p
n A.2 The space Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
o
si A.3 The domain G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
r B Stationary states and measures for Becker-Do¨ring. . . . . . . . . . . . 54
e
V References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Comment citer ce document :
Yvinec, R., Deschamps, J., Hingant, E. (2014). From a stochastic Becker-Döring model to the
Lifschitz-Slyozov equation with boundary value. arXiv preprint arXiv:1412.5025 [math.PR], 1-69.
J. Deschamps, et al./Stochastic Becker-D¨oringto Lifschitz-Slyozov 3
Summary
We dealwith the convergencein law ofthe stochastic Becker-Do¨ringprocessto
theLifschitz-Slyozovpartialdifferentialequation,upto asmallscalingparame-
ter.Theformerisaprobabilisticmodelforthe lengthening/shrinkingdynamics
ofafinitenumberanddiscretesizeclusters,whilethelatterisseenasitsinfinite
numberandcontinuoussizeextension.IntheBecker-Do¨ringmodel,theclusters
are assumed to increase or decrease their size (number of particles in a cluster)
by additionorsubtractionofonly one singleparticle atatime (stepwise coagu-
lationandfragmentation)withoutregardingthespacestructure.Moreprecisely,
inthismodel,thetransitionsareassumedtobeMarkovianandactuallyrelated
to some random Poisson point measures. The lengthening rates depend on the
size,thenumberofclustersofthissizeandthenumberoffreeparticlesthrought
aLawofMassAction.Thefragmentationratesdependonthesizeandthenum-
ber of clusters of this size, through a spontaneous shricking (exponential law).
The evolutionof the configurationof the system is then described thanks to its
empirical measure. It starts with a finite number of clusters and particles. So
that, the state space of the model is finite (but possibly large) and bounded by
the number of particles and clusters of all possible sizes up to the maximal one
(given by the total number of particles in the system).
Under an appropriate scaling of the rates parameters, the number of mono-
mersandthesizesofclusters,weconstructarescaledmeasure-valuedstochastic
process from the empirical measure of the Becker-Do¨ring model. We prove the
convergence in law of this process towards a measure solution of the Lifschitz-
t
n
Slyozov equation. This equation is of transport type with a nonlinear flux cou-
ri
p pling the particlevariable.The necessityofprescribingaboundaryvalue atthe
e
r minimalsizenaturallyappearsinthecaseofincomingcharacteristics.Thevalue
p
n of the latter is still an open-debated question for this continuous model. The
o
si probabilistic approach of this work allows us to rigorously derive a boundary
r valueasaresultofa particularscaling(asopposedtoa modelingchoice)ofthe
e
V originaldiscretemodel.The proofofthis resultismainly basedonanadiabatic
procedure,theboundaryconditionbeingtheresultofaseparationoftime scale
and an averaging of a fast (fluctuating) variable.
Comment citer ce document :
Yvinec, R., Deschamps, J., Hingant, E. (2014). From a stochastic Becker-Döring model to the
Lifschitz-Slyozov equation with boundary value. arXiv preprint arXiv:1412.5025 [math.PR], 1-69.
J. Deschamps, et al./Stochastic Becker-D¨oringto Lifschitz-Slyozov 4
1. Introduction
The self-assembly of macromolecules and particles is a fundamental process in
manyphysicalandchemicalsystems.Althoughparticlenucleationandassembly
have been studied for many decades, the interest in this field has been recently
intensified due to engineering, biotechnological and imaging advances at the
nanoscale level. Hence, this area of research is still very active [51]. Applica-
tions range from industrial material design, physics, chemistry to biology. In
particular, the understanding of a large class of biological phenomena, such as
the rare protein assembly in neuro-degenerative diseases, requires to develop
stochastic self-assembly model. The interested reader is referredto [39, 48], the
introduction in [61] and references therein.
