Exponentially-ergodic Markovian noise perturbations of delay differential equations at Hopf bifurcation

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by  Nishanth Lingala

EXPONENTIALLY-ERGODIC MARKOVIAN NOISE PERTURBATIONS OF
DELAY DIFFERENTIAL EQUATIONS AT HOPF BIFURCATION.
N. LINGALA,N. SRINAMACHCHIVAYA,AND V.WIHSTUTZ
Abstract. Weconsidernoiseperturbationsofdelaydifferentialequations(DDE)experiencing
Hopf bifurcation. The noise is assumed to be exponentially ergodic, i.e. transition density
converges to stationary density exponentially fast uniformly in the initial condition. We show
that, underan appropriatechange of time scale, asthestrength of theperturbationsdecreases
7
to zero, the law of the critical eigenmodes converges to the law of a diffusion process (without
1
delay). WeprovetheresultonlyforscalarDDE.Forvector-valuedDDEwithoutproofssee[1].
0
2

n
a
J 1. Introduction

7
1 Delay differential equations (DDEs) arise in a variety of areas such as manufacturing systems,

biological systems, and control systems. Deterministic DDEs have been the focus of intense
]
R study by many authors in the past three decades—see [2, 3] and the references therein. In
P some of these systems, variation of a parameter would result in loss of stability through Hopf
. bifurcation. For example, oscillators of the form
h
t
a (1) q¨(t)+2ζq˙(t)+q(t)= −κ [q(t)−q(t−τ)]+κ [q(t)−q(t−τ)]2
1 2
m
arise in machining processes—here q represents the position of a tool cutting a workpiece that
[
is rotating with time period τ, and κi depend on the width of the cut. There exists a threshold
1
κ beyond which the fixed point q = 0 loses stability through Hopf bifurcation and oscillations
v
7 arise [4]. This oscillatory behaviour is called regenerative chatter and results in poor surface
0 finish of the workpiece. Hopf bifurcation is also found in biological systems—for example [5]
6
discusses a model for oscillations in the area of eye-pupil as a response to incident light. The
4
0 model has similar qualitative behaviour to Mackey-Glass equation [6]
.
1 d cθn
0 xˆ(t) = −αxˆ(t)+ ,
dt θn+xˆn(t−τ)
7
1
which exhibits a Hopf bifurcation as the parameter n is varied.
:
v Typically these systems are also influenced by noise, for example, inhomogenity in the mate-
i
X rial properties of workpiece in machining processes [7], and unmodeled dynamics in biological
r systems. Therefore, it is important to study the effect of noise in the models of such systems.
a
Theaimofthispaperistostudytheeffectofnonlinearandrandomperturbationsonthosedelay
systems whose fixed points are on the verge of a Hopf bifurcation. With appropriate scaling of
coordinates, the dynamics close to the fixed point can be casted in the form of a linear DDE
perturbed by small noise and small nonlinearities. First we briefly describe the mathematical
set-up. Statements would be proved only for scalar DDE.
Let x(t) be a R-valued process governed by a DDE with maximum delay r. The evolution of
x at each time t requires the history of the process in the time interval [t−r,t]. So, the state
space can be taken as C := C([−r,0];R), the space of continuous functions on [−r,0]. Equipped
with sup norm, ||η|| = sup |η(θ)|, the space C is a Banach space. At each time t, denote
θ∈[−r,0]
2010 Mathematics Subject Classification. 34K06, 34K27, 34K33, 34K50.
Key words and phrases. Delay differential equations; Hopf bifurcation; Diffusion approximation; Martingale
problem.
1
2 N.LINGALA,N.SRINAMACHCHIVAYA,ANDV.WIHSTUTZ
the [t−r,t] segment of x as Π x, i.e. Π x ∈ C and
t t
Π x(θ)= x(t+θ), for θ ∈ [−r,0].
t
Now, a linear DDE can be represented in the following form
x˙(t) =L (Π x), t ≥ 0,
0 t
(2)
(Π0x = ζ ∈ C,
where L : C → R is a continuous linear mapping on C and ζ is the initial history required.
0
For every such L there exists a bounded function µ : [−r,0] → R, continuous from the left on
0
(−r,0) and normalized with µ(0) = 0, such that
(3) L η = dµ(θ)η(θ), ∀η ∈ C.
0
Z[−r,0]
To reflect the Hopf bifurcation scenario, we consider operators L which are such that the
0
unperturbed system (2) is on the verge of instability, i.e., we assume L satisfies the following
0
assumption.
Assumption1.1. Define∆(λ) := λI −L (eλ·) = λI − dµ(θ)eλθ.Thecharacteristic
n×n 0 n×n [−r,0]
equation
R
(4) det(∆(λ)) = 0, λ ∈C
has a pair of purely imaginary solutions ±iω (critical eigenvalues) and all other solutions have
c
negative real parts (stable eigenvalues).
