Large deviations theory for Markov jump models of chemical reaction networks

0.55 MB·

by  Andrea Agazzi

LARGE DEVIATIONS THEORY FOR MARKOV JUMP MODELS OF
CHEMICAL REACTION NETWORKS
BY ANDREA AGAZZI†,‡,§
AMIR DEMBO† AND JEAN-PIERRE ECKMANN‡
StanfordUniversity† andUniversityofGeneva‡
Abstract
7 WeproveasamplepathLargeDeviationPrinciple(LDP)foraclassofjump
1 processeswhoseratesarenotuniformlyLipschitzcontinuousinphasespace.
0 BuildingonitwefurtherestablishthecorrespondingWentzell-Freidlin(W-F)
2
(infinitetimehorizon)asymptotictheory.TheseresultsapplytojumpMarkov

n processesthatmodelthedynamicsofchemicalreactionnetworksundermass
a actionkinetics,onamicroscopicscale.Weprovidenaturalsufficienttopo-
J logical conditions for the applicability of our LDP and W-F results. This

9 thenjustifiesthecomputationofnon-equilibriumpotentialandexponential
transitiontimeestimatesbetweendifferentattractorsinthelargevolumelimit,
]
R forsystemsthatarebeyondthereachofstandardchemicalreactionnetwork
theory.
P
.
h
t 1. Introduction. Thedynamicsofchemicalreactionsareusuallymodeledbymass-action
a
m equations:Asystemofapolynomialordinarydifferentialequationswhichrelatetheevolutions
[ ofconcentrationsofchemicalcompounds.Thesesystemsofequationsinherittheirstructurefrom

1 the topologyof the ChemicalReaction Network (CRN) they model,and the interplaybetween
v topologyanddynamicsofmassactionsystemsistheobjectofstudyofchemicalreactionnetwork
6
theory [1, 9, 17]. These sets of ODEs approximate the interactions of the individual molecules
2
1 involved.Thediscretenatureofchemicalreactionsystemscanbecapturedbydiscretemodels
2 wherethestateofthesystemisgivenbythenumberofmoleculesofeachtypethatarepresent
0
inthereactor.Inthisframework,whenareactionoccurs,theinputmoleculescombinetoform
.
1 the output ones, and the system jumps to a new state. The dynamics of such systems are in
0
generalmodeledstochasticallyasapurejumpMarkovprocess[8,Sec.11,ExampleC]whose
7
1 jumpratesareapproximationsofthereactionratesfoundindeterministicmassactionmodels.
:
v Finally,assumingthatthesystemhasvolumev,onecanstudyhowthestochasticdynamicsofthe
i processXv describingtheconcentrationofthedifferentchemicalspeciesattimetscalewiththe
X t
parameterv.Thisistheobjectofstudyofthispaper.
r
a Similardiscretestochasticmassactionkineticsmodelshavebeenappliedtodiseasepropagation
dynamics [24], genetic algorithms [21], and for the simulation of noisy biochemical reaction
†ResearchpartiallysupportedbyNSFgrantDMS-1613091.
‡ResearchsupportedbyERCadvancedgrant290843.
§ResearchpartiallysupportedbySNSFgrant161866.
MSC2010subjectclassifications:Primary60F10,80A30.Secondary37B25,60J75.
Keywordsandphrases: large deviation principle, Wentzell-Freidlin theory, jump Markov processes, chemical
reactionnetworks,Lyapunovfunctions,toricjets.
1
2 A.AGAZZI,A.DEMBO,J.-P.ECKMANN
networks through the application of the so-called Gillespie algorithm [14]. Asymptotics such
as limit theorems on the convergence of the stochastic trajectories towards the deterministic
oneshavebeenprovenintheprobabilityliterature[8].Morerecently,resultsonproduct-form
steady state distributions for a certain class of CRNs have been obtained in [1] and conditions
for the irreducibility and ergodicity of the stochastic chemical dynamics of reaction networks
havebeenpresentedin[16,22].Ourworkextendstheseresultstothedomainoflargedeviations
theory, identifying a large class of CRNs to which that theory applies. We prove in particular
thatWentzell-Freidlinexittimeestimatescanbeappliedtosuchsystems,rigorouslyjustifying
thewidespreaduseofpotentialtheory[12,13,19]andultimatelyallowingfortheanalysisof
events that play a key role in, e.g., theoretical biochemistry [2, 3] and that are not covered by
deterministic mass action models, because deterministic models do not allow for transitions
betweendifferentattractors.
