Scaling Limits for Beam Wave Propagation in Atmospheric Turbulence

0.2 MB·English

by  Albert Fannjiang

SCALING LIMITS FOR BEAM WAVE PROPAGATION IN ATMOSPHERIC
TURBULENCE
ALBERTC. FANNJIANG∗ KNUTSOLNA†
Abstract. We prove the convergence of the solutions of the parabolic wave equation to that of
the Gaussian white-noise model widely used in the physical literature. The random medium is
4
isotropicandisassumedtohaveintegrablecorrelation coefficientinthepropagation direction. We
0
0 discuss thelimits of vanishinginner scale and divergent outer scale of theturbulent medium.
2

n
a
J
1. Introduction
9
1 The small-scale refractive index variations, called the refractive turbulence, in the atmosphere is

the result of small scale fluctuations of temperature, pressure and humidity caused by the turbu-
3
v lence of air velocities. For optical propagation in the atmosphere the influence of the temperature
4 variations on the refractive index field is dominant whereas in the microwave range, the effect of
2
the humidity variations is more important. The refractive turbulence results in the phenomena of
0
beam wander, beam broadening and intensity fluctuation (scintillation). It is important to note
5
0 that these effects depend on the length scales of the waves as well as the refractive turbulence [19].
3 The refractive turbulence is modeled on the basis of Kolmogorov theory of turbulence which
0
introducesthenotionoftheinertialrangeboundedbytheouterscaleL (oftheorderof100m−1km)
/ 0
h and the inner scale ℓ (of the order of 1−10mm). Other features of the refractive turbulence in the
p 0
open clear atmosphere include [22]: (i) small changes (typical value of 3×10−4 at sea level) in the
-
h refractive index related to small variations in temperature (on the order of 0.1−1oC), (ii) small
at scattering angle which is of the order λ/ℓ and has the typical value 3×10−4rad for λ = 0.6mn
0
m and ℓ = 2mm. Perturbation methods for solving the Maxwell equations are adequate provided
0
: that the propagation distance is less than, say, 100m, a severe limitation on their applicability to
v
i imaging or communication problems. Our motivation is mainly from laser or microwave beams but
X
our consideration and results apply equally well to ultrasound waves in atmospheric turbulence.
r
a The results are also relevant in the context of ultrasound waves penetrating through complicated
multiscale fluctuating (interface) zones in for instance human tissue.
Under the condition λ = O(ℓ ) (including the millimeter and the sub-millimeter range) the
0
depolarization term in the Helmholtz equation for the electric field is negligible [22] and one can
use the (scalar) Helmholtz equation
(1) ∇2E+k2n¯2(1+n˜)2E = 0
with appropriate boundary conditions where k is the wavenumber, n¯ is the mean refractive index
field and n˜ is the normalized fluctuation of the refractive index.
1.1. Therescaledparabolicapproximation. Thewell-knownparabolicapproximationtoequa-
tion (1) is applicable in a regime where the variations of the index of refraction are small on the
∗Department of Mathematics, University of California at Davis, Davis, CA 95616 Internet: fan-
[email protected]. Research is supported by The Centennial Fellowship of American Mathematical Society
and UC Davis Chancellor’s Fellowship.
† Department of Mathematics, Universityof California at Irvine,Internet: [email protected].
1
scale of the wavelength so that backscattering is negligible [22]. This is almost always valid for
laser beam in the atmosphere.
In this paper we study the initial value problem for the parabolic wave equation
∂Ψ(z,x) x
(2) ∇2Ψ(z,x)+2ik = −2k2n˜(z,x)Ψ(z,x), Ψ(0,x) = F ∈ L2(R2)
⊥ ∂z 0 a
(cid:16) (cid:17)
where z is the longitudinal coordinate in the direction of the propagation, x = (x ,x ) is the
1 2
transverse coordinates, ∇ is the transverse gradient and Ψ is related to the scalar wave field E by
⊥
E = Ψ(z,x)exp(ikn¯z). The initial condition has a typical width a which is the aperture. Below
we will drop the perp in denoting the derivatives in the transverse directions.
