Convergence of stochastic gene networks to hybrid piecewise deterministic processes

0.29 MB·English

by  Alina Crudu

Convergence of stochastic gene networks to hybrid piecewise
deterministic processes
A. Crudu1, A. Debussche2, A. Muller3, O. Radulescu4
1 IRMAR - UMR 6625, Université de Rennes 1, Campus de Beaulieu,
35042 Rennes, France,
2 IRMAR - UMR 6625, ENS Cachan Bretagne, Campus de Ker Lann,
1
1 35170 Bruz, France,
0 3 IECN - UMR CNRS 7502, Université Henri Poincaré
2
54506 Vandoeuvre-lès-Nancy, France
n
4DIMNP - UMR 5235 CNRS/UM1/UM2, Université de Montpellier 2, Place Eugène Bataillon,
a
J CP 107, 34095 Montpellier, France.

7
January 10, 2011

]
R
P
. Abstract
h
t
a We study the asymptotic behavior of multiscale stochastic gene networks using
m weaklimitsofMarkovjumpprocesses. Dependingonthetimeandconcentrationscales
[ of the system we distinguish four types of limits: continuous piecewise deterministic

processes(PDP)withswitching,PDPwithjumpsinthecontinuousvariables,averaged
1
PDP, andPDP with singularswitching. We justify rigorouslythe convergencefor the
v
1 four types of limits. The convergence results can be used to simplify the stochastic
3 dynamics of gene network models arising in molecular biology.
4 MSC: 60J25, 60J75,92B05.
1
Keywords: Stochasticgenenetworks,piecewisedeterministicprocesses,perturbed
.
1 test functions.
0
1
1 1 Introduction
:
v
i Modern molecular biology emphasizes the important role of the gene regulatory net-
X
works in the functioning of living organisms. Recent experimental advances in molec-
r
a ular biology show that many gene products do not follow deterministic dynamics and
shouldbemodeledasrandomvariables(Kepler and Elston(2001);Kaufmann and van Oudenaarden
(2007)).
Markov processes approaches to gene networks dynamics, originating from the
pioneering ideas of Delbrück ( M. Delbrück (1940)), capture diverse features of the
experimentally observed expression variability, such as bursting (Cai et al. (2006)),
various types of steady-state distributions of RNA and protein numbers (Kaern et al.
(2005)), noise amplification or reduction by network propagation (Paulsson (2004);
Warren et al.(2006)),clockde-synchronization(Barkai and Leibler(2000)),stochastic
transitions in cellular memory storage circuits (Kaufmann et al. (2007)).
However, the study of the full Markov dynamics of biochemical networks is a dif-
ficult task. Even the simplest Markovian model, such as the production module of a
single protein involves tens of variables and biochemical reactions and an equivalent
number of parameters(Kierzek et al.(2001); Krishna et al. (2005)). The direct simu-
lationofsuchmodelsbytheStochasticSimulationAlgorithm(SSA)(Gillespie(1976))
is extremely time consuming.
1
In order to increase computational efficiency, several accelerated simulation algo-
rithmsarehybridandtreatfastbiochemicalreactionsascontinuousvariables(Haseltine and Rawlings
(2002); Alfonsi et al. (2005); Alfonsi et al. (2004)). Similar approaches reducing fast
reactionscanbejustifiedbydiffusionapproximationsforMarkovprocesses(Ball et al.
(2006)).
A different hybrid approach is to distinguish between molecular species according
to their abundances. Species in small amounts can be treated as discrete variables,
whereas species in large amounts can be consideredcontinuous. It has been proposed
that, the dynamics of gene networks with well separated abundances, can be well
approximated by piecewise deterministic Markov processes (Radulescu et al. (2007);
Crudu et al. (2009)). Piecewise deterministic processes (PDP) are used in opera-
tional researchin relationwith optimal controland various technologicalapplications
(Boxma et al.(2005);Ghosh and Bagchi(2005);Pola et al.(2003);Bujorianu and Lygeros
(2004)). Their popularity in physical, chemical and biological sciences is also steadily
increasing as they provide a natural framework to deal with intermittent phenomena
in many contexts (Radulescu et al. (2007); Zeiser et al. (2008)).
By looking for the best PDP approximation of a stochastic network of biochem-
ical reactions, and depending on the time scales of the reaction mechanism, we can
distinguish severalcases (Crudu et al. (2009)):
• Continuous PDP with switching: continuous variables evolve according to or-
dinary differential equations. The trajectories of the continuous variables are
continuous,butthe differentialequationsdepend onone orseveraldiscretevari-
ables.