1.1. The model and its sample path LDP. We consider a set of chemical species S =
{s ,s ,...,s },whoseinteractionsaredescribedbyafinitesetofreactionsR = {r ,r ,...,r }.
1 2 d 1 2 m
Eachreactionisuniquelyidentifiedbyitssubstrates(inputspecies)andproducts(outputspecies),
and we express such a reaction as r = {cr (cid:42) cr }, with cr , cr ∈ Nd representing the
in out out in 0
multiplicityofthespeciess intheinputoroutputofthereaction.ThesetC ofcomplexesconsists
i
ofallcr (with# =“in”or“out”),andforeachreactionr ∈ Rwedefinethereactionvector
#
cr := crout−crin ∈ Zd.ACRNisthusdefinedbythetriple(S,C,R).
EXAMPLE1.1. Thesystem
r r r
∅ (cid:42)1 A+B (cid:42)2 2B (cid:42)3 A (1.1)
is a CRN with S = {A,B} and R = {r1,r2,r3}. The set of complexes of these reactions is
C = {∅,{A+B},{2B},{A}} = {(0,0),(1,1),(0,2),(1,0)}(inthebasisspannedby(A,B)).
Inthispaper,westudythebehaviorasafunctionofv ofthescaledprocess
(Xv) := v−1(N ) , i ∈ 1,...,d,
t i t i
whereN ∈ Nd representsthenumberofmoleculesofthedspeciesandXv ∈ (cid:0)v−1N (cid:1)d denotes
t 0 t 0
theirnumberdensity(inmols)attimet.Theinteractionsamongmoleculesarethendescribedby
eachreactionr ∈ RstandingforapossiblejumpoftheprocessXv → Xv +v−1cr,withcr the
t t
reaction(orjump)vectorassociatedwithr ∈ R.Correspondingly,Xv isacontinuoustimepure
t
jumpMarkovprocesswithgenerator
(L f)(x) := v(cid:88)Λ(v)(x)(cid:0)f(x+v−1cr)−f(x)(cid:1) (1.2)
v r
r∈R
forf : (cid:0)v−1N (cid:1)d → Randthevolume-normalizedjumprates
0
d (cid:18) (cid:19)
Λ(rv)(x) = krv−(cid:107)crin(cid:107)1(cid:89) (cvrxi) (crin)i! (1.3)
i=1 in i
LARGEDEVIATIONSTHEORYFORCHEMICALREACTIONNETWORKS 3
forsomereaction(rate)constantsk > 0,where(cid:0)a(cid:1)denotesthebinomialcoefficientwhichby
r b
conventioniszerowhenb ∈/ [0,a]and(cid:107)·(cid:107) denotesthe(cid:96) -norm.
1 1
REMARK1.2. Forafixedvolumev andinitialconditionXv = xv ∈ (cid:0)v−1N (cid:1)d,theprocess
0 0 0
Xv is confined to Sv := (cid:8)xv +{(cid:80) α cr : α ∈ (cid:0)v−1N (cid:1)m}(cid:9)∩Rd. Indeed, Xv cannot
· xv 0 r∈R r 0 + t
0
jumpoutsideof(cid:0)v−1N (cid:1)d sinceΛ(v)(x) = 0foranyr ∈ Rsuchthatx+v−1cr ∈/ (cid:0)v−1N (cid:1)d so
0 r 0
thecorrespondingsummandin(1.2)isthenzero(regardlessoff(·)).
REMARK 1.3. In the limit v → ∞, the sample paths of the processes Xv starting at
t
Xv = xv → x ∈ Rd almostsurelyconverge—uniformlyover[0,1]—tothesolutionζ(t)ofthe
0 0 0 +
deterministicODE
dζ (cid:88)
= λ (ζ)cr, ζ(0) = x , (1.4)
dt r 0
r∈R
havingtheasymptoticreactionrates
d
(cid:89) (cr )
λ (x) := k x in i (1.5)
r r i
i=1
(see[8,§11,Thm.2.1],wheresuchafunctionalLLNforCRNsisderived).