The difficulty in solving equation (2) lies in the random multiscale nature of n˜(z,x). First we
non-dimensionalize eq. (2) as follows. Let L be the propagation distance in the longitudinal
z
direction. Let λ be the characteristic wavelength. The corresponding central wavenumber is
0
k = 2π/λ . The Fresnel length L is then given by
0 0 f
L = L /k .
f z 0
We introduce dimensionless wave number and cpoordinates
k˜ = k/k , x˜ = x/L , z˜= z/L
0 f z
and rewrite the equation in the form
∂Ψ
(3) 2ik˜ +∆Ψ+2k˜2k L n˜(zL ,xL )Ψ = 0, Ψ(0,x) =F (γ1/2x) ∈L2(R2)
0 z z f 0
∂z
after dropping the tilde in the coordinate variables where
L 2
f
γ =
a
(cid:18) (cid:19)
is assumed to be O(1), thus the source is supported on the scale determined by the Fresnel length.
1.2. Model spectra. A widely used model for the structure function of the refractive index field
of the atmosphere is based on the Kolmogorov theory of turbulence and has the following modified
Von K’arm’an spectral density
(4) Φ (~k) = 0.033C2(|~k|2+K2)−11/6exp(−|~k|2/K2)
n n 0 m
where ~k = (ξ,k), with ξ ∈ R,k ∈ R2 the Fourier variables conjugate to the longitudinal and
transversal coordinates, respectively. Here K = 2π/L ,K = 5.92/ℓ . This spectrum has the
0 0 m 0
correct behavior only in the inertial subrange, i.e.
(5) Φ (~k) ∼ |~k|−11/3, |~k| ∈ (2πL−1,2πℓ−1).
n 0 0
Outsideofthisrange,particularlyfor|~k|≪ 2πL−1 thereisnophysicalbasisfortheirbehavior;they
0
are just mathematically convenient expressions of the cutoffs. In particular, if the wave statistics
strongly depend on ℓ or L , then the problem probably requires more accurate information on the
0 0
refractive index field outside of the inertial range [6], [12], [13]. Note that the ratio L /ℓ grows
0 0
like Re3/4 as the Reynolds number Re tends to infinity.
There are several variants of (4) arising from modeling more detailed features of the refractive
index field. One of them is the Hill spectrum [2], [15] to account for the “bump” at high wave
numbers which is known to occur near the inner scale
−11/6
(6Φ) (~k)= 0.033C2 1+1.802|~k|/K −0.254(|~k|/K )7/6 |~k|2+K2 exp(−|~k|2/K2 )
n n m m 0 m
h i(cid:16) (cid:17)
where K = 3.3/ℓ . The coefficient C2 is itself a random variable that depends on time as well as
m 0 n
the altitude. Note that in atmospheric turbulence the inner and outer scales and the exponent in
2
the power law may also have to be modeled as stochastic processes [21]. The temporal dependence
isirrelevant foropticalpropagation; thealtitude dependencehasaratherpermanent, non-universal
structure with length scales much greater than the outer scale L [19].
0
We will consider a class of spectra satisfying the upper bound
−2
(7) Φ(H,~k) ≤ K(L−2+|~k|2)−H−3/2 1+ℓ2|~k|2 ,~k = (ξ,k) ∈ R3,H ∈ (0,1)
0 0
(cid:16) (cid:17)
withsomeconstant K < ∞astheratioL /ℓ → ∞inthehighReynoldsnumberlimit. Thedetails
0 0
of the spectrum are not pertinent to our results, only the exponent H is. In particular, H = 1/3
for the modified Von K’arm’an spectrum (4).
1.3. White noise scaling. Let us introduce the non-dimensional parameters that are pertinent
to our scaling:
L L L
f f f
ε = , η = , ρ= .
sLz L0 ℓ0
In terms of the parameters and the power-law spectrum in (7) we rewrite (3) as
∂Ψε k˜2 z
(8) 2ik˜ +∆Ψε+ σ V( ,x)Ψε = 0, Ψε(0,x) = F (γ1/2x) ∈ L2(R2).
∂z ε H ε2 0
with
LH
f
(9) σ = µ
H ε3
where µ is the standard deviation of the refractive index field corresponding to Φ(H,~k). The
spectrum for the (normalized) process V is given by
−2
(10) Φ (~k) ≤ K(η2+|~k|2)−H−3/2 1+ρ−2|~k|2 ,~k = (ξ,k) ∈ R3,H ∈(0,1)
η,ρ
(cid:16) (cid:17)
which is rescaled version of (7). For high Reynolds number one has L /ℓ = ρ/η ≫ 1 which is
0 0
always the case in our study.