The object of study in this article are equations of the form
dx(t) = L (Π x)dt+εG (Π x)dt+ε2G(Π x)dt+εσ(ξ(t))F(Π x)dt, t ≥ 0,
0 t q t t t
(5)
(Π0x = ζ ∈ C,
where F,G,G : C → R satisfy assumption 1.2, ξ is a noise process satisfying assumption 1.3
q
and σ :M → R is a bounded mean-zero function of the noise ξ. For example, one can have ξ as
a finite-state markov chain. The coefficient G is assumed to satisfy a centering condition that
q
would be specified later in assumption 5.1.
Assumption 1.2. The functions F,G,G have atmost linear growth.
q
|F(η)| ≤C(1+||η||), |G(η)| ≤ C(1+||η||), |G (η)| ≤ C(1+||η||), ∀η ∈ C.
q
The functions F,G,G possess three bounded derivatives.
q
Assumption 1.3. The noise ξ is a M-valued Markov process with the transition function ν
given by
ν(t,ξ,B) = P{ξ ∈B|ξ = ξ}
t 0
for B a borel subset of M. The noise is exponentially erogdic, i.e., there exist a unique invariant
probability measure ν¯ and positive constants c and c such that for all t ≥ 0,
1 2
sup |ν(t,ξ,dζ)−ν¯(dζ)| ≤c e−c2t.
1
ξ∈M M
Z
The transition semigroup is Feller with infinitesimal generator denoted by G. The function σ is
bounded, σ(·) ∈ dom(G) and such that σ(ζ)ν¯(dζ) = 0.
M
When studying the effect of small nRoise perturbations on DDE whose fixed point is on the
verge of Hopf bifurcation, the dynamics close to the fixedpoint can becasted in the above forms
after appropriate scaling of coordinates. For example, consider x˜˙ = κx˜(t−1)−x˜3(t). When
κ = −π the fixed point x˜ = 0 is on the verge of instability. Suppose κ has small perturbations
2
about −π according to κ(t) = −π + εσ(ξ(t)) + ε2 where ξ is a noise. Then, zooming close
2 2
NOISE PERTURBATIONS OF DDE AT HOPF BIFURCATION 3
to the zero fixed point, x(t) := ε−1x˜(t) can be put in the form (5) with L (η) = −πη(−1),
0 2
F(η) = η(−1) and G(η) = −η3(0)+η(−1).
When ε = 0 in (5), using spectral theory [2], C can be decomposed as C = P ⊕Q where P
is a two-dimensional space determined by the (eigenspaces associated with the) pair of critical
eigenvalues. Theprojections ofΠ x ontoP andQareuncoupled. InQthedynamicsis governed
t
by the stable eigenvalues, and hence the sup-norm of the Q-projection of Π x decays to zero
t
exponentially fast as t → ∞. The space P is two-dimensional and a basis Φ can be chosen for
it. Let (z (t),z (t)) be the coordinates of P-projection of Π x with respect to the basis Φ. Then
1 2 t
the dynamics of z is a pure rotation with frequency ω and constant amplitude.
c
When the perturbation is added, i.e. ε> 0, the dynamics of Π x on P and Q is coupled. The
t
amplitude of the Q-projection decays exponentially fast to a “strip” of O(ε) and the dynamics
of z can be written as perturbation of a rotation. Employing a rotating coordinate system to
nullify the rotation of z, and writing the transformed coordinates as z we show that, under an
appropriate change of time scale, the law of z converges to the law of diffusion process as ε→ 0.
This result is useful because for small ε it provides an approximate two dimensional description
of countably infinite modes that the delay equation possesses.
The result is summarized in theorem 6.1. For vector-valued DDE without proofs, and
an illustration of the usefulness of these results using numerical simulations, see [1].
1.1. Related work. Systems with small noise perturbations are studied in [8, 9, 10]. They
consider systems of the form d x˜ε(τ) = εF(x˜ε(τ),ξ(τ)) + ε2G(x˜ε(τ),ξ(τ)) with F such that
dτ
for each fixed x˜, E[F(x˜,ξ)] = 0 where expectation is with respect to the invariant measure of
the noise ξ. On changing the time sacle in the above equation: t = ε2τ, xε(t) := x˜ε(t/ε2),
ξε(t) := ξ(t/ε2), we have dxε(t) = 1F(xε(t),ξε(t)) + G(xε(t),ξε(t)). It is shown in [8, 9, 10]
dt ε
(using different assumptions) that the law of xε converges weakly to that of a diffusion process
as ε → 0. Analogous results for DDE are in [11]. It considers
1
(6) x˙ε(t)= b(xε(t),xε(t−r),ξε(t)) + a(xε(t),xε(t−r),ξε(t))
ε
with the assumptions that E[b(x,x ,ξ(t))] = 0, E[a(x,x ,ξ(t))] = a¯(x,x ) ∀x,x , and
r r r r
1 T2 ∂b (x,x ,ξ(t))
E i r b (x,x ,ξ(T )) dt →¯b(x,x ) as T ,T ,T −T → ∞,
j r 1 r 1 2 2 1
(T −T ) ∂x
2 1 ZT1 j j
(cid:2)X (cid:3)
1 T2 1
E[b (x,x ,ξ(t)) b (x,x ,ξ(T ))]dt → S (x,x ) as T ,T ,T −T → ∞,
i r j r 1 ij r 1 2 2 1
(T −T ) 2
2 1 ZT1
1
∃Φ(x,x ) such that (S +ST) = ΦΦT.
r 2 ij ij
Then [11] shows that, as ε → 0, the law of xε converges weakly to that of a stochastic DDE
given by
dx(t) = a¯(x(t),x(t−r)) + ¯b(x(t),x(t−r)) dt + Φ(x(t),x(t−r))dW(t).
