Quantitative evaluation of an active Chemotaxis model in Discrete time

0.53 MB·

by  Abhishek Pal Majumder

Quantitative evaluation of an active Chemotaxis model in Discrete
time.
Abhishek Pal Majumder ∗
January 10, 2017
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0
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Abstract
a
J
A system of N particles in a chemical medium in Rd is studied in a discrete time setting. Underlying
9
interacting particle system in continuoustime can beexpressed as

R] dXi(t) = [−(I−A)Xi(t)+▽h(t,Xi(t))]dt+dWi(t), Xi(0)=xi∈Rd ∀i=1,...,N
N
P ∂ β
. ∂th(t,x) = −αh(t,x)+D△h(t,x)+ nXg(Xi(t),x), h(0,·)=h(·). (0.1)
h i=1
t
a where Xi(t) is the location of the ith particle at time t and h(t,x) is the function measuring the
m
concentration of the medium at location x with h(0,x) = h(x). In this article we describe a general
[ discrete time non-linear formulation of the model (0.1) and a strongly coupled particle system ap-

proximating it. Similar models have been studied before (Budhiraja et al.(2010)) under a restrictive
1 compactness assumption on the domain of particles. In current work the particles take values in Rd
v
and consequently the stability analysis is particularly challenging. We provide sufficient conditions
4
6 for the existence of a uniquefixed point for the dynamical system governing the large N asymptotics
0 of the particle empirical measure. We also provide uniform in time convergence rates for the particle
2 empirical measure to thecorresponding limit measure undersuitable conditions on the model.
0 AMS 2010 subject classifications: Primary 60J05, 60K35, 60F10.
.
1
0 Keywords: Weakly interacting particle system, propagation of chaos, nonlinear Markov chains,
7 Wasserstein distance, McKean-Vlasov equations, exponential concentration estimates, transportation
1 inequalities,metricentropy,stochasticdifferenceequations,longtimebehavior,uniformconcentration
: estimates.
v
i
X
r 1 Introduction
a
There have been a surge of significant research activities aimed towards understanding the dynamics of
collective behavior of a multi-agent system in the time limit. Motivations for such problems come from
various examples of self organizing systems such as consensus formation in opinion dynamics [11], active
chemotaxis [3], self organized networks [13], large communication systems [12], multi target tracking [6],
swarm robotics [14] (additional applications can be found in [15]) etc. One of the basic challenges is to
understand how a large group of autonomous agents with decentralized local interactions that gives rise
to a coherent behavior.
In this paper we consider a reduced model motivated by both [3],[5] for a system of interacting agents
in a stochastic diffusing environment, variations of which have been proposed (see [3],[14] and references
∗UniversityofCopenhagen
1
therein). Consider for each i=1,...,N X (0)=x Rd
i i
∈
N
1
dX (t) = (I A)X (t)+ h(t,X (t))+ K X (t),X (t) dt+dW (t), (1.1)
i − − i ∇ i N i j i
(cid:20) j=1,j=i (cid:21)
X6 (cid:0) (cid:1)
N
∂ β
h(t,x) = αh(t,x)+D h(t,x)+ g(X (t),x), h(0, )=h().
∂t − △ N i · ·
i=1
X
HereW ,i=1,...,N areindependentBrownianmotionsthatdrivethestateprocessX oftheN interacting
i i
particles. The interaction between the particles arises directly from the evolution equation (1.1) and
indirectly through the underlying potential field h which changes continuously according to a diffusion
equation and through the aggregated input of the N particles. One example of such an interaction is in
Chemotaxiswherecellspreferentiallymovetowardsahigherchemicalconcentrationandthemselvesrelease
chemicals into the medium, in response to the local information on the environment, thus modifying the
potential field dynamically over time. In this context, h(t,x) represents the concentration of a chemical
at time t and location x. Diffusion of the chemical in the medium is captured by the Laplacian in (1.1)
and the constant α > 0 models the rate of decay or dissipation of the chemical. The first equation in
(1.1) describes the motion of a particle in terms of diffusion process with a drift consisting of three terms.
The first term models a restoring force towards the origin where origin represents the natural rest state
of the particles. The second term is the gradient of the chemical concentration and captures the fact that
particles tend to move particularly towards regions of higher chemical concentration. Finally the third
termcapturestheinteraction(e.gattractionorrepulsion)betweentheparticles. Contributionoftheagents
to the chemical concentration field is given through the last term in the second equation. The function g
captures the agent response rules and can be used to model a wide range of phenomenon [15].
In[3]theauthorsconsideredadiscretetimemodelwhichcapturessomeofthekeyfeaturesofthedynamics
in (1.1) and studied several long time properties of the system. One aspect that greatly simplified the
analysis of [3] is that the state space of the particles is taken to be a compact set in Rd. However this
requirement is restrictive and may be unnatural for the time scales at which the particle evolution is
being modeled. In [14] authors had considered a number of variations of (1.1). The theoretical properties
obtainedinthisworkonthelongtimebehavioroftheparticlesystemcanbealsoappliedforsuchsystems
with some minor modifications.