In the beam approximation one has ε ≪ 1. The beam approximation is well within the range of
validity of the parabolic approximation. The white-noise scaling then corresponds to σ = O(1).
H
We set it to unity by absorbing the constant into V. This implies relatively weak fluctuations of
the index field, i.e.
C˜ ∼ L3/2−HL−3/2 ≪ 1, as L → ∞
n f z z
in view of the fact that H ∈ (0,1) and ε ≪ 1.
In the present paper we first study the white-noise scaling with ρ < ∞ and η > 0 fixed as ε → 0.
We then discuss the resulting white-noise model with ρ → ∞ and η → 0. For the proof, we adopt
the approach of [10] where the turbulent transport of passive scalars is studied. In [11] the white
noise limit is studied via the so called Wigner distribution.
2. Formulation and main results
2.1. Martingale formulation. We consider the weak formulation of the equation:
z 1 k˜2 z s
(11) ik˜[hΨε,θi−hΨ ,θi] = − hΨε,∆θids− Ψε,V( ,·)·θ ds
z 0 2 s ε s ε2
Z0 Z0 D E
for any test function θ ∈ C∞(R2), the space of smooth functions with compact support. The
c
tightness result (Section 4.1) implies that for L2 initial data the limiting measure P is supported in
the Skorohod space D([0,z ];L2(R2)). Here and below L2(R2) denotes the standard L2-function
0 w w
space with the weak topology.
3
For tightness as well as identification of the limit, the following infinitesimal operator Aε will
play an important role. Let Vε ≡ V(z/ε2,·), Fε the σ-algebras generated by {Vε, s ≤ z} and Eε
z z s z
the corresponding conditional expectation w.r.t. Fε. Let Mε be the space of measurable functions
z
adapted to {Fε,∀z} such that sup E|f(z)| < ∞. We say f(·) ∈ D(Aε), the domain of Aε, and
z z<z0
Aεf = g if f,g ∈Mε and for fδ(z) ≡ δ−1[Eεf(z+δ)−f(z)] we have
z
supE|fδ(z)| < ∞
z,δ
limE|fδ(z)−g(z)| = 0, ∀z.
δ→0
Consider the special class of admissible functions f(z) = φ(hΨε,θi),f′(z) = φ′(hΨε,θi),∀φ ∈
z z
C∞(R), then we have the following expression from (11) and the chain rule
1 k˜
(12) Aεf(z) = if′(z) hΨε,∆θi+ hΨε,Vεθi .
"2k˜ z ε z z #
A main property of Aε is that
z
(13) f(z)− Aεf(s)ds is a Fε-martingale, ∀f ∈ D(Aε).
z
Z0
Also,
z
(14) Eεf(z)−f(s)= EεAεf(τ)dτ ∀s < z a.s.
s s
Zs
(see [17]). We denote by Athe infinitesimaloperator correspondingto the unscaled processV (·) =
z
V(z,·).
Define
∞
(15) Γ(1)(x,y) = cos((x−y)·p)cos(sξ)Φ (ξ,p) dsdξdp
(η,ρ)
Z Z Z0
= π cos((x−y)·p)Φ (0,p) dp
(η,ρ)
Z
(16) Γ(1)(x) = Γ(1)(x,x)
0
where we have written the wavevector ~k ∈R3 as ~k= (ξ,p) with p ∈ R2.
Now we formulate the solutions for the Gaussian Markovian model as the solutions to the cor-
responding martingale problem: Find a measure P (of Ψ ) on the subspace of D([0,∞);L2(R2))
z w
whose elements have the initial condition F (γ1/2x) such that
0
z i
f(hΨ ,θi)− f′(hΨ ,θi) hΨ ,∆θi−k˜2 Ψ ,Γ(1)θ −k˜2f′′(hΨ ,θi)hθ,K θi ds
z s 2k˜ s s 0 s Ψs
is a martingaZl0e w(cid:26).r.t. the filtr(cid:20)ation of a cylindriDcal WieneEr(cid:21)process, for each f ∈ C∞(R(cid:27))
where
(17) K θ = Ψ (x)Ψ (y)Γ(1)(x,y)θ(y)dy.