Notethat(6)isat(cid:2)ime-rescaled versionofx˙(t) = εb(x(t(cid:3)),x(t−εr2),ξ(t))+ε2a(x(t),x(t−εr2),ξ(t)).
If you compare this with (5): here the delay is not a constant as ε varies, whereas in (5) delay
is a constant. Alternatively, in (6) the delay is fixed delay r, whereas a time-rescaled version
of (5) would have vanishing delay ε2r. Whereas [11] obtains an SDDE in the limit, we would
obtain an SDE without delay.
In [12] the effect of noise on evolution equations on Banach spaces is considered, and [13]
extends it to systems with fast and slow components. [13] considers
dyε(t) 1
(7) = Byε(t)+A(t/ε)yε(t),
dt ε
with the following assumptions: (i) the operator B (deterministic) generates a contraction semi-
group which is denoted by etB, (ii) B is such that etB → πˆ as t ↑ ∞, where πˆ is the projection
4 N.LINGALA,N.SRINAMACHCHIVAYA,ANDV.WIHSTUTZ
onto the kernel of B, (iii) ∃C, γ > 0 such that ||(etB −πˆ)f|| ≤ Ce−γt||f||. Under the assump-
tions (i) and (ii), we have that etBπˆ =πˆetB = πˆ and πˆBf = Bπˆf = 0. Define the operator A¯by
A¯ = lim 1 t0+T E[A(s)]ds. Write the solution of the equation (7) as yε(t) = Uε(t,0)yε(0).
T↑∞ T t0
[13] is concerned with the asymptotic behaviour of Uε(t,0) as ε ↓ 0. Under some assumptions
R
on Uε(t,0) [13] states
Theorem 3.1 in [13]: For 0 ≤ t ≤ T, lim E[Uε(t,0)πˆf]= etπˆA¯πˆf.
ε↓0
Theorem 3.2 in [13]: Suppose πˆA¯πˆ ≡ 0. Then, for 0 ≤ t ≤ T, lim E[Uε(t/ε,0)πˆf] =
ε↓0
etV¯πˆf, where V¯ = lim 1 t0+T s E[πˆA(s)(eB(s−u) −πˆ)A(u)πˆ]duds.
T↑∞ T t0 t0
Theaboveresulttheorem3.2isinfactnotcorrect. Whenthefastcomponentispresentthetheo-
R R
rem gives only the critical(slow)-stable(fast) interaction, but not the critical(slow)-critical(slow)
interaction. In fact doing the computations in [13] carefully shows that the correct result is
1 t0+T s
V¯ = lim E[πˆA(s)eB(s−u)A(u)πˆ]duds.
T↑∞ T Zt0 Zt0
Thedelay equation (5) could beputin the framework of [13]. However it is difficultto satisfy all
the assumptions and so we choose the easier route of using the martingale problem technique.
Thecaseof σ(ζ)ν¯(dζ) 6= 0correspondstothecaseoftheorem3.1in[13]and σ(ζ)ν¯(dζ) = 0
M M
corresponds to the case of theorem 3.2 in [13].
R R
[14] considers equations of the form (5) with σ(ζ)ν¯(dζ) 6= 0 and a different ε scaling;
M
for example: dx(t) = L (Π x)dt+εG(Π x)dt+εσ(ξ(t))F(Π x)dt. Let z be the coordinates of
0 t t t
R
P-projection of Π x with respect to basis Φ. Let z be the transformed process obtained after
t
nullifying the rotation of z. [14] shows that the probability law of z(t/ε) converges to that of
a deterministic ODE and that the norm of Q-projection decays exponentially fast. If the zero
fixedpointof thelimitODE is exponentially stable, then itis proven thatxis also exponentially
stable in the moments.
[15, 16] considers equations of the form x˙(t) = L (Π x) + εσ(ξ(t))L (Π x) with σ a mean
0 t 1 t
zero function of the noise process ξ and L is a bounded linear operator on C. Define the
1
exponentialgrowthrateλε := limsup 1log|xε(t)|andexpanditasλε = λ +ελ +ε2λ +....
t→∞ t 0 1 2
Using perturbation methods and Furstenberg-Khasminskii representation, [15, 16] show that
λ = λ = 0 and give explicit expression for λ .