We now give a general description of the N- particle system that gives a discrete time approximation of
the mechanismoutlinedabove. The spaceofrealvaluedboundedmeasurablefunctions onS isdenotedas
BM(S). Borel σ field on a metric space will be denoted as (S). (S) denotes the space of all bounded
b
and continuous functions f :S R. For a measurable spacBe S, (CS) denotes the space of all probability
measures on S. For k N, let →(Rd) be the space of µ (Rd)Psuch that
k
∈ P ∈P
1
k
µ := xkdµ(x) < .
k
k k | | ∞
(cid:18)Z (cid:19)
Consider a system of N interacting particles that evolve in Rd governed by a random dynamic chemical
fieldaccordingtothefollowingdiscretetimestochasticevolutionequationgivenonsomeprobabilityspace
(Ω,F,P). Suppose that the chemical field at time instant n is given by a nonnegative C1(i.e continuously
differentiable)realfunctiononRd satisfying η(x)dx=1. Then,giventhatparticlestateattimeinstant
Rd
n isxandthe empiricalmeasureofthe particlestatesattime nis µ,the particlestate X+ attime (n+1)
R
is given as
X+ =Ax+δf( η(x),µ,x,ǫ)+B(ǫ), (1.2)
∇
where A is a d d matrix, δ is a small parameter, ǫ is a Rm valued random variable with probability law
θ and f : Rd × (Rd) Rd Rm Rd is a measurable function. Here we consider a somewhat more
×P × × −→
general form of dependence of the particle evolution on the concentration profile than the additive form
thatappearsin(1.1). AdditionalassumptionsonA,θ,f willbeintroducedshortly. Nonlinearity(modeled
byf andB)ofthesystemcanbeverygeneralandasdescribedbelow. DenotebyXi Xi,N (aRd valued
n ≡ n
random variable) the state of the i-th particle (i = 1,...,N) and by ηN the chemical concentration field
n
2
at time instant n. Let µN := 1 N δ be the empirical measure of the particle values at time instant
n N i=1 Xni
n. The stochastic evaluation equation for the N-particle system is given as
P
Xi = AXi +δf( ηN(Xi),µN,Xi,ǫi )+B(ǫi ), i=1,...,N, n N . (1.3)
n+1 n ∇ n n n n n+1 n+1 ∈ 0
In(1.3) ǫi,i=1,...,N, n 1 isani.i.darrayofRm valuedrandomvariableswithcommonprobability
{ n ≥ }
law θ. Here Xi,i = 1,...,N are assumed to be exchangeable with common distribution µ where
µ (Rd). N{ot0e that in the n}otation we have suppressed the dependence of the sequence Xi 0on N.
0 ∈P1 { n}
We now describe the evolution of the chemical field approximating the second equation in (1.1) and its
interaction with the particle system. A transition probability kernel on S is a map P : S (S) [0,1]
×B →
such that P(x, ) (S) x S and for each A (S), P(,A) BM(S). Given the concentration
profile at time n· i∈s aPC1 pr∀oba∈bility density functio∈n ηBon Rd a·nd t∈he empirical measure of the state of
N-particlesattime instantn is µ, the concentrationprobabilitydensity η+ attime (n+1)is givenby the
relation
η+(y)= η(x)Rα(x,y)l(dx) (1.4)
µ
ZRd
where l denotes the Lebesgue measure on Rd, and Rα(x,y) is the Radon-Nikodym derivative of the tran-
µ
sition probability kernel with respect to the Lebesgue measure l(dy) on Rd. The kernel Rα is given as
µ
follows. We considered the same model as introduced in [3]. Let P and P betwo transition probability
′
kernels on Rd. For µ (Rd) and α (0,1) define the transition probability kernel Rα on Rd as
∈P ∈ µ
Rα(x,C):=(1 α)P(x,C)+αµP (C), x Rd,C (Rd).
µ − ′ ∈ ∈B
Here P represents the backgrounddiffusion of the chemical concentration while δ P captures the contri-
x ′
bution to the field by a particle with location x. So the kernel P gives a spike at origin which can be
′
approximated by a smooth density function as P(x,dy) = 1 e−(x2−λy2)2dy with very small λ > 0. The
√2πλ
parameterαgivesaconvenientwayforcombiningthe contributionfromthe backgrounddiffusionandthe
individual particles. For each x Rd, both P(x, ) and P (x, ) are assumed to be absolutely continuous
′
∈ · ·
with respect to Lebesgue measure and throughout this article we will denote the corresponding Radon-
Nikodym derivatives with the same notations P(x, ) and P (x, ) respectively. Additional properties of P
′
· ·
and P will be specified shortly. The evolution equation for the chemical field is then given as
′
ηN (y)= ηN(x)Rα (x,y)l(dx). (1.5)
n+1 n µN
ZRd n
Incontrasttothemodelstudiedin[5],thesituationhereissomewhatmoreinvolved. Notethat X (N) :=
n n 0
(X1,N,X2,N,...,XN,N) is not a Markovprocess and in order to get a Markovianstate de{scriptor}on≥e
n n n n 0
needs to consider X (N)≥,ηN which is a discrete time Markov chain with values in (Rd)N (Rd).
{ n n}n≥0 ×P
We will show that as N → ∞ (µNn,ηnN)n∈N0 converges to a deterministic nonlinear dynamical system
(µn,ηn)n∈N0 with methods followed in [3]. We established further sharp quantitative bounds (with tech-
niquesusedin[10]and[5])forweaklyinteractingparticlesystemjointlywiththe stochasticfieldpotential
to the nonlinear system of interest. For both polynomial and exponential concentration bound it requires
further constraints on the tail of the transition kernels P,P used in modeling the diffusive environment.