Ψs s s
Z
The Gaussian Markovian model has been extensively studied for beam wander, broadening and
scintillation effects in the literature (see, e.g. [5], [14]). It can also been written as the Itˆo’s
4
equation
i
dΨ = ∆−k˜2Γ(1) Ψ dz+ik˜(K )1/2 dW
z 2k˜ 0 z Ψz z
(cid:18) (cid:19)
i
= ∆−k˜2Γ(1) Ψ dz+ik˜Ψ dW˜
2k˜ 0 z z z
(cid:18) (cid:19)
i
(18) = ∆Ψ dz+ik˜Ψ ◦dW˜ , Ψ (x) = F (γ1/2x)
2k˜ z z z 0 0
where ◦ stands for the Stratonovich integral, and W (x) and W˜ (x) are the Brownian fields with
z z
the spatial covariance δ(x−y) and Γ(1)(x,y), respectively.
The existence and uniqueness for the Schro¨dinger-Itoˆ eq. (18) with ρ < ∞ and η > 0 has been
studied in [8] by using the Wiener chaos expansion. Note that the following limit exists
(19) Γ¯(x,y) = lim π cos((x−y)·p)Φ (0,p) dp
(η,ρ)
ρ→∞
Z
= π cos((x−y)·p)Φ (0,p) dp, η > 0
(η,∞)
Z
By the well-posedness result of [8] and a standard weak−⋆ compactness argument one can prove
the existence of weak solution in D([0,∞);L2(R2)) for ρ= ∞,H ∈ (0,1).
w
Nextweconsiderthelimitingcaseη = 0. Thiswouldinduceuncontrollablelargescalefluctuation
the Gaussian, Markovian model which should be factored out first. Thus we consider the solution
of the form
z
Ψ(z,x) = Ψ′(z,x)exp(ik˜ W˜ (0)ds)
s
Z0
and the resulting equation
i
(20) dΨ = ∆Ψ dz+ik˜Ψ ◦dW˜ ′, Ψ (x) = F (γ1/2x)
z 2k˜ z z z 0 0
where W˜ ′ is given by
z
(21) W˜ ′(x) = W˜ (x)−W˜ (0)
z z z
with the covariance function
Γ′(x,y) = π (eix·p−1)(e−iy·p−1)Φ (0,p)dp.
(0,∞)
Z
Note that the above integral is convergent only if
H < 1/2;
in particular, the limit exists for the modified Von K’arm’an spectrum H = 1/3. Since H < 1/2,
the limiting model is only H¨older continuous in the transverse coordinates.
Again by the well-posedness result of [8] and a standard weak−⋆ compactness argument one can
prove the existence of weak solution in D([0,∞);L2(R2)) for ρ = ∞,η = 0,H ∈(0,1/2).
w
2.2. Uniqueness. Because of the non-smoothness (when ρ= ∞) and the non-homogeneity (when
η = 0) of the white-noise potential in the transverse coordinates the uniqueness argument of [8]
does not apply here.
Taking the function f(r)= rn in the martingale formulation, we arrive after some algebra at the
following equation
(n)
∂F
(22) z = C F(n)+C F(n)
∂z 1 z 2 z
5
for the n−point correlation function
F(n)(x ,...,x ) ≡ E[Ψ (x )···Ψ (x )]
z 1 n z 1 z n
where
n
i
(23) C = ∆
1 2k˜ xj
j=1
X
n
(24) C = −k˜2 Γ(x ,x ),
2 j k
j,k=1
X
n
(25) or C = −k˜2 Γ′(x ,x )
2 j k
j,k=1
X
(26)
We will now establish the uniqueness for eq. (22) with the initial data
F(n)(x ,...,x ) = E[Ψ (x )···Ψ (x )], Ψ ∈ L2(R2).
0 1 n 0 1 0 n 0
In the former case (24) C is a bounded, H¨older continuous function and we rewrite eq. (22) in
2
the mild formulation
z
F(n) = exp(zC )F(n)+ exp[(z−s)C ]C F(n) ds
z 1 0 1 2 s
Z0
whose local existence and uniqueness can be easily established by straightforward application of
the contraction mapping principle. By linearity, local well-posedness can be extended to global
well-posedness.