We show in Section 2 that under the following mild assumption on the generator L of the
v
scaledprocess,thesolutionXv tothecorrespondingmartingaleproblemsatisfiesasamplepath
t
LDPinthesupremumnorm,withanexplicitratefunction(seeTheorem1.6).
ASSUMPTION A.1. Let Xv be the solution of the martingale problem generated by the
t
generatorL of(1.2).Weassume
v
(a) Thereexistb < ∞andacontinuous,positivefunctionU(x)ofcompactlevelsets,suchthat
forsomenon-decreasingfunctionv(cid:48) : R → R ,
+ +
(L Uv)(x) ≤ ebv ∀v > v(cid:48)((cid:107)x(cid:107) ), x ∈ (v−1N )d . (1.6)
v 1 0
(b) ThereexistsapathwithinNd startingatzeroandleadingtosomestrictlypositivevectorn ,
0 •
r r
whileusingonlyincrements−c k followedbyc k .
in out
REMARK 1.4. TheexistenceofasolutionXv tothemartingaleproblemgeneratedbyL
t v
with initial condition xv ∈ (cid:0)v−1N (cid:1)d is guaranteed by standard theory (see [23, Thm. 8.3]),
0 0
up to the possibility of explosion. In Lemma 2.1 we show that this possibility is ruled out by
AssumptionA.1.
REMARK1.5. AssumptionA.1(b)requiresthatallchemicalspeciescanbecreated,atleast
indirectly,startingfromzero,hencefromanyotherpossiblestateofthesystem.Inparticular,there
mustexistatleastonechemicalreactionwithoutsubstrates,namely,withcr = 0.Suchconstant
in
4 A.AGAZZI,A.DEMBO,J.-P.ECKMANN
rate reactions are used, e.g., in mass action models of cellular dynamics [1] and continuous-
flow stirred-tank chemical reactors [9], to model inflow of chemicals from the environment
(correspondingly,theseCRNsoftenalsohavecertainproductsexitthenetwork,reflectedbyamass
lossinsomereactions).ItispossibletohaveanLDPwithoutAssumptionA.1(b),buttheneven
whenstartingatxv → x whichisstrictlypositive,wemayhaveapathoffiniteratethatleads
0 0
to∂Rd andstaysthereforever.ThiswouldcreateproblemsestablishingtheWentzell-Freidlin
+
estimates.
Proceeding to state our sample path LDP, hereafter D0,T (cid:0)Rd+(cid:1) denotes the space of ca`dla`g
functions z : [0,T] → Rd equipped with the topology of uniform convergence. For z(·) in
+
thesubspaceAC (cid:0)Rd(cid:1)ofabsolutelycontinuousfunctionsfrom[0,T]toRd,letz(cid:48)(·)denote
0,T + +
itsRadon-NikodymderivativewithrespecttoLebesguemeasure.Further,forλ = (λ ) ∈ Rm,
r
ξ ∈ Rd andcr ∈ Rd,let
L(λ,ξ) := sup (cid:110)(cid:104)θ,ξ(cid:105)− (cid:88)λ (cid:2)exp((cid:104)θ,cr(cid:105))−1(cid:3)(cid:111)
r
θ∈Rd r∈R
= inf(cid:110)(cid:88)(cid:2)λ −q +q log qr(cid:3) : q ∈ Q (ξ)(cid:111), (1.7)
r r r λ r R
r
r∈R
whereQ (ξ) := {q ≥ 0 : (cid:80) q cr = ξ}and(cid:104)θ,ξ(cid:105)istheinnerproductofθ,ξ ∈ Rd.
R r r∈R r
THEOREM1.6. Forλ (·)of(1.5)andanyxv → x ∈ Rd,underAssumptionA.1thesample
r 0 0 +
paths{Xtv : t ∈ [0,T]}withX0v = xv0,satisfytheLDPinD0,T (cid:0)Rd+(cid:1)withratev andthegood
ratefunction
(cid:40)(cid:82)T L(λ(z(t)),z(cid:48)(t))dt ifz(0) = x &z ∈ AC (cid:0)Rd(cid:1),
I (z) := 0 0 0,T + (1.8)
x ,T
0 ∞ otherwise.