′
One major motivation of cthe current article is giving a sharp uniform in time quantitative estimate for
theparticlesystem(µN,ηN)tothenon-linearsystemofinterest(µ ,η )sothatanyfunctionaloftheform
n n n n
φ ,µ + φ ,η can be approximated by 1 N φ (Xi)+ φ ,ηN with desired precision. Previous
1 n 2 n N i=1 1 n 2 n
work on concentration bounds for similar particle system in discrete time includes [8] but that involves a
(cid:10) (cid:11) (cid:10) (cid:11) P (cid:10) (cid:11)
Dobrushin type stability condition which is not very effective if the particles are assumed to come from a
non-compactdomain. A veryrecentwork[4]addressesseveralquantitativebounds forChemotaxis model
motivated by Patlak-keller-segeltype non-linear equations.
Thefollowingnotationswillbeusedinthis article. Rd willdenotethe ddimensionalEuclideanspacewith
theusualEuclideannorm . Thesetofnaturalnumbers(resp. wholenumbers)isdenotedbyN(resp. N ).
0
Cardinalityofa finite set|S·|is denotedby S . Forx Rd, δ is the Diracdelta measureonRd thatputs a
x
| | ∈
3
unit mass at location x. The supremum norm of a function f :S R is f =sup f(x). When S
isametricspace,theLipschitzseminormoff isdefinedbykfk1 =→supx6=yk|f(kdx∞()x−,fy)(y)| wxh∈eSre| dis|themetric
onthe spaceS. ForaboundedLipschitzfunctionf onS wedefine f := f + f . Lip (S)(resp.
BL (S) ) denotes the class of Lipschitz (resp. bounded Lipschitz)kfuknBctLionskf k:1S k Rk∞with 1f (resp.
1 1
→ k k
f ) bounded by 1. Occasionally we will suppress S from the notation and write Lip and BL when
k kBL 1 1
clear from the context. For a Polish space S, (S) is equipped with the topology of weak convergence. A
P
convenientmetricmetrizingthistopologyon (S)isgivenasβ(µ,γ)=sup fdµ fdγ : f 1
for µ,γ (S). For a signed measure γ onPRd, we define f,γ := fd{γ|whene−ver the|intkegkrBalL1m≤ake}s
sense. T∈hePspace (Rd) will be equipped with the Wassersthein-1idistance thRat is deRfined as follows:
1
P R
(µ ,γ ):= inf E X Y , µ ,ν (Rd),
1 0 0 0 0 1
W X,Y | − | ∈P
where the infimum is taken over all Rd valued random variables X,Y defined on a common probability
space and where the marginals of X,Y are µ and γ respectively. From Kantorovich-Rubensteinduality
0 0
(cf. [17]) one sees the Wasserstein-1 distance has the following characterization
(µ ,γ )= sup f,µ γ , µ ,ν (Rd). (1.6)
1 0 0 0 0 0 0 1
W |h − i| ∈P
f Lip (Rd)
∈ 1
Fora signedmeasureµ on(S, (S)),the totalvariationnormofµ is definedas µ :=sup f,µ .
Probability distribution of a BS valued random variable X will be denoted a|s|TV(X). Co||nfv||e∞r≤ge1nhce iin
L
distributionofaS valuedsequence X toaS valuedrandomvariableX willbewrittenasX X.
n n 1 n
{ } ≥ ⇒
A finite collection Y ,Y ,...,Y of S valued random variables is called exchangeable if
1 2 N
{ }
(Y ,Y ,...,Y )= (Y ,Y ,...,Y )
1 2 N π(1) π(2) π(N)
L L
for every permutation π on the N symbols 1,2,...,N . Let YN,i = 1,...,N be a collection
of S valued random variables, such that for{every N, }YN,YN{,.i..,YN is excha}nNg≥e1able. Let ν =
{ 1 2 N } N
(YN,YN,...,YN). Thesequence ν iscalledν -chaotic(cf. [16])foraν ( ),ifforanyk 1,
L 1 2 N { N}N≥1 ∈P S ≥
f ,f ,...,f ( ), one has
1 2 k b
∈C S
k
lim f f ... f 1... 1,ν = f ,ν . (1.7)
1 2 k N i
N h ⊗ ⊗ ⊗ ⊗ ⊗ i h i
→∞ i=1
Y
Denoting the marginaldistribution on first k coordinates of ν by νk, equation (1.7) says that, for every
N N
k 1, νk ν k. The gradient of a real differentiable function f on Rd denoted by f is defined as the
≥ N → ⊗ ∇
d dimensional vector field f :=( ∂f , ∂f ,..., ∂f ). For a function f :Rd Rm R
∇ ∂x1 ∂x2 ∂xd ′ × →
∂f ∂f ∂f ′
f(x,y):= , ,..., .