0 1 2
1.2. Organization of this paper. In sections 2 we collect the results on spectral properties
of linear DDE that would be useful to us. In section 3 we arrive at coupled equations for the
evolution of projections of Π x of the system (5) on to the critical and stable eigenspaces. The
t
sections 2 and 3 are justrecalling the set-up from [16] which draws from [2] and [3]. In Section 4
we prove the weak convergence result following [10], using the technique of martingale problem.
For the brevity of notation in section 4 we work with G = G = 0. In section 5 we consider the
q
effect of G and G . For convenience of the reader, the final result is summarized in section 6.
q
2. The unperturbed deterministic system
Here we are just recalling the set-up from [16] which draws from [2] and [3].
The solution of (2) gives rise to the strongly continuous semigroup T(t) : C → C, t ≥ 0,
defined by T(t)Π x = Π x. The generator A of the semigroup is given by
0 t
d
(8) Aϕ= ϕ, dom(A) = D(A)= {ϕ ∈ C1|ϕ′(0) = L ϕ}
0
dθ
(C1 is the linear space of continuously differentiable functions on [−r,0], and ′ = d ). The
dθ
equation (2) with the initial condition ϕ in D(A), is equivalent to the abstract differential
equation
d
(9) Π x = AΠ x, t ≥ 0; Π x = ζ ∈ D(A),
t t 0
dt
where the differentiation with respect to t is taken in the sense of the sup-norm in C.
NOISE PERTURBATIONS OF DDE AT HOPF BIFURCATION 5
The state space C splits in the form C = P ⊕Q where P = span {Φ , Φ } where (recall ±iω
R 1 2 c
are the critical eigenvalues)
Φ (θ)= cos(ω θ), Φ (θ)= sin(ω θ), θ ∈[−r,0].
1 c 2 c
Write Φ = [Φ , Φ ]. Using the identity cos(ω (t+·)) = cos(ω t)cos(ω ·)−sin(ω t)sin(ω ·) and
1 2 c c c c c
the linearity of L , it can be shown that
0
0 ω
(10) T(t)Φ(·) = Φ(·)eBt, B = c ,
−ω 0
c
(cid:20) (cid:21)
with the derivative
d
(11) AΦ(·)= T(t)Φ(·)| = Φ(·)B.
t=0
dt
Let π denote the projection of C onto P along Q, i.e. π :C → P with π2 = π and π(η) = 0 for
η ∈ Q. Theoperatorπ canberepresentedusingabilinearformh·,·ionC([−r,0],R)×C([0,r],R)
given by
·
(12) hφ,ψi := φ(0)ψ(0)−L ( φ(u)ψ(u−·)du)
0
Z0
Ψ (·)
and functions Ψ(·) = 1 , where Ψ are linear combinations of cos(ω·) and sin(ω·) and are
Ψ (·) i
2
(cid:20) (cid:21)
such that hΦ ,Ψ i = δ . We have for the projection π :C → P,
i j ij
(13) π(η) = Φhη,Ψi = hη,Ψ iΦ +hη,Ψ iΦ ,
1 1 2 2
and Q = ker(π) = {η ∈C|π(η) = 0}. There exists positive constants κ and K such that
(14) ||T(t)φ|| ≤ Ke−κt||φ||, ∀φ∈ Q.
Write the solution to (2) (with initial condition in dom(A)) as Π x = πΠ x+(I −π)Π x and
t t t
define z,y by Φz(t) = πΠ x and y := (I −π)Π x. Here z(t) is R2-valued. Then, using (11),
t t t
equation (9) can be replaced by the following system of equations
d
(15) z˙(t)= Bz(t), y = Ay
t t
dt
with initial values z(0) and y (·) given by Π x(·) = ζ(·) = Φ(·)z(0)+y (·).
0 0 0
From (15) it can be noted that z oscillate with frequency ω and constant amplitude. From
c
(14) it can be noted that ||y || decays exponentially fast.
t
We aim at investigating the noise perturbed system (5), that is, comparing the unperturbed
system with the perturbedone within the same framework. But if noise is present the boundary
condition (8), in general, cannot be satisfied: for example, when G = G = 0, the equation
q
d ζ(0) = L ζ +εF(ζ)σ(ξ (ω)) cannot hold for all chance elements ω. Therefore, jumps must
dθ 0 0
be allowed as result of differentiation. In other words, we need to extend the space C together
with the operators A and T(t). See chapter 6, 7 of [2], and [3] for the extension. See [16] for a
summary of the results that follow [17].
Let Cˆ = spanR1{0} ⊕ L∞ where L∞ = L∞([−r,0],B([−r,0]),Leb;R). The norm on Cˆ is
||γ1 +φ|| =|γ|+||φ|| . Let the extension of the semigroup T to Cˆbe denoted by Tˆ. The
{0} esssup
generator A is extended to
d d
(16) Aˆφ= (L φ−φ′(0))1 + φ, φ ∈Lip, ′ ≡
0 {0} dθ dθ
where Lip = ∪α∈RLip(α) and Lip(α) is the equivalence class in L∞ which contain at least one
Lipschitz-continuous function φ with φ(0) = α. If φ is not continuous at 0, we represent the
equivalence class [φ] by a function φ with φ(0) = 0. Note that at any rate (Aˆφ)(0) = L φ.
