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arXiv:cond-mat/0206095v1 [cond-mat.dis-nn] 6 Jun 2002
STATISTICAL STABILITY IN TIME REVERSAL
GEORGE PAPANICOLAOU ∗, LEONID RYZHIK †,AND KNUT SØLNA ‡
Abstract. When a signal is emitted from a source, recorded by an array of transducers, time reversed
and re-emitted into the medium, it will refocus approximately on the source location. We analyze the refo-
cusing resolution in a high frequency, remote sensing regime, and show that, because of multiple scattering,
in an inhomogeneous or random medium it can improve beyond the diffraction limit. We also show that
the back-propagated signal from a spatially localized narrow-band source is self-averaging, or statistically
stable, and relate this to the self-averaging properties of functionals of the Wigner distribution in phase
space. Time reversal from spatially distributed sources is self-averaging only for broad-band signals. The
array of transducers operates in a remote-sensing regime so we analyze time reversal with the parabolic or
paraxial wave equation.
Key words. wave propagation, random medium, Liouville-Ito equation, stochastic-flow, time reversal
AMS subject classifications. 35L05, 60H15, 35Q60
1. Introduction. In time reversal experiments a signal emitted by a localized source
is recorded by an array and then re-emitted into the medium time-reversed, that is, the tail
of the recorded signal is sent back first. In the absence of absorption the re-emitted signal
propagates back toward the source and focuses approximately on it. This phenomenon
has numerous applications in medicine, underwater acoustics and elsewhere and has been
extensively studied in the literature, both from the experimental and theoretical points of
view [12, 13, 14, 15, 16, 20, 24, 25, 31]. Recently time reversal has been also the subject
of active mathematical research in the context of wave propagation and imaging in random
media [2, 3, 4, 7, 8, 9, 32]. A schematic description of a time reversal experiment is presented
in Figure 1.1.
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λL
ae
L
a
Fig. 1.1.A pulse propagates toward a time reversal array of size a. The propagation distance L
is large compared to a. The ambient medium has a randomly varying index of refraction with a typical
correlation length that is small compared to a. The signal is time reversed at the array and sent back into
the medium. The back propagated signal refocuses with spot size λL/ae, where aeis the effective aperture
of the array (Section 3.3).
∗Department of Mathematics, Stanford University, Stanford CA, 94305; papanico@math.stanford.edu,
†Department of Mathematics, University of Chicago, Chicago IL, 60637; ryzhik@math.uchicago.edu,
‡Department of Mathematics, University of California, Irvine CA, 92697; ksolna@math.uci.edu.
1
2G. PAPANICOLAOU, L. RYZHIK AND K. SØLNA
For a point source in a homogeneous medium, the size of the refocused spot is approx-
imately λL/a, where λis the central wavelength of the emitted signal, Lis the distance
between the source and the transducer array and ais the size of the array. We assume here
that the array is operating in the remote-sensing regime a≪L. Multiple scattering in a
randomly inhomogeneous medium creates multipathing, which means that the transducer
array can capture waves that were initially moving away from it but get scattered onto it
by the inhomogeneities. As a result, the array captures a wider aperture of rays emanating
from the original source and appears to be larger than its physical size. Therefore, somewhat
contrary to intuition, the inhomogeneities of the medium do not destroy the refocusing but
enhance its resolution. The refocused spot is now λL/ae, where ae> a is the effective
size of the array in the randomly scattering medium, and depends on L. The enhancement
of refocusing resolution by multipathing is called super-resolution [7]. The time reversed
pulse is also self averaging and refocusing near the source is therefore statistically stable,
which means that it does not depend on the particular realization of the random medium.
There is some loss of energy in the refocused signal because of scattering away from the
array but this can be overcome by amplification, up to a point.
The purpose of this paper is to explore in detail the mathematical basis of pulse sta-
bilization, beyond what was done in [7]. We want to explore in particular in what regime
of parameters statistical stability is observed in time reversal. We show here that for high
frequency waves in a remote sensing regime, spatially localized sources lead to statistically
stable super-resolution in time reversal, even for narrow-band signals. We also show that
when the source is spatially distributed, only for broad-band signals do we have statistical
stability in time reversal. The regime where our analysis holds is a high frequency one, more
appropriate to optical or infrared time reversal than to ultrasound, sonar or microwave radar.
In this regime we can make precise what spatially localized or distributed means (see Section
3.1). The numerical simulations in [7] and [8], which are set in ultrasound or underwater
sound regime, indicate that time reversal is not statistically stable for narrow-band signals
even for localized sources. Only for broad-band signals is time reversal statistically stable
in the regime of ultrasound experiments or sonar.