∇x ∂x ∂x ∂x
(cid:18) 1 2 d(cid:19)
Thefunction f(x,y)isdefinedsimilarly. Absolutecontinuityofameasureµwithrespecttoameasureν
y
∇
willbedenotedbyµ ν.WewilldenotetheRadon-Nikodymderivativeofµwithrespecttoν by dµ. For
≪ dν
f BM( ) and a transition probability kernel P on S, define Pf BM( ) as Pf() = f(y)P(,dy).
∈ S ∈ S · S ·
ForanyclosedsubsetB S,andµ (B),defineµP (S)asµP(A)= P(x,A)µ(dx). Foramatrix
∈ ∈P ∈P B R
B the usual operator norm is denoted by B .
k k R
2 Description of the nonlinear system:
Wenowdescribethe nonlineardynamicalsystemobtainedontakingthelimitN of(µN,ηN). Given
a C1 density function ρ on Rd and µ (Rd), define a transition probability ker→nel∞Qρ,µ onnRdnas
∈P
Qρ,µ(x,C)= 1 θ(dz), (x,C) Rd (Rd).
Ax+δf( ρ(x),µ,x,z)+B(z) C
ZRm { ∇ ∈ } ∈ ×B
4
With an abuse of notation we will also denote by Qρ,µ the map from BM(Rd) to itself, defined as
Qρ,µφ(x)= φ(y)Qρ,µ(x,dy), φ BM(Rd),x Rd.
ZRd ∈ ∈
For µ,µ1 (Rd), let µQρ,µ1 (Rd) be defined as
∈P ∈P
µQρ,µ1(C)= Qρ,µ1(x,C)µ(dx), C (Rd). (2.1)
ZRd ∈B
Note that µQρ,µ1 = AX +δf( ρ(X),µ1,X,ǫ)+B(ǫ) where (X,ǫ)=µ θ.
L ∇ L ⊗
Define (Rd):= µ (cid:0) (Rd):µ l,dµ iscontinuously(cid:1)differentiableand dµ < .Fornotational
simplicPity1∗ we will{ide∈ntPify1 an elem≪ent idnl (Rd) with its density and denokt∇e bdolkth1 by∞th}e same symbol.
Define the map Ψ: (Rd) (Rd) (PR1∗d) (Rd) as
P ×P1∗ →P ×P
Ψ(µ,η) = (µQη,µ,ηRα), (µ,η) (Rd) (Rd). (2.2)
µ ∈P ×P1∗
Under suitable assumptions (which will be introduced in Section 3) it will follow that for every (µ,η)
(Rd) (Rd), η+ defined by (1.4) is in (Rd) and µQη,µ defined by (2.1) is in (Rd). Thu∈s
(Pu1nder t×hoPse1∗assumptions) Ψ is a map from P(1∗Rd) (Rd) to itself. Using the above nPo1tation we see
that (X1,...,XN),µN,ηN is a (Rd)N P1(Rd)×P1∗(Rd) valued discrete time Markov chain defined
recur{sivelny as fonllowsn. Lnet}nX≥0(N) (X1,×XP2,1...,XN×)P,1∗and ηN be the initial chemical field which is a
random element of (Rd). Lekt =≡σ Xk (Nk),ηN .kThen, for0k 1
P1∗ F0 { 0 0 } ≥
P(Xk(N)∈C|FkN−1)= Ni=1(δXkj−1QηkN−1,µNk−1)(C) ∀C ∈B(RdN),
µηkNNk ==ηN1kN−P1RNi=µα1Nk−δ1X,ki, N (2.3)
N =σ ηN,X (N) N .
We will call thisFpkarticle{sykstemk as I}P∨SF.k−W1e next describe a nonlinear dynamical system which is the
1
formal Vlasov-Mckean limit of the above system, as N . Given (µ ,η ) (Rd) (Rd) define a
sequence (µ ,η ) in (Rd) (Rd) as →∞ 0 0 ∈P1 ×P1∗
{ n n }n≥0 P1 ×P1∗
µ =µ Qηn,µn, η =η Rα , n 0. (2.4)
n+1 n n+1 n µn ≥
Using (2.2) the above evolution can be represented as
(µ ,η )=Ψ(µ ,η ), n N . (2.5)
n+1 n+1 n n 0
∈
As in [5],the starting pointofour investigationonlongtime asymptoticsof the aboveinteracting particle
system will be to study the stability properties of (2.4). We identify η,η (Rd) that are equal a.e
′
under the Lebesgue measure on Rd. From a computational point of view we∈arPe approximating (µ ,η )
n n
by (µN,ηN) uniformly in time parameter n, with explicit uniform concentrationbounds. Such results are
n n
particularly important for developing sampling methods for approximating the steady state distribution
of the mean field models such as in (2.4).
The third equation in (2.3) makes the simulation of IPS numerically challenging. In section 3 we will
1
mention another particle system (based on the second particle system in [3]) referred to as IPS which
2
also gives an asymptotically consistent approximation of (2.4) and is computationally more tractable.