In the latter case (25) C is unbounded, H¨older continuous function with sub-Lipschitz growth.
2
We first note that C is non-positive everywhere since
2
n
Γ′(xj,xk)= π (eixj·p−1) (eixk·p−1)Φ(0,∞)(p)dp ≥0.
j,k=1 Z j k
X X X
Hence both C and C are generators of one-parameter contraction semigroups on L2(R2n), thus by
1 2
the product formula (Theorem 3.30, [7]) we have
z z m
lim exp( C )exp( C ) F = exp[z(C +C )]F
1 2 1 2
m→∞ m m
for all F ∈ L2(R2n), which thhen gives rise to a uniqiue semigroup on L2(R2n).
2.3. Mainassumptionsandtheorem. LetV beaz-stationary,x-homogeneoussquare-integrable
z
process whose spectral density satisfies the upper bound (10).
Let F and F+ bethe sigma-algebras generated by {V : ∀s≤ z} and {V : ∀s≥ z}, respectively.
z z s s
Define the correlation coefficient
(26) ρ(t) = sup sup E[hg].
E[h]=h0∈,EF[zh2]=1 g∈Fz++t
E[g]=0,E[g2]=1
Assumption 1. The correlation coefficient ρ(t) is integrable
When V is a Gaussian process, the correlation coefficient ρ(t) equals the linear correlation
z
coefficient r(t) which has the following useful expression
(27) r(t) = sup R(t−τ −τ ,k)g (τ ,k)g (τ ,k)dkdτ dτ
1 2 1 1 2 2 1 2
g1,g2Z
6
where
R(t,k) = eitξΦ(ξ,k)dξ
Z
and the supremum is taken over all g ,g ∈ L2(Rd+1) which are supported on (−∞,0]×Rd and
1 2
satisfy the constraint
(28) R(t−t′,k)g (t,k)g¯ (t′,k)dtdt′dk = R(t−t′,k)g (t,k)g¯ (t′,k)dtdt′dk = 1.
1 1 2 2
Z Z
There are various criteria for the decay rate (e.g., exponential decay) of the linear correlation
coefficients, see [16].
Lemma 1. Assumption 1 implies that the random field
∞
V˜ (x) = E [V (x)]dt
z z t
Z0
is x-homogeneous and has a finite second moment which satisfies the upper bound:
∞ ∞
E[V˜2] ≤ |E[E [V ]E [V ]]|dsdt
z z t z s
Zz Zz
∞ 2
≤ E[V2] ρ(t)dt .
z
(cid:18)Z0 (cid:19)
Proof. Consider in the definition of the correlation coefficient
h = E (V ) ∈ L2(Ω,P,F )
1 z s z
h = V ∈ L2(Ω,P,F+).
2 t t
We then have
|E[E [V (x)]E [V (x)]]| = |E[E [V (x)]V (x)]|≤ ρ(t−z)E1/2 E2[V ] E1/2 V2 .
z s z t z s t z s t
Hence by setting s = t first and the Cauchy-Schwartz inequality we(cid:2)have (cid:3) (cid:2) (cid:3)
E E2[V ] ≤ ρ2(s−z)E[V2]
z s z
|E[E [V (x)]E [V (x)]]| ≤ ρ(t−z)ρ(s−z)E[V2], s,t ≥ z.
z s z(cid:2) t (cid:3) z
Therefore
∞ ∞
E[V˜2] ≤ |E[E [V ]E [V ]]|dsdt
z z t z s
Zz Zz
∞ 2
(21) ≤ E[V2] ρ(t)dt
z
(cid:18)Z0 (cid:19)
which, together with the integrability of ρ(t), implies a finite second order moment of V˜ .
z
(cid:3)
Corollary 1. For each L,z < ∞ and ρ < ∞,η > 0 there exists a constant C˜ such that
0
2
sup E ∆V˜ ≤ C˜
λz
z<z0
|x|≤L h i
for all H ∈(0,1),λ ≥ 1.