0
6 N.LINGALA,N.SRINAMACHCHIVAYA,ANDV.WIHSTUTZ
Aˆhas the same spectrum as A and the decomposition of C is lifted to the decomposition of Cˆ
(see [3], page 100) with help of the extension of π to the projection πˆ : Cˆ→ P . Let Ψˆ = Ψ(0)
Λ
and Ψˆ = Ψ (0). Explicitly, we only need
i i
πˆ(γ1 )= γΦΨˆ = γ(Ψˆ Φ +Ψˆ Φ ).
{0} 1 1 2 2
We have Cˆ= P ⊕Qˆ where Qˆ = ker(πˆ). The spaces P and Qˆ are Aˆ-invariant and the commuta-
tivity property πˆAˆ= Aˆπˆ holds on D(Aˆ) = Lip. There exists positive constants κ and K such
that
(17) ||T(t)φ|| ≤ Ke−κt||φ||, ∀φ∈ Qˆ.
3. The randomly perturbed system
We are now prepared to study the randomly perturbed system (5). To keep the notation
simple we first deal with the case G = G = 0. The effect of G and G are studied in section 5.
q q
So we consider
dx(t) = L (Π x)dt+εσ(ξ(t))F(Π x)dt, t ≥ 0,
0 t t
(18)
(Π0x = ζ ∈ Lip,
The equation (18) is equivalent to the abstract differential equation
d
(19) Π x = AˆΠ x + εσ(ξ )F(Π x)1 , Π x = ζ ∈ Lip.
t t t t {0} 0
dt
Writing Π x = Φz(t)+y with Φz(t) = πˆΠ x and y := (I −πˆ)Π x, and using the facts that
t t t t t
Aˆ(Φz) = ΦBz, Aˆ commutes with πˆ on D(Aˆ) and πˆ1 =ΦΨˆ, we have
{0}
(20) z˙(t) = Bz(t)+εσ(ξ )F(Φz(t)+y )Ψˆ,
t t
d
(21) y = Aˆy +εσ(ξ )F(Φz(t)+y )(I −πˆ)1 ,
t t t t {0}
dt
with initial conditions z(0) ∈ R2 and y ∈ Qˆ ∩Lip such that Π x = ζ = Φz(0)+y .
0 0 0
From (20) it can be noted that dynamics of z is small perturbation of a rotation with fre-
quency ω . To study the effect of perturbation itself, we need to employ a coordinate system
c
which nullifies the rotation of z. Further, the noise perturbations take order O(1/ε2) time to
significantly affect the dynamics. So, we employ the following transformation:
zεt = e−Bt/ε2z(t/ε2), ξtε = ξt/ε2, ytε := yt/ε2, τtε := t/ε2,
and write the evolution equations as
d 1
(22) zε = σ(ξε)F(ΦeBτtεzε+yε)e−BτtεΨˆ,
dt t ε t t t
d 1 1
(23) yε = Aˆyε+ σ(ξε)F(ΦeBτtεzε+yε)(I −πˆ)1 ,
dt t ε2 t ε t t t {0}
The process (zε,τε,yε,ξε) is a Markov process on S := R2×(R+ ∪{0})×(Qˆ ∩Lip)×M.
t t t t Λ
Our goal is to show that the probability law of zε converges to the probability law of a two-
dimensional diffusion process. For the proof we use the technique of martingale problem. The
procedure we employ is as follows. Consider the following truncated process:
d ϑ (zε,n)
(24) zε,n = n t σ(ξε)F(ΦeBτtεzε,n+yε,n)e−BτtεΨˆ,
dt t ε t t t
d 1 ϑ (zε,n)
(25) yε,n = Aˆyε,n+ n t σ(ξε)F(ΦeBτtεzε,n+yε,n)(I −πˆ)1 ,
dt t ε2 t ε t t t {0}
NOISE PERTURBATIONS OF DDE AT HOPF BIFURCATION 7
with ϑ :R2 → R a smooth function given by
n
1 for ||z|| ≤ n,
2
ϑ (z)=
n
(0 for ||z||2 ≥ n+1.
Define the stopping time e := inf{t ≥ 0 : ||zε|| ≥ n}. Then the law of zε agrees with law of
n t 2
zε,n until e . We identify some drift b(n) and diffusion coefficient a(n), and show that as ε → 0,
n
the law of the truncated processes zε,n converge to the unique solution of martingale problem
with diffusion and drift coefficients (a(n),b(n)). We identify (a,b) so that a ≡ a(n), b ≡ b(n) on
{z ∈ R2 : ||z|| ≤ n} and show that there exists unique solution for the martingale problem of
2
(a,b). By corollary 10.1.2 and lemma 11.1.1 of [18] we then have that the law of zε converges as
ε→ 0 to the law of diffusion process with diffusion and drift coefficients (a,b).