• PDP with jumps inthe continuousvariables: the same as the previous case,but
the continuous variables can jump as well as the discrete variables.
• Averaged PDP: some discrete variables have rapid transitions and can be aver-
aged. The resulting approximationis an averagedPDP.
• Discontinuous PDP with singular switching: the continuous variable has two
time scaling. The switch between the two regimes is commanded by a discrete
variable. The rapid parts of the trajectory of the continuous variable can be
approximated by discontinuities.
In this paper, we justify rigorously these approximations that were illustrated by
modelsofstochasticgeneexpressioninCrudu et al.(2009). Moreprecisely,wepresent
several theorems on the weak convergence of biochemical reactions processes towards
piecewise deterministic processes of the type specified above. The resulting piecewise
deterministic processes can be used for more efficient simulation algorithms, also, in
certaincases,canleadtoanalyticresultsforthestochasticbehaviorofgenenetworks.
Higher orderapproximationsofmultiscale stochasticchemicalkinetics,corresponding
to stochastic differential equations with jumps, though not discussed in this paper,
represent straightforwardextensions of our results.
Thestructureofthisarticleisasfollows. Insection2,wepresentthePDP,auseful
theorem on the uniqueness of the solution of a martingale problem and the Markov
jump model for stochastic regulatory networks. The four remaining sections discuss
the asymptotic behaviors of the models, corresponding to the four cases presented
above.
2 Piecewise Deterministic Processes
We begin with a brief description of Piecewise Deterministic Processes (PDP) and
collect useful results on these. We do not consider PDPs in their full generaliy. The
reader is refered to Davis (1993) for further results.
Standard conditions:
2
In this article, a PDP taking values in E = Rn ×Nd is a process x = (y ,ν ),
t t t
determined by its three local characteristics :
1. For all ν ∈ Nd, a Lipschitz continuous vector field in Rn, denoted by F , which
ν
determines a unique global flow φ (t,y) in Rn such that, for t>0,
ν
d
φ (t,y)=F (φ (t,y)), φ (0,y)=y, ∀y ∈Rn.
ν ν ν ν
dt
We also use the notation: F(y,ν)=F (y).
ν
2. Ajumprateλ:E →R+ suchthat,foreachx=(y,ν)∈E,thereexistsǫ(x)>0
such that
ǫ(x)
λ(φ (t,y),ν)dt<∞.
ν
Z0
3. A transition measure Q : E → P(E), x 7→ Q(·;x), where P(E) denotes the set
of probability measures on E. We assume that Q({x};x)=0 for each x∈E.
From these standard conditions, a right-continuous sample path {x : t >0} starting
t
at x=(y,ν)∈E may be constructed as follows. Define
x (ω):=φ (t,y), for 0≤t<T (ω),
t ν 1
whereT (ω)istherealizationofthefirstjumptimeT ,withthefollowingdistribution:
1 1
t
P (T >t)=exp − λ(φ (s,y),ν)ds =:H(t,x), t∈R+.
x 1 ν
(cid:16) Z0 (cid:17)
We have then x (ω) = (φ (T (ω),y),ν), and the post-jump state x (ω) has
T1−(ω) ν 1 T1(ω)
the distribution given by :
P (x ∈A|T =t)=Q(A;(φ (t,y),ν))
x T1 1 ν
on the Borel sets A of E.
We then restart the process at x (ω) and proceed recursively according to
T1(ω)
the same procedure to obtain a sequence of jump-time realizations T (ω),T (ω),....
1 2
Between each of two consecutive jumps, x (ω) follows a deterministic motion, given
t
by the flow corresponding to the vector field F.
Such a process x is called a PDP. The number of jumps that occurr between the
t
times 0 and t is denoted by
N (ω)= 1 (ω).
t t≥Tk
k
X
It can be shown that x is a strong Markov process with right-continuous, left-
t
limited sample paths (see Davis (1993)). The generator A of the process is formally
given by
Af(x)=F (x)·∇ f(x)+λ(x) (f(z)−f(x))Q(dz;x) (1)
ν y
ZE
for each x = (y,ν) ∈ E, we have denoted by ∇ the gradient with respect to the
y
variable y ∈ Rn. The domain of A is described precisely in Davis (1993). We do not
need such a precise description and just note that A is well defined for f ∈E, the set
of functions f :E →R such that :
E1. f is bounded,
E2. for all ν ∈Nd, f(·,ν)∈C1(Rn),
E3. its derivatives are bounded uniformly in E.