We show in THeorem 3.2 that under conditions that include a Lipschitz property of f (Assumptions 1
and 2), smoothness assumptions on the transition kernels of the background diffusion of the chemical
medium (Assumption 4) the Wasserstein-1( ) distance between the occupation measure of the particles
1
W
along with the chemicalmedium (µN,ηN) and (µ ,η ) convergesto 0, for everytime instantn. Under an
n n n n
additional condition on the contractivity of A and δ,α being sufficiently small we show that the nonlinear
system (2.5) has a unique fixed point and starting from an arbitrary initial condition, convergence to the
fixed point occurs at a geometric rate. Using these results we next argue in Theorem 1 that under some
integrabilityconditions(Assumption7-8),asN ,theempiricaloccupationmeasureoftheN-particles
→∞
5
and density of the chemical medium at time instant n, namely (µN,ηN) converges to (µ ,η ) in the
n n n n W1
distance, in L1, uniformly in n. This result in particular shows that the distance between (µN,ηN)
W1 n n
and the unique fixed point (µ ,η ) of (2.5) converges to zero as n and N in any order.
We next show that for each N∞, th∞ere is unique invariant measure ΘN→of∞the N-par→ticle∞dynamics with
integrable first moment and this sequence of measures is µ -chaotic,∞namely as N , the projection
of ΘN on the first k-coordinates convergesto µ k for every∞k 1. This propagation→of∞chaos property all
⊗
the w∞ay to n = crucially relies on the unifo∞rm in time con≥vergence of (µN,ηN) to (µ ,η ). Such a
∞ n n ∞ ∞
result is important since it says that the steady state of a N-dimensional fully coupled Markoviansystem
has asimple approximatedescriptioninterms ofaproduct measurewhenN is large. This resultis key in
developing particle based numerical schemes for approximating the fixed point of the evolution equation
(2.5). Next we present some uniform in time concentrationbounds of (µN,µ )+ (ηN,η ). Proofis
W1 n n W1 n n
very similar to that of Theorem 3.8 of [5] so we only provide a sketch after showing necessary conditions.
3 Main Results:
We now introduce our main assumptions on the problem data. Recall that Xi,i = 1,...N is assumed
to be exchangeable with common distribution µ . We assume further (µ ,η{) 0 (Rd) }(Rd). For a
0 0 0 ∈ P1 ×P1∗
d×d matrix B we denote its norm by kBk, i.e. kBk=supx∈Rd\{0} |B|xx||.
Assumption 1 The error distribution θ is such that A (z)θ(dz):=σ (0, ) where
1
∈ ∞
R f(y ,µ ,x ,ǫ) f(y ,µ ,x ,ǫ)
A (ǫ) := sup | 1 1 1 − 2 2 2 | . (3.1)
1 {x1,x2,y1,y2∈Rd,µ1,µ2∈P1(Rd):µ16=µ2,x16=x2,y16=y2}|x1−x2|+|y1−y2|+W1(µ1,µ2)
It follows that x,y Rd,µ (Rd),
1
∀ ∈ ∈P
f(y,µ,x,ǫ) (y + µ + x)A (ǫ)+A (ǫ) (3.2)
1 1 2
| |≤ | | k k | |
where A (ǫ):=f(0,0,ǫ).
2
Recall the function B :Rm Rd introduced in (1.2).
→
Assumption 2 The error distribution θ is such that
A (z)+ B(z) θ(dz)< .
2
ZRm(cid:16) | |(cid:17) ∞
Assumption 3 ηN (the density function) is a Lipschitz function on Rd and ηN (Rd) .
0 0 ∈P1∗
Assumptions 4 and 5 on the kernels P and P hold quite generally. In particular, they are satisfied for
′
Gaussian kernels.
Assumption 4 There exist l (0,1] and l (0, ) such that for all x,y,x,y Rd
P∇ ∈ P∇′ ∈ ∞ ′ ′ ∈
P(x,y) P(x,y ) l (y y + x x ) (3.3)
|∇y −∇y ′ ′ | ≤ P∇ | − ′| | − ′|
|∇yP′(x,y)−∇yP′(x′,y′)| ≤ lP∇′(|y−y′|+|x−x′|). (3.4)
Furthermore
sup P(x,0) P (x,0) < . (3.5)
y y ′
x Rd{|∇ |∨|∇ |} ∞
∈
Using the Lipschitz property in (3.3) and the growth condition (3.6) one has the linear growth property for
some M (0, )
P∇ ∈ ∞
supx∈Rd|∇yP(x,y)|≤MP∇(1+|y|). (3.6)
A similar inequality holds for P from (3.4) with M (0, ).
′ P∇′ ∈ ∞
6
Denote (1−α)lP∇+αlP∇′ by lP∇P,α′.
Assumption 5 For every f Lip (Rd), Pf and P f are also Lipschitz and
∈ 1 ′
Pf(x) Pf(y)
sup sup − :=l(P)<
f∈Lip1(Rd)x6=y∈Rd |x−y| ∞
Also l(P ) defined as above for P is finite.
′ ′
Assumption 6 Both P(x, ) and P (x, ) are such that for any compact set K Rd, the families of
′
· · ⊂
probability measures P(x, ):x K and P (x, ):x K are both uniformly integrable.
′
{ · ∈ } { · ∈ }
Let max l(P),l(P ) =l .