In the next section we show that the law of truncated process zε,n converges as ε → 0 to that
of a diffusion process. The following notation would be used.
• For f : Cˆ→ R and η ∈Cˆ, ζ ∈ Cˆ, let (ζ.∇)f(η) denote the Frechet derivative of f at η in
the direction ζ, i.e.
1
(ζ.∇)f(η) := lim f(η+δζ)−f(η) .
δ→0 δ
• For f :R2 → R such that f ∈C1(R2;R),(cid:0)and v ∈ R2, (cid:1)
1
(v.∇z)f(z) = lim f(z+δv)−f(z) .
δ→0 δ
• For f :(Qˆ ∩Lip)→ R differentiable, and(cid:0) y˜∈ Qˆ ∩Lip, (cid:1)
1
(y˜.∇y)f(y) = lim f(y+δy˜)−f(y) .
δ→0 δ
(cid:0) (cid:1)
4. Convergence of the law of {zε,n}
ε>0
Let N ∈ N such that for the initial condition Π x = ζ both ||z(0)|| = ||zε|| < N and
∗ 0 2 0 2 ∗
||(I −π)ζ||= ||yε|| < N holds. We consider only n ≥ N .
0 ∗ ∗
From (24) it can be seen that
(26) ||zε,n|| ≤ n+1, t ≥ 0.
t
Employing this fact in the variation-of-constants formula
t−s t t−s
yε,n = Tˆ( )yε,n+ Tˆ( )(1−πˆ)1 σ(ξε)F(ΦeBτsεzε,n+yε,n)ds,
t ε2 0 ε2 {0} s s s
Z0
using the exponential decay (17) and Gronwall inequality, it can be shown that
(27) ||yε,n|| ≤ K||y ||e−κt/ε2 +εC(n+||y ||) ≤ Cn, t ≥0.
t 0 0
Thus, once n is fixed, zε,n and yε,n are bounded with the bound independent of ε.
t t
The truncated process (zε,n,τε,yε,n,ξε) is a Markov process on
t t t t
S := (R2) ×(R+∪{0})×(Qˆ ∩Lip) ×M
n n Λ n
where
(R2) := {z ∈ R2 : ||z|| ≤ n+1}, (Qˆ ∩Lip) = {y ∈ Qˆ ∩Lip : ||y|| ≤ Cn}
n 2 Λ n Λ
where C is from (27).
The infinitesimal generator Lε of (zε,n,τε,yε,n,ξε) is defined as follows. Let
t t t t
∂
L = G+(Aˆy).∇y+ ,
0
∂τ
L = ϑ (z)σ(ξ)F(ΦeBτz+y)(e−BτΨˆ).∇z,
1,1 n
L = ϑ (z)σ(ξ)F(ΦeBτz+y)((I −πˆ)1 ).∇y
1,1 n {0}
8 N.LINGALA,N.SRINAMACHCHIVAYA,ANDV.WIHSTUTZ
Then, the infinitesimal generator Lε is
1 1 1 1
(28) Lε = L + L = L + (L +L ).
ε2 0 ε 1 ε2 0 ε 1,1 1,2
Theorem 4.1. Let Lε be the generator of (zε,n,τε,yε,n,ξε) process and let L† be as in equa-
t t t t n
tion (33) (more explicitly written in remark 4.1). Then for any g ∈ C3(R;R) with bounded
derivatives, there exists functions (“correctors”) ψ(k)(z,τ,y,ξ), k=1,2, bounded on S such that
g n
(29) Lε g+εψ(1) +ε2ψ(2) = L†g+εL ψ(2)
g g n 1 g
(cid:16) (cid:17)
The correctors ψ(k) are given in equations (31)–(32). Further, L ψ(2) is bounded on S .
g 1 g n
Proof. First we set-up some notation. For functions of (z,τ,y,ξ), define
(etL0f)(z,τ,y,ξ) := f(z,τ +t,Tˆ(t)y,ζ)ν(t,ξ,dζ).
M
Z
For functions of (z,τ,y,ξ), define operator Ξ by
[Ξf](z,τ,y) = f(z,τ,y,ξ)ν¯(dξ).
M
Z
For functions of (z,τ,y), define operator Y by
[Yf](z,τ)= f(z,τ,0).
For functions of (z,τ), define operator T by
1 2π/ωc
[Tf](z)= f(z,τ)dτ.
2π/ω
c Z0
Now, expanding LHS of (29), and noting that L g =0 and L g = 0, we get
0 1,2
1
(30) L ψ(1) + L g + L ψ(2) + L ψ(1) + ε L ψ(2) .