If the aperture of the transducer array is small a/L ≪1, the Fresnel number L/(ka2)
is of order one, and the random inhomogeneities are weak, which is often the case, we may
analyze wave propagation in the paraxial or parabolic approximation [29]. The wave field is
then given approximately by
u(t, x, z) = 1
2πZeiω(z/c0−t)ψ(z , x;ω/c0)dω(1.1)
where the complex amplitude ψsatisfies the parabolic or Schr¨odinger equation
2ikψz+ ∆xψ+k2(n2−1)ψ= 0.(1.2)
Here x= (x, y) are the coordinates transverse to the direction of propagation z, the wave
number k=ω/c0and n(x, z) = c0/c(x, z) is the random index of refraction relative to a
reference speed c0. The fluctuations of the refraction index
σµ(x
l,z
l) = n2(x, z)−1(1.3)
are assumed to be a stationary random field with mean zero, variance σ2, correlation length
land normalized covariance with dimensionless arguments
R(x, z) = E{µ(x+x′, z +z′)µ(x′, z′)}.(1.4)
Parabolic approximation and time reversal 3
A convenient tool for the analysis of wave propagation in a random medium is the
Wigner distribution [19, 28] defined by
W(z, x,p) = 1
(2π)dZRd
eip·yψ(x−y
2, z)ψ(x+y
2, z)dy(1.5)
where d= 1 or 2 is the transverse dimension and the bar denotes complex conjugate. The
Wigner distribution may be interpreted as phase space wave energy and is particularly well
suited for high frequency asymptotics and random media [28]. The quantity of principal
interest in time reversal, the time-reversed and back-propagated wave field, can be also
expressed in terms of the Wigner distribution (see Section 3.1). The self-averaging properties
of the back-propagated field are related to the self-averaging properties of functionals of the
Wigner distribution in the form of integrals of Wover the wave numbers p.
In the next Section we introduce a precise scaling that corresponds to (a) high frequency,
(b) long propagation distance, (c) narrow beam propagation, and (d) weak random fluctua-
tions. In the asymptotic limit where the small parameters go to zero the Wigner distribution
satisfies a stochastic partial differential equation (SPDE), a Liouville-Ito equation, that has
the form
dW (z, x,p;k) = −p
k· ∇xW+k2D
2∆pWdz −k
2∇pW·dB(x, z)(1.6)
where B(x, z) is a vector-valued Brownian field with covariance
E{Bi(x1, z1)Bj(x2, z2)}=−∂2R0((x1−x2))
∂xi∂xjz1∧z2,(1.7)
where z1∧z2= min{z1, z2}, and in the isotropic case
D=−R′′
0(0)
4, R0(x) = Z∞
−∞
R(x, s)ds.(1.8)
In Section 2.5 we analyze this SPDE in the asymptotic limit of small correlation length
for B(x, z) in the transverse variables x, and show that W(z, x,p;k)’s with different wave
vectors pare uncorrelated. From this decorrelation property we deduce that for localized
sources the time-reversed, back-propagated field is self-averaging, even for narrow-band sig-
nals. For distributed sources it is self-averaging only for broad-band signals. We show in
detail in Section 3 how the asymptotic theory is used in time reversal. In Appendix A we
introduce other scalings which lead to the same averaged SPDE but we do not analyze them
in detail.
Throughout the paper we define the Fourier transform by
ˆ
f(k) = Zdxe−ik·xf(x)
so that
f(x) = Zdk
(2π)deik·xˆ
f(k).
G. Papanicolaou was supported in part by grants AFOSR F49620-01-1-0465, NSF DMS-
9971972 and ONR N00014-02-1-0088, L. Ryzhik by NSF grant DMS-9971742, an Alfred P.
Sloan Fellowship and ONR grant N00014-02-1-0089. K. Solna by NSF grant DMS-0093992
and ONR grant N00014-02-1-0090.
4G. PAPANICOLAOU, L. RYZHIK AND K. SØLNA
2. Scaling and asymptotics.
2.1. The rescaled problem. To carry out the asymptotic analysis we begin by rewrit-
ing the Schr¨odinger equation (1.2) in dimensionless form. Let Lzand Lxbe characteristic
length scales in the propagation direction, as, for example, the distance Lbetween the source
and the transducer array for Lzand the array size afor Lx. We introduce a dimensionless
wave number k′=k/k0with k0=ω0/c0and ω0a central frequency. We rescale xand zby
x=Lxx′,z=Lzz′and rewrite (1.2) in the new coordinates dropping primes:
2ik ∂ψ
∂z +Lz
k0L2
x
∆ψ+k2k0Lzσµ xLx
l,zLz
lψ= 0.(2.1)
The physical parameters that characterize the propagation problem are: (a) the central
wave number k0, (b) the strength of the fluctuations σ, and (c) the correlation length l. We
introduce now three dimensionless variables
δ=l
Lx
, ε =l
Lz
, γ =1
k0l
(2.2)
which are the reciprocals of the transverse scale relative to correlation length, the recip-
rocal of the propagation distance relative to correlation length, and the central wave
length relative to the correlation length. We will assume that the dimensionless parameters
γ,σ,εand δare small
γ≪1; σ≪1; δ≪1; ε≪1.(2.3)
This is a regime of parameters where super-resolution phenomena can be observed.