Thatis,foranysetΓ ⊂ D (Rd)wehave
0,T +
limsup 1 logP (cid:2)Xv ∈ Γ¯(cid:3) ≤ −inf I (z),
v→∞ v xv0 t z∈Γ¯ x0,T
1
liminf logP [Xv ∈ Γo] ≥ − inf I (z).
v→∞ v xv0 t z∈Γo x0,T
REMARK 1.7. Theidentity(1.7)iswellknown(see[25,Thm.5.26]),andclearlythenon-
negativeLagrangianL(λ,ξ)of(1.7)iszeroiffξ = (cid:80) λ cr.Hence,therateI (z)of(1.8)
r∈R r x ,T
0
iszeroiffz(·)solvestheODE(1.4)startingatz(0) = x0.
LARGEDEVIATIONSTHEORYFORCHEMICALREACTIONNETWORKS 5
FIGURE1.TheRd+-diagramforExample1.1.Avectorwinthespaceofcomplexesandthecorrespondingorthogonal
hyperplanehasbeendrawninredtoidentifythew-maximalsubsetoftheinputcomplexes(C ) :thecomplexA+B.
in w
1.2. Topologicalstabilityandstronglyendotacticnetworks. Standardlargedeviationstheory
is not directly applicable for proving Theorem 1.6, because we need to deal with jump rates
that are neither bounded away from zero, nor globally Lipschitz continuous. The diminishing
jumpratesattheboundaryarehandledbyadaptingoursystemtotheframeworkofmean-field
interactingparticlesystems,andtherebyapplying[6,Thm.3.9],whereasLemma2.1takescare
ofthelackofglobalLipschitzcontinuitybyemployingLyapunovstabilitytheorytoestablish
exponentialtightness.Indoingso,amostimportantchallengeistophraseastabilitycondition
strong enough for such exponential tightness, and a sufficient condition for escape from the
boundary(inextensionof[26]),thatarebothapplicabletoabroadcollectionofCRNs.
Thisispreciselywhatwedonext,withourtopologicalconditionssummarizedbyAssump-
tionA.2below.Specifically,givenafinitesetQ ⊂ Rd andavectorw ∈ Rd,wecall
Q := {c ∈ Q : (cid:104)w,c−c(cid:48)(cid:105) ≥ 0 forallc(cid:48) ∈ Q} ,
w
thew-maximalsubsetofQandconsiderthefollowingcollectionofCRNs.
DEFINITION1.8. [15]Thenetwork(S,C,R)iscalledstronglyendotacticifforanynon-zero
w ∈ Rd, the set R ⊆ R of reactions such that cr ∈ (C ) contains at least one reaction
w in in w
satisfying(cid:104)w,cr(cid:105) < 0andnoreactionwith(cid:104)w,cr(cid:105) > 0.
ThisclassofCRNsiswellknown(see[15]),andalgorithmstodetermineifanetworkisstrongly
endotacticaredevisedin[18](usingvariantsofthesimplexalgorithm).
Example 1.1 (continued) The network of Example 1.1 is represented as in Fig. 1, where we
identify(C ) bysweepingRd withahyper-planeorthogonaltow ∈ Rd (hereford = 2,drawn
in w +
inred),andtakingthelastpointofC thatsuchhyper-planeintersected.Itiseasytoseethatour
in
specificnetworksatisfiestherequirementsofDef.1.8andisthereforestronglyendotactic.
6 A.AGAZZI,A.DEMBO,J.-P.ECKMANN
Whileinastronglyendotacticreactionnetwork,allreactions“pointinward”withrespecttothe
facesoftheconvexhullofCin (etymologicallyendo-tactic:inward-arranged),ourLDPrequires
addressingthefollowingadditionalboundaryconcept.
DEFINITION1.9. Anon-emptysubsetP ⊆ S iscalledasiphonifeveryreactionr ∈ Rwith
atleastoneoutputfromP alsohassomeinputspeciesfromP.
EXAMPLE 1.10. It is readily checked that the sets P = {A},{A,B} are siphons of the
network
A (cid:42) 2A (cid:42) 3A+2B (cid:42) A,
whereas{B}isnot.