ε 0 g 1,1 0 g 1 g 1 g
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
The function
∞
(31) ψ(1) = esL0L gds
g 1,1
Z0
(1)
solves L ψ + L g = 0. The integral from 0 to ∞ is well-defined because (i) by mean-zero
0 g 1,1
and bounded nature of σ and assumption 1.3 we have the exponential decy:
σ(ζ)ν(t,ξ,dζ) = σ(ζ)(ν(t,ξ,dζ)−ν¯(dζ)) ≤ |σ| sup |ν(t,ξ,dζ)−ν¯(dζ)| ≤ |σ|c e−c2t,
1
(cid:12)ZM (cid:12) (cid:12)ZM (cid:12) ζ∈MZM
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)and (ii) L g is bo(cid:12)und(cid:12)ed on S due to atmost line(cid:12)ar growth of F.
1,1 n
(cid:12) (cid:12) (cid:12) (cid:12)
(2) (1)
Nowwetrytofindψ . However, notethatL ψ isnotmean-zero. Inordertohaveanexpo-
g 1 g
nentialdecay, wechooseψ(2,a) = ∞esL0 L ψ(1) −Ξ(L ψ(1)) ds. Thenψ(2,a) solves L ψ(2,a)+
g 0 1 g 1 g g 0 g
L ψ(1) −Ξ(L ψ(1)) = 0. We theRn choose(cid:16)ψ(2,b) = ∞esL0 Ξ(cid:17)(L ψ(1))−YΞ(L ψ(1)) ds. This
1 g 1 g g 0 1 g 1 g
time, the exponential decay would be provided by bounded(cid:16)derivatives of F and th(cid:17)e decay at
R
(17). Then ψ(2,b) solves L ψ(2,b) +Ξ(L ψ(1))−YΞ(L ψ(1)) = 0. We then choose ψ(2,c)(z,τ) =
g 0 g 1 g 1 g g
− τ YΞ(L ψ(1))−TYΞ(L ψ(1)) | ds. The integrand here is bounded and is periodic in
0 1 g 1 g (z,s)
τ Rwit(cid:16)h average zero and so ψg(2,c(cid:17)) is bounded. Now, ψg(2,c) solves L0ψg(2,c) + YΞ(L1ψg(1)) −
TYΞ(L ψ(1))= 0. Let
1 g
(32) ψ(2) := ψ(2,a) +ψ(2,b) +ψ(2,c).
g g g g
Then ψ(2) solves L ψ(2) +L ψ(1) −TYΞ(L ψ(1))= 0.
g 0 g 1 g 1 g
NOISE PERTURBATIONS OF DDE AT HOPF BIFURCATION 9
†
Define L by
n
(33) L†g = TYΞ(L ψ(1)).
n 1 g
Collectingalltheabove, wehave(29). Boundedderivatives ofF andg ensurethatthecorrectors
are bounded on S and that L ψ(2) is bounded on S . (cid:3)
n 1 g n
Remark 4.1. The generator L† can be explicitly written in the following form:
n
2 ∂ 1 2 ∂2
(34) L† = b(n)(z) + a(n)(z)
n i ∂z 2 ij ∂z ∂z
i i j
i=1 i,j=1
X X
where a(n),b(n) are as written as follows in terms of the auto-correlation function R of the noise:
(35) R(t) := σ(ξ) σ(ζ)ν(t,ξ,dζ) ν¯(dξ).
M M
Z (cid:18)Z (cid:19)
Let
1 2π/ωc ∞
(36) a (z)= dτ R(s)F(ΦeτBz)F(Φe(τ+s)Bz) e−τBΨˆ e−(τ+s)BΨˆ ds,
ij 2π/ω i j
c Z0 Z0
(cid:0) (cid:1) (cid:0) (cid:1)
b (z) = bF(z) = bF,P(z)+bF,Q(z),
i i i i
(37)
1 2π/ωc ∞
bF,P(z) = dτ R(s)F(ΦeτBz) ΦesBΨˆ .∇ F(Φe(τ+s)Bz) e−(τ+s)BΨˆ ds,
i 2π/ω i
c Z0 Z0 (cid:18) (cid:19)
(38) (cid:0) (cid:1) (cid:0) (cid:1)
1 2π/ωc ∞
bF,Q(z) = dτ R(s)F(ΦeτBz) Tˆ(s)(I −πˆ)1 .∇ F(Φe(τ+s)Bz) e−(τ+s)BΨˆ ds.
i 2π/ω {0} i
c Z0 Z0 (cid:18) (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
Then,
a(n) = ϑ2a , b(n) = ϑ2bF +b(n),F,ϑ,
ij n ij i n i i
where
ϑ (z) 2π/ωc ∞
b(n),F,ϑ(z) = n dτ R(s)F(ΦeτBz)F(Φe(τ+s)Bz) e−τBΨˆ .∇z ϑ (z) e−(τ+s)BΨˆ ds.
i 2π/ω n i
c Z0 Z0 (cid:18) (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
Note that (a(n),b(n)) agree with (a,b) on the set {z ∈R2 : ||z|| ≤ n}. (cid:3)
2
WeassumetheS-valuedprocesses(zε,n,τε,yε,n,ξε)aredefinedonprobabilitytriples(Ωε,n,Fε,n,Pε,n).
t t t t t
Let Pε be probability law of (zε,τε,yε,ξε), i.e. without truncation. Also define the following
t t t t
canonical set-up.