7
Proof. Analogous to (21) we have
∞ 2
E[∆V˜ ]2 ≤ E[∆V ]2 ρ(t)dt .
z z
(cid:18)Z0 (cid:19)
A straightforward spectral calculation shows that
E[∆V (x)]2 = O(ρ4−2H), ∀x,z.
z
(cid:3)
It is easy to see that
AV˜ = −V
z z
and that
Γ(1)(x,y) = E V˜ (x)V (y)
z z
where Γ(1) is given by (15). h i
Next we assume the following quasi-Gaussian property:
Assumption 2.
(21) sup E[Vε(y)]4 ≤ C sup E2[Vε]2(y)
z 1 z
|y|≤L |y|≤L|
4 2
(22) sup E V˜ε (y) ≤ C sup E2 V˜ε (y)
z 2 z
|y|≤L |y|≤L
h i h i
4 2
(23) sup E [Vε]2 V˜ε (y) ≤ C sup E[Vε]2(y) sup E2 V˜ε (y)
z z 3 z z
|y|≤L (cid:20) (cid:21) ( |y|≤L ! |y|≤L !
h i h i
2
+ sup E2 VεV˜ε (y) sup E V˜ε (y)
z z z
|y|≤L ! |y|≤L !)
h i h i
for all L < ∞ where the constants C ,C and C are independent of ε,η,ρ,γ.
1 2 3
Assumption 3. For any fixed η > 0 and every θ ∈ C∞(R2)
c
z 1
(23) sup kθV˜( ,·)k = o , ∀ε≤ 1 ≤ ρ
ε2 2 ε
z<z0 (cid:18) (cid:19)
with a random constant of finite moments independent of ρ and ε.
When V is Gaussian, V˜ is also Gaussian and condition (23) is always satisfied
z z
(24) sup kθV˜( ,·)k ≤ C˜log 0
ε2 2 ε2
z<z0
h i
where the random constant C˜ has a Gaussian-like tail by a simple application of Borell’s inequality
[1].
Theorem 1. Let V satisfy Assumptions 1, 2 and 3. Let η > 0 and ρ < ∞ be fixed as ε → 0. Then
the weak solution Ψε of (11) converges in the space of D([0,∞);L2(R2)) to that of the Gaussian
w
white-noise model with the covariance functions Γ(1) and Γ(1).
0
Notethatinthelimitingmodelwithρ = ∞thewhite-noisevelocityfieldhastransverseregularity
of H¨older exponent H +1/2.
The convergence of the white-noise limit has been established in [3] and [4] for a refractive index
field that is a function of z only.
8
3. Proof of Theorem 1
3.1. Tightness. In the sequel we will adopt the following notation
f(z) ≡ f(hΨε,θi), f′(z) ≡ f′(hΨε,θi), f′′(z) ≡ f′′(hΨε,θi), ∀f ∈C∞(R).
z z z
Namely, the prime stands for the differentiation w.r.t. the original argument (not z) of f,f′ etc.
A family of processes {Ψε,0 < ε < 1} ⊂ D([0,∞);L2(R2)) is tight if and only if the family
w
of processes {hΨε,θi,0 < ε < 1} ⊂ D([0,∞);L2(R2)) is tight for all θ ∈ C∞(R2). We use the
w c
tightness criterion of [18] (Chap. 3, Theorem 4), namely, we will prove: Firstly,
(25) lim limsupP{sup|hΨε,θi| ≥ N}= 0, ∀z < ∞.
z 0
N→∞ ε→0 z<z0
Secondly, for each f ∈ C∞(R) there is a sequence fε(z) ∈ D(Aε) such that for each z < ∞
0
{Aεfε(z),0 < ε <1,0 < z <z } is uniformly integrable and
0
(26) limP{sup|fε(z)−f(hΨε,θi)| ≥ δ} = 0, ∀δ > 0.
ε→0 z<z0
Then it follows that the laws of {hΨε,θi,0 < ε < 1} are tight in the space of D([0,∞);L2(R2))
w
Condition (25) is satisfied because the L2-norm is preserved. Let
ik˜ ∞
fε(z) ≡ Eεf′(z)hΨε,Vεθi ds
1 ε z z s
Zz
be the 1-st perturbation of f(z). Let
1 ∞
V˜ε = EεVεds.
z ε2 z s
Zz
We obtain
(26) fε(z) = ik˜εf′(z) Ψε,V˜εθ .