WemakethefollowingassumptiononthetopologicalstructureofCRNs.Wecall(S,C,R) an
AsiphonicStronglyEndotactic(ASE)networkifitsatisfies
ASSUMPTIONA.2. TheCRN(S,C,R)hastheproperties:
(a) Itisstronglyendotactic,asinDef.1.8,
(b) IthasnosiphonP ⊆ S.
REMARK1.11. AssumptionA.2(b)isequivalenttofinding,foranynon-emptyP ⊆ S,some
reaction from R that produces at least one output in P while requiring no input species from
P. When this holds, then, for any state x on the P-boundary of Rd (namely, having x = 0
+ i
for all s ∈ P), there is some reaction of non-vanishing rate that brings the system back to a
i
higher-dimensionalsubspaceofRd.Followingasequenceofsuchjumpsweconcludethatany
+
asiphonicCRNsatisfiesAssumptionA.1(b).
CombiningthefollowingresultwithRemark1.11yieldstheLDPofTheorem1.6fortheASE
networksofAssumptionA.2.
PROPOSITION1.12(ExistenceofaLyapunovfunction). IfthenetworkisASE,thegenerator
L of(1.2)satisfiesAssumptionA.1(a)forthechemicalLyapunov(continuous)function
v
d
(cid:88)
U(x) := d+1+ x (logx −1) : Rd → R . (1.9)
i i + ≥1
i=1
The connection between Lyapunov stability analysis and large deviations rate functions is
an active area of research (see for example [4]). Also, the problem of stability of mass action
kineticssystemshasbeenaddressedin[1,9,15,17]andsufficientconditionsfortheexistence
of a globally attracting steady state for the deterministic dynamics of such systems have been
establishedin[9]and[17].Inparticular,theexistenceofaglobalattractorforacertainclassof
LARGEDEVIATIONSTHEORYFORCHEMICALREACTIONNETWORKS 7
CRNsisprovenin[17]usingthechemicalLyapunovfunctionof(1.9).Theseresultshavebeen
extendedin[15]wherethesamefunctionisusedforshowingtheexistenceofacompactattracting
setforstronglyendotactic CRNs.However,noneofthereferencesabovedealdirectlywiththe
generatorL ,usingthechemicalLyapunovfunctiontoestablishexponentialtailestimatesfor
v
thefinite-timedistributionsofsuchstochasticprocesses,aswedoinSection3(whereweprove
Proposition1.12byverifying(1.6)forthisfunction).
REMARK1.13. Proposition1.12impliesthatitissufficienttocheckasetofalgebraiccondi-
tionstoguaranteetheapplicabilityofaLDPtothedynamicsofCRNs.Thisismostadvantageous
forapplicationsine.g.,biochemistry,wheretypicallyd > 100andquantitativeestimateslike
(1.6)wouldbeprohibitivetocheck.Notefurthermorethatourconditionsdonotdependonthe
reactionrateconstantsk ,whichareoftenverydifficulttodetermine.
r
1.3. Quasi-potential and exit time asymptotic. Following the Wentzell-Freidlin approach,
we utilize our sample path LDP to define the corresponding quasi-potential (as in [11]), and
provideasymptoticanalysisoveraninfinitetimehorizon,forquantitiesofinterestsuchasthe
exittimefromsomedomainD ⊂ Rd,orthetransitiontimebetweendifferentattractorsof(1.4)
+
(asproposedby[12]).Todoso,wefirstassumethatthedomainofinterestD hasthefollowing
mildregularityproperties.
ASSUMPTION A.3. ThecompactD ⊂ Rd istheclosureofitsinterior,withboundary∂D
+
that is a piecewise twice continuously differentiable sub-manifold of Rd. Furthermore, there
+
existsaballB ⊂ D sothatforallx ∈ B andy ∈ D thesetD containsthelinesegmentbetween
xandy.
DEFINITION1.14. Thequasi-potentialbetweenanyx,y ∈ Rd is
+
V (x,y) := inf inf {I (z)} ,
D x,t
t≥0 {z(·)∈D,z(t)=y}
forI (z)ofTheorem1.6.Wesaythatx,y areD-equivalent(denotedx ∼ y),ifV (x,y) =
x,t D D
V (y,x) = 0.Wefurtherdefine
D
V (A,B) := inf V (x,y), ∀A,B ⊆ D,
D D
x∈A,y∈B
anduseV(·,·)forV (·,·).