′ PP′
{ }
Remark 3.1 Assumption 5 is satisfied if P,P are given as follows. For any f (Rd), let
′ b
∈C
Pf():=Ef(g (,ε )), P f():=Ef(g (,ε )) (3.7)
1 1 ′ 2 2
· · · ·
where ε ,ε are Rm valued random variables and ε ,ε and g ,g :Rd Rm Rd are maps with following
1 2 1 2 1 2
× →
properties:
E(G (ε )) l(P) and E(G (ε )) l(P ), (3.8)
1 1 2 2 ′
≤ ≤
where
g (x ,y) g (x ,y) g (x ,y) g (x ,y)
G (y):= sup 1 1 − 1 2 and G (y):= sup 2 1 − 2 2 . (3.9)
1 x x 2 x x
x16=x2 | 1− 2| x16=x2 | 1− 2|
Simulation of the system is numerically intractable due to the step that involves the updating of ηN to
n 1
ηN. This requires computing the integral in (1.4) which, since Rα is a mixture of two transition ker−nels,
n µ
over time leads to an explosion of terms in the mixture that need to be updated. An approach (proposed
in [3]) that addressesthis difficulty is, without directly updating ηN , to use the empiricaldistributionof
n 1
the observations drawn independently from ηN . −
n 1
−
Denote X¯ (N) by (X¯1,N,...,X¯N,N) a sample of size N from µ . Let M N. The new particle scheme
wsoimllebeprdoebs0acrbiiblietdyasspaac0feamdeifilyne(Xd¯kr0e(Ncu)r,sµ¯ivNkel,yη¯kMas)kfo∈lNlo0wosf.(RSedt)NX¯×(PN()R0=d)×(X¯P1∗,N(R∈,.d.).v,aX¯luNe,dNr),aηn¯Mdom=eηle,m¯eMnt,Nso=n
σ(X¯N(0)). For k 1 0 0 0 0 0 F0
≥
µ¯Nk = N1 Ni=1δX¯ki,
Pη¯¯kM(MX¯,=Nk((N=1)P−σ∈αη¯C)M(|SF,XMk¯M−(,(η1¯NNkM)−)=1)PN)¯Ni+=M1α,(Nδµ¯XNk¯−kj−11PQ′,η¯kM−1,µ¯Nk−1)(C) ∀C ∈B(Rd)N, (3.10)
Fk { k k }∨Fk 1
where SM(η¯M) is the random measure defin−ed as 1 M δ where Yi,M conditionally on
k−1 M i=1 Yki−,M1 { k−1}i=1,...,M
Ft¯hkMa−t,1No,urarneotMatioi.ni.disdniosttraibcucuterdataecscionrcdeinbgotthotηh¯kMe−q1u.aWnteitPwieisllµ¯cNa,llη¯MthidseppaerntdicolensMys,tNem. TahseIPsuSp2e.rsWcreiprtesmoanrlyk
k k
describethenumberofparticles/samplesusedintheproceduretocombinethem. NotethatlikeIPS ,here
1
(X¯ (N),η¯M) is not a Markovchainon(Rd)N (Rd) anymore. Rather (X¯N(k),η¯M,SM(η¯M)) is
a dkiscretektimke≥0Markov chain on (Rd)N (Rd)×P1∗ (Rd). k k k≥0
×P1∗ ×P1
For any randomvariableZ we denote E Z M,N by EM,N Z . The followingresult showsthat the par-
Fk k
ticlesystemsin(2.3)and(3.10)approximatethedynamicalsystemin(2.4)asN (respectivelymin M,N
for IPS ) becomes large for a fixed time(cid:2)ins(cid:12)(cid:12)tant. (cid:3) (cid:2) (cid:3) { }
2
Proposition 3.2 Suppose Assumptions 1,2,4 and 5 hold.
7
(a) Consider the particle system IPS in (1.3,1.5). Suppose the sampling of the exchangeable data-
1
Epoints(ηXN,0η(N)) ≡0(aXs01N,X02,....,TXh0Nen),iasseNxchangeable and {L(X0(N))}N∈N is µ0- chaotic. Suppose
W1 0 0 → →∞ →∞
E (µN,µ )+ (ηN,η ) 0 (3.11)
W1 n n W1 n n →
for all n 0 where µ ,η are as(cid:2)in (2.4). (cid:3)
n n
≥
(b) Consider the second particle system IPS . Suppose that in addition Assumption 6 holds. Suppose the
2
sisamµp-linchgaooftitch.eTexhcehnanagseambilnedNat,aMpointsX¯0(N)≡(X¯01,X¯02,...,X¯0N)isexchangeableand{L(X¯0(N))}N∈N
0
{ }→∞
E (µ¯N,µ )+ (η¯M,η ) 0 (3.12)
W1 n n W1 n n →
for all n 0. (cid:2) (cid:3)
≥
As a consequence of Proposition 3.2, we have a finite time propagation of chaos result of the following
form. Let νN = (X1,N,X2,N,...,XN,N).
n L n n n
Corollary 3.3 Under Assumptions as in Proposition 3.2 the family νN is µ chaotic for every
{ n }N≥1 n
n 1.
≥
As noted in introduction, the primary goal is studying long time properties of (1.3) and the non-linear
dynamical system (2.4). Following proposition identifies the range of values of the modeling parameters
that leads to stability of the system.
Proposition 3.4 Suppose Assumptions (1)-(5) hold. Then there exist ω ,α ,δ (0,1) such that for all
0 0 0
∈
A < ω ,α (0,α ), and δ (0,δ ). The map Ψ defined in (2.2) has a unique fixed point (µ ,η ) in
0 0 0
k (kRd) (∈Rd). ∈ ∞ ∞
P1 ×P1∗
Now we will givemore stringrentconditions under whicha non-asymptotic boundon convergenceratesof
the particle system to the deterministic nonlinear dynamics and their consequences for the steady state
behavior can be established.