3
For f ∈E, we denote by
L = sup ||D f|| = sup kD f(y,ν)k, (2)
f y ∞ y
y∈Rn (y,ν)∈E
the Lipschitz constant of f with respect to the variable y.
The space E is a Banach space when endowed with the norm
||f|| =||f|| +L . (3)
E ∞ f
If Z is a Banach space, B (Z) is the set of bounded Borel measurable functions
b
on Z ; Ck(Z) is the set of Ck-differentiable functions on Z, such that the derivatives,
b
until the k-th order, are bounded ; C (Z) is the set of bounded continuous functions
b
on Z. Also D(R+;Z) is the set of process defined on R+ with right-continuous, left-
limited samplepaths definedon R+ andtakingvaluesinZ andC(R+;Z)is the setof
continuous process defined on R+ and taking values in Z.
The PDPs considered in this paper will always satisfy the following property :
Hypothesis 2.1 The three local characteristics of the PDP satisfy the standard con-
ditions given above. The jump rate λ is C1-differentiable with respect to the variable
y ∈Rn. For every starting point x=(y,ν)∈E and t∈R+, we suppose E(N )<∞.
t
Remark 2.2 E(N ) < ∞ implies in particular that T (ω) → ∞ almost surely. This
t k
assumption is usually quite easy to check in applications, but it is hard to formulate
general conditions under which it holds, becauseof the complicated interaction between
F,λ and Q. It can be shown for instance that if λ is bounded, then E(N ) < ∞ (cf.
t
Davis (1993)).
For some results, we need the following stronger property :
Hypothesis 2.3 The functions F, λ and x 7→ λ(x) f(z)Q(dz;x), with f ∈ E,
E
are bounded on E, C1-differentiable with respect to the variable y ∈ Rn and their
R
derivatives with respect to y are also bounded.
When Hypothesis 2.3 is satisfied, we set
M =kFk , L = sup kD Fk ,
F ∞ F y ∞
y∈Rn
M =kλk , L = sup kD λk .
λ ∞ λ y ∞
y∈Rn
and L a constant such that, for all f ∈E and x=(y,ν)∈E :
Q
D λ(x) f(z)Q(dz;x) ≤L kfk
y Q E
(cid:13) (cid:18) ZE (cid:19)(cid:13)∞
(cid:13) (cid:13)
(cid:13) (cid:13)
Denote by (Pt)t≥0 the(cid:13)transition semigroupassoci(cid:13)atedto the PDP constructed above
and by P the law of the PDP starting from x ∈ E. Then, P is a solution of the
x x
martingale problem associated to A in the following sense:
t
f(x )−f(x)− Af(x )ds
t s
Z0
is a local martingale for any f ∈ E (see Davis (1993)). Moreover, if Hypothesis 2.3
holds,itis aboundedmartingale. As usual,wehavedenotedby(x ) the canonical
t t≥0
process on D(R+;E).
The following results gives a uniqueness property for this martingale problem. It
will enable us to characterize the asymptotic behavior of our stochastic regulatory
networks.
4
Theorem 2.4 IfHypothesis2.3issatisfied, thenthelawofthePDPdeterminedbyF,
λ, and Q is the unique solution of the martingale problem associated to the generator
A.
The proof of the theorem is given in the appendix.
Markov jump model for stochastic regulatory networks: known results
WeconsiderasetofchemicalreactionsR ,r ∈R;Rissupposedtobefinite. These
r
reactions involve species indexed by a set S = 1,...,M, the number of molecules of
the specie i is denoted by n and X ∈ NM is the vector consisting of the n ’s. Each
i i
reaction R has a rate λ (X) which depends on the state of the system, described by
r r
X and corresponds to a change X →X +γ , γ ∈ZM.
r r
Mathematically, this evolution can be described by the following Markov jump
process. It is based on a sequence (τ ) of random waiting times with exponential
k k≥1
distribution. Setting T =0, T =τ +···+τ , X is constant on [T ,T ) and has a
0 i 1 i i−1 i
jump at T . The parameter of τ is given by λ (X(T )):
i i r∈R r i−1
P(τ >t)=exp − Pλ (X(T ))t .
i r i−1
(cid:16) rX∈R (cid:17)
AttimeT ,areactionr ∈Rischosenwithprobabilityλ (X(T ))/ λ (X(T ))
i r i−1 r∈R r i−1
and the state changes according to X →X +γ :
r
P
X(T )=X(T )+γ .
i i−1 r
This Markov process has the following generator (see Ethier and Kurtz (1986)):
Af(X)= [f(X+γ )−f(X)]λ (X).
r r
r∈R
X
We do not need a precise description of the domain of A, the above definition holds
for instance for functions in C (RM).
b
Intheapplicationswehaveinmind,thenumbersofmoleculeshavedifferentscales.