Definition 4.2. Define Ω† := C([0,∞),R2) equipped with the metric
∞ 1 sup |ω(t)−ω′(t)|
D(ω,ω′)= t∈[0,n] , ω,ω′ ∈ Ω†.
2n 1+sup |ω(t)−ω′(t)|
n=1 t∈[0,n]
X
Define the coordinate functions X†(ω) = ω(t) for all t ≥ 0 and all ω ∈ Ω†. For each t ≥ 0, define
t
F† := σ{X†;0 ≤ s ≤ t} and define a σ-algebra on Ω† by F† = ∨ F†. Let B denote the Borel
t s t≥0 t
σ-algebra on Ω† and define the induced probabilities
(39) Pε,n,†(A) = Pε,n{zε,n ∈ A}, Pε,†(A) = Pε{zε ∈ A}, A∈ B.
10 N.LINGALA,N.SRINAMACHCHIVAYA,ANDV.WIHSTUTZ
Let C (Ω†), equipped with sup norm, be the space of bounded continuous functions on Ω†. Let
b
P(Ω†) be the space of probability measures on Ω† equipped with weak∗ topology when P(Ω†) is
considered as dual of C (Ω†).
b
Remark 4.2. The metric space (Ω†,D) is Polish and the convergence induced by the metric D
is uniform convergence on compacts. Also, F† = B. The topology on P(Ω†) is same as the one
induced by the Prohorov metric. See for example [18]. (cid:3)
Theorem 4.3. Let Pn,† be the unique solution of the martingale problem for L†, with initial
n
condition z such that Φz = Π x. As ε → 0, the measures Pε,n,† tend to Pn,†.
0 0 0
Proof. We follow the approach of [10] (also see [19]).
Three bounded derivatives of F ensure that the coefficients (a(n),b(n)) defined in remark 4.1
have two boundedderivatives. Further, (a(n),b(n)) are boundedon R2. By corollary 6.3.3 of [18]
†
thesolution of martingale problem for L is well-posed. In particular, thesolution of martingale
n
problem for L† with initial condition z exists and is unique.
n 0
Let gε : S → R be such that it is bounded and has continuous bounded derivatives with
n
respect to z, τ and y (in Frechet sense) and lies in the domain of G. Define
t
(40) Mgε := gε(zε,n,τε,yε,n,ξε)−gε(zε,n,τε,yε,n,ξε)− (Lεgε)(zε,n,τε,yε,n,ξε)ds,
t t t t t 0 0 0 0 s s s s
Z0
gε ε,n
then M is a martingale with respect to the filtration F .
t t
For g ∈ C3(R2;R) bounded with bounded derivatives, let gε = g+εψ(1) +ε2ψ(2) where ψ(k)
g g g
are “correctors” as given in theorem 4.1. Applying (40) to this gε and using equation (29) we
have
(41)
t
g(zε,n)−g(zε,n)− (L†g)(zε,n)du = Mgε −Mgε
t s n u t s
Zs
(cid:0) t (cid:1)
+ε (Lεψ(2))(zε,n,τε,yε,n,ξε)du
g u u u u
Zs
2
− εk ψ(k)(zε,n,τε,yε,n,ξε)−ψ(k)(zε,n,τε,yε,n,ξε) ,
g t t t t g s s s s
k=1
X (cid:0) (cid:1)
where Mgε is a Fε,n martingale. By theorem 4.1, ψ(k) and Lεψ(2) are bounded. Hence we have
t t g g
t
(42) |g(zε,n)−g(zε,n)| ≤ | L†g(zε,n)du|+|Mgε −Mgε|+εC +εC |t−s|+ε2C .
t s n u t s 1 2 3
Zs
Let Hgε(z,τ,y,ξ) = Lε(gε)2−2gεLεgε (z,τ,y,ξ). Then hMgεit = 0tHgε(zεs,n,τsε,ysε,n,ξsε)ds.
Direct computation yields
(cid:0) (cid:1) R
Hgε = [G(ψg(1))2−2ψg(1)Gψg(1)]
+2ε[G(ψ(1)ψ(2))−ψ(1)Gψ(2) −ψ(2)Gψ(1)]+ε2[G(ψ(2))2−2ψ(2)Gψ(2)]
g g g g g g g g g
showingthatHgε isaboundedfunction. Writetheinequality (42)forg(z) = (z)1 andg(z) = (z)2.
Squaring both the inequalities and adding them and taking expectations, then using
t
Eε,n[|Mtgε −Msgε|2|Fsε,n]= Eε,n[hMgεit −hMgεis|Fsε,n] = Eε,n[Hgε(zεu,n,τuε,yuε,n,ξuε)|Fsε,n]du
Zs
together with the fact that Hgε is bounded, it can be shown that
lim lim sup supEε,n ||zε,n−zε,n||2|Fε,n = 0
t s s
δ↓0 ε↓0 |t−s|≤δ
(cid:2) (cid:3)