The mathematical study of theoretical models for self-assembly have a long
story.Often,these modelsconsider the mean-fieldconcentrationsofclusters for
each possible discrete size (number of particles in a cluster) and describe their
evolution using the so-called Law of Mass-Action. Probably one of the most
common model used is the celebrated Becker-Do¨ringmodel in its deterministic
version. The well-posedness theory and long-time behaviour have been exten-
sivelystudied,seee.g.[3,4,11,53,60].Forareviewoftheseresults,wereferto
[46],whilein[15,59]the readerwillfindconnectionto othermass-actiondeter-
ministic coagulation-fragmentation models. Nevertheless many open-questions
still remain,particularlyonthe long-time behaviour,as the computationof the
rate of convergence towards equilibrium [13, 30] or the precise and rigorous
description of the transient metastability phenomena [8, 14, 22, 23, 32, 44, 55].
t
n
Probabilisticapproacheshavebeenalsoinvestigatedasgeneralfinite-particle
ri
p stochastic coagulation models introduced in [37, 38] or the stochastic counter-
e
r partoftheBecker-Do¨ringmodel,bywhichwestart.But,forinstance,thelatter
p
n hasreceivedmuchlessattentionthantheirdeterministicanalogues.Initialanal-
o
si yses, numerical simulations and interesting open-questions have been raised in
r [6, 52] on this model. More recently, stationary states and first passage times
e
V have been partly characterized in [20, 61], emphasizing striking finite-size ef-
fects that arise in the stochastic the Becker-Do¨ring model. We also mention an
interesting work in [47] which relates stochastic modeling to metastability.
On the other side, insteadof a discrete size, the clusters canbe describedby
a continuous size (radius, length, etc). In this case an equivalent of the Becker-
D¨oring model would be the celebrated too Lifschitz-Slyozovmodel. Theoretical
analyses of this equation have also been extensively developed. Particularly,
the well-posedness has been laid down in [28, 34, 35, 43], while the long-time
behaviourhasbeenanalyzedtheoreticallyin[16,17]andnumericallyin[12,56].
We finally mention a review [42] on the Lifschitz-Slyozov theory together with
open-questions.
An interesting problem is to link mathematically the stochastic and deter-
ministic modelsand/orthe discretewiththe continuous-sizemodels.This leads
to many questions, particularly on the domain of validity of each model, the
scaling law between them, etc. Some of them found rigorous answers, here, we
intent to go further in this direction.
Comment citer ce document :
Yvinec, R., Deschamps, J., Hingant, E. (2014). From a stochastic Becker-Döring model to the
Lifschitz-Slyozov equation with boundary value. arXiv preprint arXiv:1412.5025 [math.PR], 1-69.
J. Deschamps, et al./Stochastic Becker-D¨oringto Lifschitz-Slyozov 5
The link between stochastic and deterministic coagulation models is stud-
ied since the review [1]. The seminal work [31] consists of deriving a law of
large number (mean-field limit) for discrete-size generalstochastic coagulation-
fragmentation models, including the stochastic Becker-Do¨ring model as a par-
ticular case. This approach is useful to derive results on existence of solutions
ofthedeterministicmodel,andonexplosion(gelation)times.Sincethen,tothe
best of our knowledge,much of the works relatedto stochastic models focus on
the pure coagulation model, e.g. [25, 26].
Discrete and continuous-size models have been linked and studied within
the context of deterministic models. Two main approaches are used. The first
considers the large time behaviour of the Becker-Do¨ringmodel, and relates the
dynamics of large clusters to solutions of various version of Lifschitz-Slyozov
equations. It is the so-called theory of Ostwald ripening, see [40, 41, 45, 54,
57, 58]. A second approach considers an initial condition with a large excess
of particles. Then, an appropriate re-scaling of the initial condition and the
ratefunctions leadsto solutions“closed”to the Lifschitz-Slyozovdynamics,see
[18, 21, 36].
Here,wewillfollowthelatterapproach,butwithastochasticmodel.Indeed,
our approach is intended to define a general scaling between the stochastic
Becker-Do¨ring and the deterministic Lifschitz-Slyozov. It seems it is the first
time a rigorouslink from discrete-stochastic to continuous-deterministic is pro-
posed.Our method recoverssimilar results knownyetin the pure deterministic
case[18,36]andageneralexistenceresultforalargeclassofrates.Thenovelty
ofourresultisthatwerigorouslyidentify,forgeneralscaling,aboundary-value
t
n
in the Lifschitz-Slyozov equation. It was conjectured e.g. in [18, 48] but never
ri
p proved. Historically, there was no need of boundary-value in Lifschitz-Slyozov
e
r since the problem was wellposed under physical assumptions (when small clus-
p
n terstendtofragment).But,recentapplicationsinBiologyhaveraisedthisprob-
o
si lemtoincludenucleationinthisequation,forinstancein[29,48].Theoriginality
r of the work resides in the proof too in order to identify the boundary. Indeed,
e
V we carefully introduce particular measure spaces and their topology. We adapt
toourcontextthe toolsdevelopedin[33]aboutaveragingtoobtainthe limitof
some fast (fluctuating) variables. Our results are illustrated by simulations at
the ends.