Some of the molecules are in small numbers and some are in large numbers. Accord-
ingly, we split the set of species into two sets C and D with cardinals M and M .
C D
This induces the decomposition X = (X ,X ), γ = (γC,γD). For i ∈ D, n is of
C D r r r i
order1whilefori∈C,n isproportionaltoN whereN isalargenumber. Fori∈C,
i
1
setting n˜ =n /N, n˜ is of order 1. We define x = X and x=(x ,X ).
i i i C C C D
N
We also decompose the set of reactions according to the species involved. We set
R = R ∪R ∪R . A reaction in R (resp. R ) produces or consumes only
D C DC D C
species in D (resp. C). Also, the rate of a reaction in R (resp. R ) depends only
D C
on X (resp. x ). A reactionin R has a rate depending on both x and X and
D C DC C D
produces or consumes, among others, species from C or D.
The rate of a reaction in r ∈R is also large and of order N and we set λ˜ = λr.
C r N
In general, reactions in R or R have a rate of order 1.
D DC
Introducing the new scaled variables, the generator has the form:
1
A˜f(x ,X ) = f(x + γC,X )−f(x ,X ) Nλ˜ (x )
C D C N r D C D r C
rX∈RC(cid:20) (cid:21)
1
+ f(x + γC,X +γD)−f(x ,X ) λ (x ,X )
C N r D r C D r C D
r∈XRDC(cid:20) (cid:21)
+ f(x ,X +γD)−f(x ,X ) λ (X ).
C D r C D r D
rX∈RD(cid:2) (cid:3)
Assuming that the scaled rates λ˜ are C1 with respect to x , it is not difficult to see
r C
that if N → ∞, the two sets of species decouple. Indeed, reactions in R do not
DC
5
happen sufficiently often and they do not change x in a sufficiently large manner.
C
The limit would simply give a set of differential equations for the continuous variable
x , which evolves without influence of X . The discrete variable would have its own
C D
dynamic made of jumps. These results have been shown by Kurtz (1971) and Kurtz
(1978).
In the following sections, we consider more general systems where other types of
reactions may happen and which yield different limiting systems
3 Continuous piecewise deterministic process
In this section, we assume that some of the reactions in a subset S of R are such
1 DC
that their rate is large and scales with N. We again set λ˜ = 1λ for r ∈ S . We
r N r 1
assume that these equations do not change X , in other words
D
γD =0, r ∈S . (4)
r 1
However, the rate λ depends on X . Note that this is possible and even frequent in
r D
molecular biology, meaning that reactions of the type S recover the reactant, like in
1
the reaction A→A+B, with A and B discrete and continuous species, respectively.
The more complicated case γD 6=0 is treated in section 5.
r
The scaled generator has now the form
1
A˜ f(x ,X ) = f(x + γC,X )−f(x ,X ) Nλ˜ (x )
N C D C N r D C D r C
rX∈RC(cid:20) (cid:21)
1
+ f(x + γC,X )−f(x ,X ) Nλ˜ (x ,X )
C N r D C D r C D
rX∈S1(cid:20) (cid:21)
1
+ f(x + γC,X +γD)−f(x ,X ) λ (x ,X )
C N r D r C D r C D
r∈RXDC\S1(cid:20) (cid:21)
+ f(x ,X +γD)−f(x ,X ) λ (X ).
C D r C D r D
rX∈RD(cid:2) (cid:3)
(5)
For f ∈C1(E), we may let N →∞ and obtain the limit generator
b
A f(x ,X ) = λ˜ (x )γC + λ˜ (x ,X )γC ·∇ f(x ,X )
∞ c D r C r r C D r xC C D
!
rX∈RC rX∈S1
+ f(x ,X +γD)−f(x ,X ) λ (x ,X )
C D r C D r C D
r∈RXDC\S1(cid:2) (cid:3)
+ f(x ,X +γD)−f(x ,X ) λ (X ).