Assumption 7 For some τ >0,
1+τ
µ (Rd), A (z)1+τθ(dz):=σ (τ)< A (z)+ B(z) θ(dz):=σ (τ)< .
0 1+τ 1 1 2 2
∈P ∞ | | ∞
Z Z (cid:16) (cid:17)
We need to impose the following condition on P,P for uniform in time convergence.
′
Assumption 8 For some x1+τ,η < . There exist m (P) and m (P ) in R+ such that following
holds for all x Rd | | 0 ∞ τ τ ′
∈ (cid:10) (cid:11)
y 1+τP(x,dy) m (P) 1+ x1+τ , and y 1+τP (x,dy) m (P ) 1+ x1+τ .
ZRd| | ≤ τ | | ZRd| | ′ ≤ τ ′ | |
(cid:0) (cid:1) (cid:0) (cid:1)
Now we state a generalization of the Proposition 3.2, which gives the convergence rate of
E (µ¯N,µ )+ (η¯M,η ) 0
W1 n n W1 n n →
uniformly over all n 0 in a nonasym(cid:8)ptotic manner. (cid:9)
≥
Recall lP∇,lP∇′ introducedin Assumption 3. For α∈(0,1), let lP∇P,α′ =(1−α)lP∇+αlP∇′. With the notations
of Assumption 1 we define
1 A
a := −k k .
0 σ(2+lP∇P,α′)
For (µ ,η ),(µ ,η ) (Rd) (Rd) define the following distance on (Rd) (Rd)
n n ′n n′ ∈P1 ×P1∗ P1 ×P1∗
((µ ,η ),(µ ,η )):= (µ ,µ )+ (η ,η ).
W1 n n ′n n′ W1 n ′n W1 n n′
8
Theorem 1 Consider the particle system IPS . Suppose Assumptions (1)-(5) and Assumptions (7),(8)
2
hold for some τ >0. Let N :=min M,N . Also assume δ (0,a ), (1 α)m (P)<1 and
1 0 τ
{ } ∈ −
max kAk+δσ(2+lP∇P,α′)+αl(P′) ,(1−α)l(P) +δσmax αlP∇′,(1−α)lP∇ <1, .
n(cid:16) (cid:17) o (cid:8) (cid:9)
Then there exists θ <1, and a (0, ) such that for each n 0, the upperbound b(N ,τ,d) of
1
∈ ∞ ≥
E (µ¯N,η¯M),(µ ,η ) aθnE (µ¯N,η¯M),(µ ,η )
W1 n n n n − W1 0 0 0 0
can be expressed as (cid:0) (cid:1) (cid:0) (cid:1)
max 1, τ
N− {2 1+τ} if d=1,τ =1,
1 6
1
N1−2 logN1 if d=1,τ =1,
b(N1,τ,d)=CNN11−−1122(llooggNN11+)2N1−1+ττ iiff dd==22,,ττ =6=11,, . (3.13)
N−max{d1,1+ττ} if d>2,τ = 1 ,
where the value of the consNta11−ntd1ClogwNill1vary for each ofitfhe cadse>s.2,τ =6 dd−−111,
Remark 3.5 For the first particle system (1.3-1.5) similar results hold where the explicit bounds are
given in terms of number of particles N instead of N . For IPS if the initial sampling scheme of X¯ (N)
1 2 0
(X¯1,X¯2,...,X¯N) is µ -chaotic then using the fact E (µ¯N,µ ) 0 as N , it follows from th≡e
0 0 0 0 W1 0 0 → → ∞
conclusion of the Theorem 1
supE (µ¯N,η¯M),(µ ,η ) 0
W1 n n n n →
n 0
≥
(cid:0) (cid:1)
as min N,M . For the first particle system in (1.3-1.5), if E (ηN,η ) 0 as N , and
{ } → ∞ W1 0 0 → → ∞
X (N) (X1,X2,...,XN) is µ -chaotic then following
0 ≡ 0 0 0 0
supE (µN,ηN),(µ ,η ) 0
W1 n n n n →
n 0
≥
(cid:0) (cid:1)
holds for N .
→∞
One consequence of above theorem and Proposition 3.4 will be the following interchange of limit results
which is analogous to Corollary 3.5 from [5].
Corollary 3.6 Under conditions of the Theorem 1
limsup limsupE ((µ¯N,η¯M),(µ ,η )) = limsup limsup E ((µ¯N,η¯M),(µ ,η ))
min N,M n W1 n n ∞ ∞ n min N,M W1 n n ∞ ∞
{ }→∞ →∞ →∞ { }→∞
= 0. (3.14)
SupposeAssumptionsofTheorem1holdandlet(µ ,η )bethefixedpointofthemapΨof(2.5). Weare
interestedinestablishinga propagationofchaosres∞ult∞forn= . RecallforIPS ,SM(η¯M)is the random
∞ 2 n
measure defined as 1 M δ where Yi,M conditionally on M,N, are M i.i.d distributed
M i=1 Yni,M { n }i=1,...,M Fn
Rd valued random variables according to η¯M . Denote Y (M):=(Y1,M,...,YM,M).