Rd
+
Recall Remark 1.7 that V (x,y) = 0 iff the solution of (1.4) starting at ζ(0) = x passes
D
through y while remaining within D. Thus, the equivalence x ∼ y reveals the structure of
D
attractorsfor(1.4)withinD,aboutwhichweassumethefollowing.
ASSUMPTIONA.4. [11,ConditionA,§6.2]Thereexist(cid:96)compactsetsK ⊂ D suchthat:
i
(a) everyω-limitsetof(1.4)lyingentirelyinD isfullycontainedwithinoneK ,
i
(b) foranyx ∈ K wehavex ∼ y ifandonlyify ∈ K .
i D i
8 A.AGAZZI,A.DEMBO,J.-P.ECKMANN
WefurtherassumethattheconichullCo{cr} ofvectors{cr} isRd.
r∈R r∈R
SuchK arecalledstableifV(Kδ,(Kδ)c) > 0forδ > 0smallenough(whereBδ denotesthe
i i i
δ-neighborhoodofthesetB inthe(cid:107)·(cid:107) -norm).Themostprobabletransitionsbetween{Kδ}for
1 i
smallδ > 0andv → ∞,defineadeterministicdynamiconthefinitecollectionofstablecompact
sets.Suchdynamiccanbepartitionedintodisjointcycles,witheachcycleπ consistingofasingle
transitivepoint(π = {i})oraperiodicorbitπ = {i → i → ··· → i → i }(c.f.[11,§6.6]for
1 2 j 1
theprecisedefinition).ThankstoAssumptionA.3andA.4,adaptingthemachineryof[11]toour
setup,wetransferinSection4thesamplepathLDPtothefollowingresultaboutthetimeittakes
theCRNtoexitD oracycleπ andtheprobabilitycostofrelevantexitpaths.
THEOREM1.15. [11,Thm.5.1,5.3and6.2,§6]
ConsideraCRNsatisfyingAssumptionA.1andtheprocesst (cid:55)→ Xtv startingatxv0 → x ∈ Do.Let
τ denoteitsexittimefromasetDthatsatisfiesAssumptionA.3andA.4andletτ itsfirsthitting
D π
timeof∪ Kδ foracycleπ ⊆ {1,...,(cid:96)}andsufficientlysmallδ > 0.Then,withnon-random
j∈/π j
M (x), W and W (x,y) as in [11, §6], we have that for any x in a compact F ⊂ Do and
D D D
y ∈ ∂D
1 (cid:104) (cid:105)
lim lim logP (cid:107)Xv −y(cid:107) < δ = W −W (x,y), (1.10)
δ→0v→∞ v xv0 τD 1 D D
1
lim logE [τ ] = W −M (x). (1.11)
v→∞ v xv0 D D D
Furthermore,withC(π) < ∞asin[11,§6.6],anyγ > 0anduniformlyinx ∈ ∪ (K )δ/2,
i∈π i
lim Pxv (cid:2)(cid:12)(cid:12)v−1logτπ −C(π)(cid:12)(cid:12) ≤ γ(cid:3) = 1. (1.12)
v→∞ 0
REMARK 1.16. Notethatmodelsincellbiology[3]usuallyhavesignificantlylargerdimen-
siondthanmanyotherapplicationsofWentzell-Freidlintheory.
2. ProofofTheorem1.6. WestartbyshowingthatAssumptionA.1(a)yieldsexponentially
negligibleexitprobabilityfromthecompactlevelsetsofthefunctionU(·).