C D r C D r D
rX∈RD(cid:2) (cid:3)
Thisformalargumentindicatesthat,asN →∞,theprocessconvergestoacontinuous
PDP (see (1)). The state is described by a continuous variable x and a discrete
C
variable X . The discrete variable is a jump process and is piecewise constant. The
D
continuousvariableevolvesaccordingto differentialequations depending on X . Itis
D
continuous but the vector field describing its evolution changes when X jumps.
D
This is rigorouslyjustified by the following theorem.
Theorem 3.1 Let xN = (xN,XN) be a jump Markov process as above, starting at
C D
xN(0) = (xN(0),XN(0)). Assume that the jump rates λ˜ , r ∈ R ∪ S and λ ,
C D r C 1 r
6
r ∈ RDC \S1 are C1 functions of xC ∈ RMC. We define Px0 the law of the PDP
starting at x =(x ,X ) whose jump intensities are:
0 C,0 D,0
λ(x)= λ (x),
r
r∈RD∪XRDC\S1
the transition measure is defined by:
f(z)Q(dz;x)
E
=R 1 f(x ,X +γD)λ (x ,X )+ f(x ,X +γD)λ (X ) ,
λ(x) C D r r C D C D r r D 
r∈RXDC\S1 rX∈RD
 
for x=(x ,X ), and the vectors fields are given by:
C D
F (x )= γCλ˜ (x )+ γCλ˜ (x ,X ).
XD C r r C r r C D
rX∈RC rX∈S1
Assume that Hypothesis 2.1 is satisfied and xN(0) converges in distribution to x ,
0
then xN converges in distribution to the PDP whose law is P .
x0
Proof:
In the following, we work only with scaled variables and simplify the notation by
omitting the tildes. In other words, we use λ to denote the rate of all reactions.
r
The proof is divided into three steps. We begin our proof by supposing that the
jump rates and their derivatives with respect to x are bounded. Hypothesis 2.3 is
C
then satisfied. We then prove Theorem 3.1 by a truncation argument.
Step 1: Tightness for bounded reaction rates.
We first assume that all rates λ are bounded as well as their derivatives with
r
respect to x .
C
Let xN be a Markov jump process whose generator is given by A˜ .
N
Without loss of generality,we assume that the initial value ofthe processis deter-
ministic: xN(0)=(xNC(0),XDN(0))andconvergestox0 =(xC,0,XD,0)inRMC×NMD.
Let (Y ) be a sequence of independent standard Poisson processes. By Propo-
r r∈R
sition 1.7, Part4, and Theorem 4.1,Part 6, of Ethier and Kurtz (1986) we know that
there exists stochastic processes (x˜N)N∈N in D(R+;E) such that
t
x˜N(t)=xN(0)+ γ Y λ (x˜N(s))ds , t≥0.
r r r
r∈R (cid:18)Z0 (cid:19)
X
Moreover,for eachN, xN and x˜N have the same distribution. Since we consider only
the distributions of the processes, we only consider x˜N in the following and use the
same notation for both processes.
Using the decomposition xN =(xN,XN), we have
C D
1 t
xN(t) =xN(0)+ γCY N λ˜ (xN(s))ds
C C N r r r C
rX∈RC (cid:18) Z0 (cid:19)
1 t
+ γCY N λ˜ (xN(s),XN(s))ds
N r r r C D
rX∈S1 (cid:18) Z0 (cid:19)
1 t
+ γCY λ (xN(s),XN(s))ds
N r r r C D
r∈RXDC\S1 (cid:18)Z0 (cid:19)
7
and
t
XN(t) =XN(0)+ γDY λ (xN(s),XN(s))ds
D D r r r C D
r∈RXDC\S1 (cid:18)Z0 (cid:19)
t
+ γDY λ (XN(s))ds.
r r r D
rX∈RD (cid:18)Z0 (cid:19)
We easily prove tightness in D(R+;RMD) of the laws of (XDN)N∈N by the same proof
as for Proposition 3.1 in chapter 6 of Ethier and Kurtz (1986) and by using the fact
that the law of Y is tight in D(R+;N), for every r ∈R, according to Theorem 1.4 of
r
Billingsley (1999).
Toprovethatthelawsof(xNC)N∈NaretightinC(R+;RMC),weadapttheargument
of section 2 chapter 11 in Ethier and Kurtz (1986).