P k 1 n n n
−
Theorem 2 Consider the second particle system IPS . Suppose Assumptions 1,2,4,5 hold with conditions
2
∞
δ (0,a ), (1 α)i y P Pi(0,dy)< .
0 ′
∈ i=0 − ZRd| | ∞
X
Then for every N,M 1, the Markov process X¯N(n),η¯M,SM(η¯M) on (Rd)N (Rd) (Rd) has
≥ n n n 0 ×P1∗ ×P
a unique invariant measure ΘN,M if following holds ≥
(cid:0) (cid:1)
∞
max kAk+δσ(2+lP∇P,α′)+αl(P′) ,(1−α)l(P) +δσmax αlP∇′,(1−α)lP∇ <1.
n(cid:16) (cid:17) o (cid:8) (cid:9)
9
Let Θ1,N,M be the marginal distribution on (Rd)N of the first co-ordinate of ΘN,M. Suppose additionally
Assum∞ption 4,3 and Assumption 7,8 hold with further condition for some τ >0∞
(1 α)l (P)<1.
− τ
Then Θ1,N,M is µ - chaotic, where µ is defined in Proposition 3.4.
∞ ∞ ∞
Remark 3.7 For first particle system (IPS ) similar steady state resultholds for the discrete timeMarkov
1
chain X¯N(n),η¯N on (Rd)N (Rd).
n n 0 ×P1∗
≥
(cid:0) (cid:1)
3.1 Concentration Bounds:
In order to obtain uniform in time concentration bounds of (µN,ηN),(µ ,η ) we proceed according
W1 n n n n
to those in Theorem 3.7 and Theorem 3.8 of [5] respectively. Here we establish two different types of
(cid:0) (cid:1)
concentration bounds. The first one is with initial non iid (i.e initial samples are µ chaotic) assumption
0
and the second one is without that.
Assumption 9 (i) For some K (1, ), A (x) K for θ a.e. x Rm.
1
∈ ∞ ≤ ∈
(ii) There exists α (0, ) such that eαxµ (dx)< and there exists α(δ) (0,α) such that
| | 0
∈ ∞ ∞ ∈
R eα(δ) A2(z)+|B(δz)| θ(dz)< .
ZRm (cid:0) (cid:1) ∞
Assumption 10 Supposethereexistsfunctionsh (),h (), h (),h (),h (),h ()(h ,h ,h ,h arenon-
1 · 2 · ′1 · ′2 · 3 · ′3 · 2 ′2 3 ′3
decreasing with h (0)=0, h (0)=0;), and constants l (0,1], l (0, ) such that h (x), and h (x)
2 ′2 h1 ∈ h′1 ∈ ∞ 1 ′1
are respectively l and l Lipschitz. There exists α (0, ) such that following hold for all α (0,α)
h1 h′1 ∈ ∞ 1 ∈
eα1|y|P(x,dy) eh2(α1) eα1|h1(x)|+eh3(α1) , eα1|y|P′(x,dy) eh′2(α1)(eα1|h′1(x)|+eh′3(α1)).(3.15)
≤ ≤
Z Z
(cid:0) (cid:1)
Remark 3.8 (a) For Gaussian transtion kernel P(x,dy)= 1 e−(x2−λy2)2dy, one has
√2πλ
eα1|y|P(x,dy)=eα212λ2 e−αxΦ x αλ +eαxΦ αλ+ x ,
λ − λ
Z h (cid:0) (cid:1) (cid:0) (cid:1)i
whereΦ()is thecumulativedistribution function ofNormaldistribution. So (3.15)holds withh (x)=
1
·
x, h ()=0, h (α )= λ2α21.
3 · 2 1 2
(b) For Bi-exponential kernel P(x,dy)= 21λe−|x−λy|dy one has
1
eα1|y|P(x,dy)=eα1x 1 α2λ2 .
Z h − 1 i
So (3.15) holds under condition α < 1 with h (x) = x, h () = 0, h (α ) = log 1 . Note
1 λ1 1 3 · 2 1 1−α21λ2
that any kernel with tail lighter than exponential (like Gaussian) will satisfy (3.15) fohr all α1i, where
for kernels with exponential like tail will have a specific restriction on α .
1
(c) We worked here only for l =1 as the upper bound. It only influences in the choice of α for which
h1 1
sup sup E eα1|x|,η¯nM <∞. (3.16)
n 0M,N 1
≥ ≥ D E
For l = 1 one has a definite upper bound of α . More precisely denoting α h (0) i lj +
h1 1 1 1 j=0 h1
i h (α lj ) by g(i) if g(i) is linear in i (happens only for l = 1) then there exists α such that
j=0 2 1 h1 h1 P∗
(Pw3i.l1l6r)emhoalidnsfifonritαe1fo<r αal∗l.αOn>th0e. oItfhger(i)hainsdexifpogn(e·)ntiisalboiunnide(dw,htehnenl su>pn≥1)0stuhepnM,tNhe≥1uEppeerαb1o|xu|,nη¯dnMof
1 h1 (cid:10) (cid:11)
supn 0supM,N 1E eα1|x|,η¯nM will diverge.
≥ ≥
(cid:10) (cid:11)
10