LEMMA2.1. Let{Xv}beaMarkovjumpprocesswithgenerator(1.2)andinitialcondition
t
xv ∈ (cid:0)v−1N (cid:1)d.UnderAssumptionA.1(a),thereis,foreveryα,β,γ,afinite(cid:37) ,sothat
0 0 α,β,γ
limsup 1 log(cid:16) sup P (cid:2) sup (cid:107)Xv(cid:107) > (cid:37) (cid:3)(cid:17) ≤ −α. (2.1)
v→∞ v (cid:107)xv(cid:107) ≤γ xv0 t∈[0,eβv] t 1 α,β,γ
0 1
PROOF. Foreach(cid:96)thereisa(cid:37) = (cid:37)((cid:96))sothat{x : U(x) ≤ (cid:96)}isasubsetoftheball
K(cid:101) := {x ∈ Rd : (cid:107)x(cid:107) ≤ (cid:37)}. (2.2)
(cid:37) + 1
LARGEDEVIATIONSTHEORYFORCHEMICALREACTIONNETWORKS 9
Consideringthev-dependentstoppingtimes
σ := inf{t > 0 : Xv ∈/ K(cid:101) }, (2.3)
(cid:37) t (cid:37)
andthestoppedprocessesX(cid:98)v,(cid:37) := Xv ,itfollowsbyMarkov’sinequalitythatforanyT,
t σ ∧t
(cid:37)
(cid:104) (cid:105) (cid:104) (cid:105)
Pxv sup (cid:107)Xtv(cid:107)1 > (cid:37) = Pxv (cid:107)X(cid:98)Tv,(cid:37)(cid:107)1 > (cid:37)
0 0
t∈[0,T]
(cid:104) (cid:105) (cid:104) (cid:105)
≤ Pxv U(cid:0)X(cid:98)Tv,(cid:37)(cid:1) > (cid:96) ≤ (cid:96)−vExv Uv(cid:0)X(cid:98)Tv,(cid:37)(cid:1) ,
0 0
fromwhichweget(2.1)onceweshowthat
s(cid:37)u≥pγlivm→s∞up v1 log(cid:107)xv(cid:107) ≤suγp,T≤eβvExv0 (cid:104)Uv(cid:0)X(cid:98)Tv,(cid:37)(cid:1)(cid:105) < ∞. (2.4)
0 1
Tothisend,asU(·)iscontinuous,sup {U(x)} ≤ eκ forsomeκ = κ(γ) < ∞.Further,
(cid:107)x(cid:107) ≤γ
1
when(cid:107)Xv(cid:107) ≤ (cid:37),theMarkovprocessX(cid:98)v,(cid:37) hasthegeneratorL of(1.2)restrictedtoK(cid:101) and
0 1 t v (cid:37)
is confined for any v ≥ 1 to a compact (K(cid:101) )c¯ with c¯ := sup (cid:107)cr(cid:107) < ∞. Thus, combining
(cid:37) r 1
Dynkin’sformula[7,§5.1]withAssumptionA.1(a)wefindthatforsomev ∈ [1,∞),allv > v
(cid:37) (cid:37)
and(cid:107)xv(cid:107) ≤ γ ≤ (cid:37),
0 1
(cid:34) (cid:35)
(cid:104) (cid:105) (cid:90) σ ∧T
Exv Uv(X(cid:98)Tv,(cid:37)) ≤ Uv(xv0)+Exv (cid:37) (LvUv)(Xsv)ds ≤ eκv +Tebv. (2.5)
0 0
0
ConsideringforT ≤ eβv thelimitasv → ∞ofv−1 timesthelogarithmof(2.5)leadsto(2.4)
andtherebyconcludestheproof.
REMARK 2.2. Lemma2.1canbealternativelyprovedbydefiningasuper-martingalefrom
thecondition(1.6)onourgenerator,andapplying[10,Thm.4.20]toit.
The Markov jump process Xv corresponds to the generator of (1.2), now with reaction
tT
constantsTk forwhichAssumptionA.1continuestohold.Thischangesλ(·)of(1.5)toTλ(·),
r
hencetransformsIx ,T(z(t))intoIx ,1(z(tT))(sinceL(Tλ,y) = TL(λ,T−1y)).Thus,WLOG,
0 0
wetakehereafterT = 1andproceedtoestablishtheexponentialtightnessofanexponentially
equivalentprocessX(cid:101)v.
t
LEMMA 2.3. Under Assumption A.1(a), the C (Rd)-valued processes X(cid:101)v obtained by
0,1 + t
linearly interpolating the jump points of t (cid:55)→ Xv, form an exponentially tight family in the
t
uniform topology, which for uniformly bounded (cid:107)xv(cid:107) is further exponentially equivalent to
0 1
{Xv}intheuniformtopologyonD (Rd).