LetY˜ (u)=Y (u)−u be the standardPoissonprocesscenteredatitsexpectation,
r r
we have:
1 t
xN(t) =xN(0)+ γCY˜ N λ˜ (xN(s))ds
C C N r r r C
rX∈RC (cid:18) Z0 (cid:19)
1 t
+ γCY˜ N λ˜ (xN(s),XN(s))ds
N r r r C D
rX∈S1 (cid:18) Z0 (cid:19)
t
+ F(xN(s),XN(s))ds
C D
Z0
1 t
+ γCY λ (xN(s),XN(s))ds
N r r r C D
r∈RXDC\S1 (cid:18)Z0 (cid:19)
Observe that
1
sup Y˜ (Nu)→0, a.s.
r
N
u∈[0,A]
for any A≥0. Since λ are bounded, it follows that, for all T >0,
r
1 t 1 t
sup γCY˜ N λ˜ (xN(s))ds + γCY˜ N λ˜ (xN(s),XN(s))ds
t∈[0,T](cid:12)(cid:12)rX∈RC N r r(cid:18) Z0 r C (cid:19) rX∈S1 N r r(cid:18) Z0 r C D (cid:19)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
→0, a(cid:12).s. when N →∞. (cid:12)
Clearly
1 t
sup γCY λ (xN(s),XN(s))ds →0, a.s.
(cid:12) N r r r C D (cid:12)
t∈[0,T](cid:12)(cid:12)r∈RXDC\S1 (cid:18)Z0 (cid:19)(cid:12)(cid:12)
(cid:12) (cid:12)
Itfollowsthatther(cid:12)eexistsarandomconstantK goingtozeros(cid:12)uchthat,fort,t ,t ∈
(cid:12) N (cid:12) 1 2
[0,T], and kFk =sup |F(x)|.
∞ x∈RMC×NMD
|xN(t)|≤|xN(0)|+kFk t+K , a.s.
C C ∞ N
and
|xN(t )−xN(t )|≤kFk |t −t |+2K , a.s.
C 1 C 2 ∞ 1 2 N
Tightness of(xNC)N∈N in C(R+;RMC) follows by classicalcriteria(see for instance
Jacod and Shiryaev (1987), chapter 6, section 3b).
Weconclude,fromJacod and Shiryaev(1987)(chapter6,section3b),that{xN} =
N
{(xN,XN)} is tight in D(R+;E).
C D N
Step 2: Identification of limit points for bounded reaction rates.
8
Letx=(x ) be the canonicalprocesson D(R+;E), andP the law of(xN)
t t≥0 N t t≥0
on this space, for each N ∈N.
We know that for each N ∈N and ϕ∈E.
t
ϕ(x )−ϕ(x )− A˜ ϕ(x )ds
t 0 N s
Z0
is a P -martingale. Equivalently, for each n ∈ N, t ,...,t ∈ [0,r], t ≥ r ≥ 0,
N 1 n
ψ ∈(C (E))n and ϕ∈E
b
t
E ϕ(x )−ϕ(x )− A˜ ϕ(x )ds ψ(x ,...,x )
PN t 0 N s t1 tn
(cid:18)(cid:18) Z0 r (cid:19) (cid:19) (6)
=E ϕ(x )−ϕ(x )− A˜ ϕ(x )ds ψ(x ,...,x ) .
PN r 0 N s t1 tn
(cid:18)(cid:18) Z0 (cid:19) (cid:19)
Let(P ) beasubsequencewhichconvergesweaklytoameasureP onD(R+;E).
Nk k
WeknowthatxisP almostsurelycontinuousateverytexceptforacountablesetD
P
and that for t ,...,t outside D , P π−1 converges weakly to Pπ−1 where
π isthep1rojectinonthatcarrPiesthNekpto1i,n..t.,txn∈D(R+;E)tothepoint(t1x,...,,t.n..,x )
t1,...,tn t1 tn
of Rn.
Therefore,it is easy, using dominated convergencetheorem and weak convergence
properties, to let k →∞ in (6) and obtain for t, t ,...,t ,r outside D :
1 n P
t
E ϕ(x )−ϕ(x )− A ϕ(x )ds ψ(x ,...,x )
P t 0 ∞ s t1 tn
(cid:18)(cid:18) Z0 r (cid:19) (cid:19) (7)
=E ϕ(x )−ϕ(x )− A ϕ(x )ds ψ(x ,...,x ) .