1 z z
D E
Proposition 1.
lim sup E|fε(z)| = 0, lim sup |fε(z)| = 0 in probability
1 1
ε→0z<z0 ε→0z<z0
.
Proof. We have
(27) E[|fε(z)|] ≤ εkf′k kΨ k EkθV˜εk
1 ∞ 0 2 z 2
and
(28) sup |fε(z)| ≤ εkf′k kΨ k supkθV˜εk .
1 ∞ 0 2 z 2
z<z0 z<z0
Therightside of (27) is O(ε) by Lemma 1whilethat of (28)is o(1) in probability by Assumption3.
(cid:3)
Set fε(z) = f(z)+fε(z). A straightforward calculation yields
1
k˜2
Aεfε = −εf′′(z) hΨε,∆θi+ hΨε,Vεθi Ψε,V˜εθ
1 z ε z z z z
" #
D E
1 k˜2 ik˜
−εf′(z) Ψε,∆(V˜εθ) + Ψε,VεV˜εθ − f′(z)hΨε,Vεθi
2 z z ε z z z ε z z
" #
D E D E
9
and, hence
i
(27) Aεfε(z) = f′(z)hΨε,∆θi−k˜2f′(z) Ψε,VεV˜εθ −k˜2f′′(z)hΨε,Vεθi Ψε,V˜εθ
2k˜ z z z z z z z z
ε D E D E
− f′(z) Ψε,∆(V˜εθ) +f′′(z)hΨε,∆θi Ψε,V˜εθ
2 z z z z z
= Aε(zh)+ADε(z)+Aε(z)E+Aε(z) D Ei
1 2 3 4
where Aε(z) and Aε(z) are the O(1) statistical coupling terms.
2 3
For the tightness criterion stated in the beginnings of the section, it remains to show
Proposition 2. {Aεfε} are uniformly integrable and
lim sup E|Aε(z)| = 0
4
ε→0z<z0
.
Proof. We show that {Aε},i = 1,2,3,4 are uniformly integrable. To see this, we have the following
i
estimates.
1
|Aε(z)| ≤ kf′k kΨ k k∆θk
1 2k˜ ∞ 0 2 2
|Aε(z)| ≤ k˜2kf′k kΨ k kVεV˜εθk
2 ∞ 0 2 z z 2
|Aε(z)| ≤ k˜2kf′′k kΨ k2kVεθk kV˜εθk .
3 ∞ 0 2 z 2 z 2
For fixed η, the second moments of the right hand side of the above expressions are uniformly
bounded as ε→ 0 and hence Aε(z),Aε(z),Aε(z) are uniformly integrable.
1 2 3
ε
|Aε| ≤ kf′′k kΨ k2k∆θk kV˜εθk +kf′k kΨεk k∆(V˜εθ)k .
4 2 ∞ 0 2 2 z 2 ∞ z 2 z 2
h i
By Lemma 1 and Corollary 1 Aε is uniformly integrable. Finally, it is clear that
4
lim supE|Aε(z)| = 0.
4
ε→0z<z0
(cid:3)
3.2. Identification of the limit. Once the tightness is established we can use another result in
[18] (Chapter 3, Theorem 2) to identify the limit. The setting there is finite-dimensional but the
argument is entirely applicable to the infinite-dimensional setting here (cf. [11]).
Let A¯ be a diffusion or jump diffusion operator such that there is a unique solution ω in the
z
space D([0,∞);L2(R2)) such that
w
z
(22) f(ω )− A¯f(ω )ds
z s
Z0
is a martingale. We shall show that for each f ∈ C∞(R) there exists fε ∈D(Aε) such that
(23) sup E|fε(z)−f(hΨε,θi)| < ∞
z
z<z0,ε
(24) limE|fε(z)−f(hΨε,θi)| = 0, ∀z < z
z 0
ε→0
(25) sup E|Aεfε(z)−A¯f(hΨε,θi)| < ∞
z
z<z0,ε
(26) limE|Aεfε(z)−A¯f(hΨε,θi)| = 0, ∀z < z .
z 0
ε→0
Then it follows that any tight processes hΨε,θi converges in law to the unique process generated
z
by A¯. As before we adopt the notation f(z) = f(hΨε,θi).
z
10