t 0,1 +
PROOF. ForanyconsecutivejumpsofXv occurringat(random)timest < t weset
t 1 2
t−t
X(cid:101)v := Xv + 1 (Xv −Xv).
t t1 t −t t2 t1
2 1
10 A.AGAZZI,A.DEMBO,J.-P.ECKMANN
Hence,(cid:107)Xv−X(cid:101)v(cid:107) ≤ v−1c¯forfinitec¯:= sup (cid:107)cr(cid:107) ,allt ≥ 0,andv,yieldingtheexponential
t t 1 r 1
equivalenceof{X(cid:101)v}and{Xv}(intheuniformtopology).Asfortheexponentialtightnessof
t t
{X(cid:101)v}inC (Rd),notethatforanyt > s,
t 0,1 +
(cid:107)X(cid:101)v −X(cid:101)v(cid:107) ≤ v−1c¯N (Xv),
t s 1 [s,t]
where N (Xv) counts the number of jumps by Xv in the time interval [s,t]. Further, as
[s,t] ·
Λ(v)(x) ≤ λ (x)forallx ∈ Rd,wehaveforσ of(2.3)andanyv ≥ 1themonotonecoupling
r r + (cid:37)
N (Xv) ≤ M(cid:37) on[0,σ ],whereM(cid:37) isaPoissonprocessofintensity
[s,t] [s,t] (cid:37)
(cid:110)(cid:88) (cid:111)
Λ¯(cid:37) := sup λ (x) .
r
x∈(K(cid:101)(cid:37))c¯ r∈R
InviewoftheArzela`-AscolitheoremandLemma2.1,itthussufficesforthestatedexponential
tightnessof{X(cid:101)v}toshowthat
t
(cid:34) (cid:35)
limlimsupv−1logP sup {M(cid:37) } ≥ vε = −∞, ∀(cid:37) < ∞,ε > 0. (2.6)
[s,t]
δ→0 v→∞ 0≤s≤t≤(s+δ)∧1
Tothisend,bytailestimatesforthePoisson(2δΛ¯(cid:37))law,foranyε > 0and(cid:37) < ∞,
(cid:104) (cid:105)
limlimsupv−1logP M(cid:37) ≥ vε = −∞.
[0,2δ]
δ→0 v→∞
Further,if|t−s| ≤ δ andn = [1/δ],then
M(cid:37) ≤ max {M(cid:37) } =: M¯(cid:37).
[s,t] [iδ,(i+2)δ] δ
i=0,...,n−1
Hence,applyingtheunionboundforthemaximumM¯(cid:37)ofnidenticallydistributedPoisson(2δΛ¯(cid:37))
δ
variablesyields(2.6),andtherebyconcludestheproof.
LetM (S )denotetheprobabilitysimplexoverS = {(cid:63)}∪S andcr := (cid:104)11,cr(cid:105) = (cid:107)cr (cid:107) −
1 (cid:63) (cid:63) (cid:63) out 1
(cid:107)cr (cid:107) thenumberofmoleculesgained(orlost,ifnegative)ineachchemicalreaction.For{λ (·)}
in 1 r
of(1.5)and(cid:37) > 0weconsiderµ(t)satisfyingtheODE
dµ (cid:88)
= (cid:37)−1 λ ((cid:37)µ| )(−cr,cr), µ(0) ∈ M (S ), (2.7)
dt r S (cid:63) 1 (cid:63)
r∈R
establishingastrictlypositivelowerboundon{µ(t)| }thatholdsuniformlyoverµ(0)| ≤ γ/(cid:37)
S S
witharbitrary,fixedγ andall(cid:37)largeenough.
LEMMA 2.4. Under Assumption A.1 there exist D ∈ N and b = b((cid:37)) > 0 such that any
solutionof(2.7)satisfies
µ (t) ≥ btD, ∀t ∈ [0,1], i = 1,...,d. (2.8)
s
i
Further,forsome(cid:37) (γ),if(cid:37) ≥ (cid:37) (γ)and(cid:37)µ(0)| ≤ γ,thenµ(t) ∈ M (S )fort ∈ [0,1].
0 0 S 1 (cid:63)