P r 0 ∞ s t1 tn
(cid:18)(cid:18) Z0 (cid:19) (cid:19)
If t ∈ D , we choose a sequence (tk) outside D such that tk → t with tk > t.
P P
Then Pπ−1 converges weakly to Pπ−1 since x is P-a.s. right continuous in t and x
tk t tk
converges almost surely to x . Then, we use (7) with tk instead of t, let k → ∞ and
t
deduce that (7) also holds for t ∈ D . Similarly, we show that t ,...,t ,r may be
P 1 n
taken in D .
P
This shows that the measure P is a solution of the martingale problem associated
to the generator A on the domain E.
∞
Hypothesis2.3enablesustoapplyTheorem2.4. Themartingaleproblemhasthen
a unique solution. It followsthat the limit P is equal to P , the law ofthe PDP, and
x0
that the whole sequence (P ) converges weakly to P .
N N x0
Step 3: Conclusion
Now, we prove Theorem 3.1 with a truncation argument.
Let θ ∈C∞(R+) such that
θ(x)=1, x∈[0,1],
θ(x)=0, x∈[2,∞),
(cid:26)
and, for k≥1 and r∈R, define
|x|2
θ (x)=θ , x∈E,
k k2
(cid:18) (cid:19)
and
λk(x)=θ (x)λ (x).
r k r
Then, the problem with λk instead of λ fulfills Hypothesis 2.3. We define xN =
r r k
(xN ,XN ) the jump Markov process associated to the jump intensities λk, starting
C,k D,k r
at xN(0). By the preceding result, we know that, for all k ∈ N, (xNk )N∈N converges
weakly to the PDP x in D(R+;E), whose characteristics are the jump intensities
k
9
λk, with correspondingtransitionmeasure,and vectorfields (obvious definitions as in
r
Theorem 3.1).
Then, ((xNk )k∈N)N∈N converges weakly to (xk)k∈N in D(R+;E)N.
By Skorohod representationTheorem (see Billingsley (1971), Theorem 3.3), up to
achangeofprobabilityspace,wemayassumethatthatforallk ∈N(xN) converges
k N
a.s. to x in D(R+;E).
k
Let T >0 and the stopping times
τk =inf{t∈[0,T], |x (t)|≥k},
k
with τk =T if {t∈[0,T], |x (t)|≥k}=∅.
k
Then, for k,l ∈N
x (t)=x (t), t∈[0,τl∧τk], a.s.
k l
so that τk is a.s. non-decreasing.
Moreover,if x (resp. xN) are the PDP associated to A (resp. the Markov jump
∞
process associated to A˜ ), then
N
x (t)=x , t∈[0,τk), a.s.
k t
and if
τk =inf{t∈[0,T], |xN(t)|≥k},
N k
with τk =T if {t∈[0,T], |xN(t)|≥k}=∅, then
N k
xN(t)=xN(t), t∈[0,τk), a.s.
k N
Let δ > 0. Observing that if τk−1 > T −δ and d (xN,x ) < ǫ, where d is
T−δ k k T−δ
the distance on D([0,T −δ];E), then, for enough small ǫ:
sup |xN(t)|≤k, a.s.
k
t∈[0,T−δ]
then a.s., τk ≥ T −δ and xN = xN in [0,T −δ]. Since τk ≥ τk−1 > T −δ; we have
N k
also, a.s., x =x in [0,T −δ] and
k
d (xN,x)<ǫ.
T−δ
We deduce that
∀δ >0, P d (xN,x)≥ǫ ≤P τk−1 ≤T −δ +P d (xN,x )≥ǫ .
T−δ T−δ k k
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Finally, we have
P τk−1 ≤T −δ =P( sup |x (t)|≥k−1)≤P( sup |x(t)|≥k−1).
k
t∈[0,τk] t∈[0,T−δ]
(cid:0) (cid:1)
By Hypothesis 2.1, the PDP x can not explode in finite time. Thus for k large, this
term is small. Then for large N, the second is small. We deduce that xN converges
in probability to x in the new probability space. Returning in the originalprobability
space, we obtain that xN converges in distribution to x.
4 Piecewise deterministic process with jumps
In this section, we assume that some of the reactions in a subset S of R are
2 DC
such that γC is large and scales with N. We set γ˜C = 1γC for r ∈ S . We define
r r N r 2
S =S ∪S .
1 2
10