Finally, note that such links between the discrete-size and continuous-size
models may also have interest for numerical schemes to solve the latter model
(see [5]). We also believe that our study may be helpful to understand large
deviation phenomena on the stochastic Becker-Do¨ring model.
Organization of the paper We start by introducing the stochastic Becker-
D¨oring model in the next Section 2. Then Section 3 is devoted to a measure-
valuedformulationofthe modelknownas the empiricalmeasure.We introduce
a definition of the scaling law and the statement of our two main results: con-
vergence to vague and weak solution (with boundary value). The martingale
problemofboththeoriginalandtherescaledproblemarehighlightedinSection
Comment citer ce document :
Yvinec, R., Deschamps, J., Hingant, E. (2014). From a stochastic Becker-Döring model to the
Lifschitz-Slyozov equation with boundary value. arXiv preprint arXiv:1412.5025 [math.PR], 1-69.
J. Deschamps, et al./Stochastic Becker-D¨oringto Lifschitz-Slyozov 6
4. Technical results on moment estimates and tightness properties are grouped
in Section 5 in order to prepare the proof of the main results. We emphasize
inthis sectionthe introductionof a particular(occupation)measurecontaining
theinformationontheboundaryvalue.InSection6arethetwomostimportant
theorems which yield the main results. First an identificationof the limit equa-
tion in its general form with abstract boundary value, second we identify the
boundaryvalue to the stationarymeasure ofa modified (deterministic) Becker-
D¨oring system. We then conclude by numerical illustration of the theoritical
results in Section 7 and a discussion which relates other scaling in Section 8.
2. The stochastic Becker-D¨oring process
WeconsiderafinitestochasticversionoftheBecker-Do¨ringmodel.Aspreviously
introducedin [6, 20, 52, 61], we candefine such processas a Markovchainona
finite subset of a lattice. Choose an integer i ≥2 and a (possibly random, but
0
almost surely finite) parameter M ∈ N , where N := {i∈N:i≥j} for any
i0 j
j ≥1, that gives the total mass of the system. The state space of the process is
given by
E := (P ) ⊂N : iP ≤M .
 i i∈Ni0 i 
 iX≥i0 
For each configuration (P ) ∈E, the number P represents the quantity of
i i∈Ni0 i 
clustersconsistingofiparticles,whileC =M− iP ,isthenumberoffree
i≥i0 i
nt particles. This quantity is non-negative by virtue of the definition of the state
ri space E. In the Becker-Do¨ring model, clustersPcan increase or decrease their
p
e size one-by-one,by capturing (aggregationprocess)or shedding (fragmentation
r
p process) one particle. The set of kinetics reactions that we consider can be

n
o resumed by
ersi i0C −↽k−−0−(−C−−⇀)− Pi0,
V l0(Pi0)
(1)
C+Pi −↽−−a−−0−−(i−−)−C−−P−−i−⇀− Pi+1, i≥i0.
b0(i+1)Pi+1
The first reaction is the formation/destruction of an cluster of the minimal
size i . The second reaction occurs for any i ≥ i and is the aggregation-
0 0
fragmentationprocessbetweenclustersoftwosuccessivesizes.Thissetofkinet-
ics reactions (1) completely defines a Markov chain on E. Let us briefly explain
how we build the transition matrix from the reaction (1). For the forward ag-
gregationreaction, the transition is
C,P ,··· ,P ,P ,··· 7→ C−1,P ,··· ,P −1,P +1,··· ,
i0 i i+1 i0 i i+1
(cid:16) (cid:17) (cid:16) (cid:17)
andoccurswith arate givenby a (i)CP while for the backwardfragmentation
0 i
reaction, the transition is
C,P ,··· ,P ,P ,··· 7→ C+1,P ,··· ,P +1,P −1,··· ,
i0 i i+1 i0 i i+1
(cid:16) (cid:17) (cid:16) (cid:17)
Comment citer ce document :
Yvinec, R., Deschamps, J., Hingant, E. (2014). From a stochastic Becker-Döring model to the
Lifschitz-Slyozov equation with boundary value. arXiv preprint arXiv:1412.5025 [math.PR], 1-69.
J. Deschamps, et al./Stochastic Becker-D¨oringto Lifschitz-Slyozov 7
at a rate given by b (i+1)P . Equivalently, for the formation of an cluster
0 i+1
with minimal size, the transitions is
C,P ,··· ,P ,P ,··· 7→ C−i ,P +1,··· ,P ,P ,··· ,
i0 i i+1 0 i0 i i+1
(cid:16) (cid:17) (cid:16) (cid:17)
occuringataratek(C)andforthedestructionofsuchancluster,thetransition
is
C,P ,··· ,P ,P ,··· 7→ C+i ,P −1,··· ,P ,P ,··· ,
i0 i i+1 0 i0 i i+1
(cid:16) (cid:17) (cid:16) (cid:17)
and occurs at a rate l (P ).
0 i0
The Markov chain is well-defined as long as the sum of all rates is finite,
and up to the minimal explosion time (the limit of the transition times). The
well-posedness of the model is then guaranteed by
Assumption 1. We suppose that the formation and destruction rates vanish
when there are not enough reactants, i.e.
k (c)=0, ∀c<i .
0 0
l (0)=0.
0
Moreover, all aggregation and fragmentation rates are non-negative, i.e. for any
i≥i ,
0
a (i)≥0, b (i+1)≥0.
0 0
Indeed, with such conditions, and when (P (0)) ∈ E, it is then trivial
nt i i∈Ni0
pri to see that for any time t ≥ 0 up to the minimal explosion time, (Pi(t))i∈Ni0
belongs to E. But, (P (t)) can be re-written as a Markov chain in a finite
pre state space (Card(E) i< ∞i)∈,Nfio0r which existence for all times is guaranteed (no

n explosion in finite time). A crucial property of this model is also to preserve
o
si the mass balance property (because each transition preserves it together with
er Assumption 1)
V
iP (t)+C(t)≡M. (2)
i
iX≥i0
On the set of kinetics reactions (1), we emphasize that we have chosen a Law
of Mass-Action for the aggregation and fragmentation of clusters of size larger
thani .The non-negativefunctions a andb ,definedonN andN ,stand
0 0 0 i0 i0+1
respectivelyfortheaggregationandfragmentationconstantreactionrates(that
maydependonthesizeofthecluster).Fortheformationanddestructionrateof
anclusteroftheminimalsizei ,wechooseageneralizedlaw,givenbyarbitrary
0
functions C 7→k (C) and P 7→l (P ) that satisfy Assumption 1. This choice
0 i0 0 i0
is motivated by the fact that these two latter reactionswill be re-scaledfurther
differently from the others.
Remark 1. If i = 2 with k (x) =a (1)x(x−1) and l (x) =b (2)x we recover
0 0 0 0 0
thestochasticBecker-Do¨ringmodelwiththeLawofMass-Actionuptothefirst
size. See the discussion in Section 8 for corresponding results in this case.
Comment citer ce document :
Yvinec, R., Deschamps, J., Hingant, E. (2014). From a stochastic Becker-Döring model to the
Lifschitz-Slyozov equation with boundary value. arXiv preprint arXiv:1412.5025 [math.PR], 1-69.
J. Deschamps, et al./Stochastic Becker-D¨oringto Lifschitz-Slyozov 8
3. Scaling law and main results
Notations Forthe remainderweintroducefew classicalnotationswe willuse
for sake of clarity.First, C denotes the space of continuous functions. Similarly,
C , C and C are the spaces of continuous functions which are, respectively,
b c 0
bounded,supportlycompactandvanishingatboundary(seenasaclosureofC ).
c
We denote by Ck the functions having k continuous derivatives (up to k =∞).
Similarly for the other spaces the k derivatives have the same regularity.
For a Polish space E, we denote by M(E) the set of non-negative Radon
measures on E, M (E) the set of non-negative and finite Radon measures on
b
E and P(E) the probability measures.For any ν ∈M (E) and ϕ a real-valued
b
measurable function on E, we write
hν,ϕi = ϕ(x)ν(dx).
E
ZE
When no doubtremains onthe measurablespace E,we will simply write hν,ϕi
instead of hν,ϕi .
E
3.1. The measure-valued stochastic Becker-Do¨ring process
The model described in Section 2 can be studied using classical tools from
Markov chains, such as stochastic equations, Chapman-Kolmogorov equations,
first-passage time analysis, etc. As our objective is in particular to investigate
nt the limit as M → ∞ in (2) (large numbers) and to recover a weak form of
pri a deterministic partial differential equation, it is preferable to use a measure-
e
valued stochastic process approach.The advantage is to get a fixed state space
r
p
while performing the limit M →∞. To that, we consider the set
n
o
si
er n
V M (N ):= δ : n≥0, (x ,...,x )∈Nn ⊂M ([i ,+∞)).
δ i0 xi 1 n i0 b 0
( )
i=1
X
We represent the population of clusters, with the following measure at time
t≥0
µ = P (t)δ ∈M (N ). (3)
t i i δ i0
iX≥i0
where (P (t)) is the Markov chain described in Section 2 by the set of
i i∈Ni0
kinetics reactions (1), with finite mass initial condition given by (P (0)) ∈
i i∈Ni0
E. The solution P (t) represents the number of clusters of size i at time t ≥ 0,
i
andmaybegivennowbyP (t)=hµ ,1 iwherey 7→1 (y)isthefunctionequals
i t i i
to 1 for y =i and 0 elsewhere. This point of view defines (µ ) as a measure-
t t≥0
valued stochastic process that entirely contains the information of the system.
We define below, first the probabilistic objects we use and then the stochastic
differential equation satisfied by the empirical measure (3)
Comment citer ce document :
Yvinec, R., Deschamps, J., Hingant, E. (2014). From a stochastic Becker-Döring model to the
Lifschitz-Slyozov equation with boundary value. arXiv preprint arXiv:1412.5025 [math.PR], 1-69.
J. Deschamps, et al./Stochastic Becker-D¨oringto Lifschitz-Slyozov 9
Definition 1 (Probabilistic objects). Let i ∈ N∗ and (Ω,F,P) a sufficiently
0
large probability space. E[·] denotes the expectation. We define on this space
four independent random Poisson point measures
i) The nucleationPoissonpoint measure Q (dt,du) on R ×R with inten-
1 + +
sity
E[Q (dt,du)]=dtdu.
1
ii) The de-nucleation Poisson point measure Q (dt,du) on R × R with
2 + +
intensity
E[Q (dt,du)]=dtdu.
2
iii) The aggregation Poisson point measure Q (dt,du,di) on R ×R ×N
3 + + i0
with intensity
E[Q (dt,du,di)]=dtdu # (di).
3 i0
iv) The fragmentation Poisson point measure Q (dt,du,di) on R ×R ×
4 + +
N with intensity
i0+1
E[Q (dt,du,di)]=dtdu # (di).
4 i0+1
where dt and du are Lebesgue measures on R+, and # (di) is the counting
j
measure on N . Moreover, we define two more independent (from the above)
j
random elements
v) Theinitialdistributionµ isaM ([i ,+∞))-valuedrandomvariablesuch
in b 0
that a.s. µ belongs to M (N ) and hµ ,Idi is finite, where Id is the
nt in δ i0 in
ri identity function.
p vi) TheinitialquantityofparticlesC isaR -valuedrandomvariable(a.s.fi-
e in +
r nite).
p

n
o Finally,wedefinethecanonicalfiltration(Ft)t≥0associatedtothePoissonpoint
si measure such that µ and C are F -measurable.
r in in t
e
V NowwegiveadefinitionofmeasureformulationoftheBecker-Do¨ringmodel.
Definition 2 (Measure-valued stochastic Becker-Do¨ring process). Assume the
probabilisticobjectsofDefinition1aregiven,andthattheratefunctionssatisfy
Assumption1.Ameasure-valued stochastic Becker-Do¨ring process (abbreviated
by SBD process) is a M ([i ,+∞))-valued stochastic process µ = (µ ) that
b 0 t t≥0
satisfies a.s. and for all t≥0
t
µ =µ + δ 1 Q (ds,du)
t in Z0 ZR+ i0 {u≤k0(Cs−)} 1
t
− δ 1 Q (ds,du)
Z0 ZR+ i0 {u≤l0(hµs−,1i0i)} 2 (4)
t
+ (δ −δ )1 Q (ds,du,di)
Z0 ZR+×Ni0 i+1 i {u≤a0(i)Cs−hµs−,1ii} 3
t
− (δ −δ )1 Q (ds,du,di),
Z0 ZR+×Ni0+1 i i−1 {u≤b0(i)hµs−,1ii} 4
Comment citer ce document :
Yvinec, R., Deschamps, J., Hingant, E. (2014). From a stochastic Becker-Döring model to the
Lifschitz-Slyozov equation with boundary value. arXiv preprint arXiv:1412.5025 [math.PR], 1-69.