To make the scaling more precise we introduce the Fresnel number
θ=Lz
k0L2
x
=γδ2
ε.(2.4)
We can then rewrite the Schr¨odinger equation (2.1) in the form
2ikθψz+θ2∆xψ+k2δ
ε1/2µ(x
δ,z
ε)ψ= 0.(2.5)
provided that we relate εto σand δby
ε=σ2/3δ2/3.(2.6)
One way that the asymptotic regime (2.3) can be realized is with the ordering
θ≪ε≪δ≪1,(2.7)
and γ≪σ4/3δ−2/3, corresponding to the high-frequency limit. We see from the scaled
Schr¨odinger equation (2.5) that this regime can be given the following interpretation. We
have first a high frequency limit θ→0, then a white noise limit ε→0, and then
abroad beam limit δ→0. We will analyze in detail and interpret these limits in the
following Sections. Another scaling in which (2.3) is realized is ε≪θ≪δ≪1. This
is a regime in which the white noise limit is carried out first, then the high frequency limit
and then the broad beam limit. We do not analyze this case here. Additional comments on
scaling are provided in Appendix A.
Parabolic approximation and time reversal 5
It is instructive to express the constraints (2.6) and (2.7) in terms of the dimensional
parameters of the problem. First, both the size of the transverse scale Lxand the propagation
distance Lzshould be much larger than the correlation length lof the medium. Moreover,
(2.6) implies that the longitudinal and transverse scales should be related by
Lz
Lx
=δ
σ21/3
≫1
so that we are indeed in the beam approximation. The first inequality in (2.7) implies that
Lz
Lx≪pk0l=1
√γ,
and with the above choice of Lzthis implies that
γ3/2
σ2≪Lx
l≪1
σ2.
2.2. The high frequency limit. A convenient tool for the study of the high frequency
limit, especially in random media, is the Wigner distribution. It is often used in the context
of energy propagation [19, 28] but it is also useful in analyzing time reversal phenomena
[2, 3, 7]. Let φθ(x) be a family of functions oscillating on a small scale θ. The Wigner
distribution is a function of the physical space coordinate xand wave vector pdefined as
Wθ(x,p) = Z
Rd
dy
(2π)deip·yφθ(x−θy
2)φθ(x+θy
2).(2.8)
The family Wθis bounded in the space of Schwartz distributions S′(Rd×Rd) if the functions
φθare uniformly bounded in L2(Rd). Therefore there exists a subsequence θk→0 such that
Wθkconverges weakly as k→ ∞ to a limit measure W(x,p). This limit W(x,p) is non-
negative and is customarily interpreted as the limit phase space energy density because
|φθk(x)|2→Z
Rd
W(x,p)dpas θ→0(2.9)
in the weak sense. This allows one to think of W(x,p) as a local energy density.
Let Wθ(z, x,p) be the Wigner distribution of the solution ψof the Schr¨odinger equa-
tion (2.5), in the transversal space-variable x. A straightforward calculation shows that
Wθ(z, x,p) satisfies in a weak sense the linear evolution equation
∂Wθ
∂z +p
k· ∇xWθ
(2.10)
=ikδ
2√εZeiq·x/δ ˆµq, z
εWθp−θq
2δ−Wθp+θq
2δ
θ
dq
(2π)d.
In the limit θ→0 the solution converges weakly in S′, for each realization, to the (weak)
solution of the random Liouville equation
∂W
∂z +p
k· ∇xW+k
2√ε∇xµx
δ,z
ε· ∇pW= 0.(2.11)
The initial condition at z= 0 is W(0,x,p) = WI(x,p), the limit Wigner distribution of the
initial wave function.
6G. PAPANICOLAOU, L. RYZHIK AND K. SØLNA
2.3. The white noise limit. In this Section we take the white noise limit ε→0 in
the random Liouville equation (2.11) whose solution we now denote by Wε. We can do this
using the asymptotic theory of stochastic differential equations and flows [22, 6, 21, 26] as
follows. Using the method of characteristics, the solution of the Liouville equation (2.11)
may be written in the form
Wε(t, x,p) = WI(Xε(t;x,p),Pε(t;x,p)),
where the processes Xε(t;x,p) and Pε(t;x,p) are solutions of the characteristic equations
dXε
dz =−1
kPε;dPε
dz =−k
2√ε∇xµXε
δ,z
ε
with the initial conditions Xε(0) = xand Pε(0) = p. The asymptotic theory of random
differential equations with rapidly oscillating coefficients implies that, under suitable condi-
tions on µ, in the limit ε→0 the processes Xε,Pεconverge weakly (in the probabilistic
sense), and uniformly on compact sets in x,pto the limit processes X(t), P(t) that satisfy
a system of stochastic differential equations
dP=−k
2dB(z), dX=−1
kPdz, X(0) = x,P(0) = p.
The random process B(z) is a Brownian motion with the covariance function
E{Bi(z1)Bj(z2)}=−∂2R0(0)
∂xi∂xj
dsz1∧z2
(2.12)
=δij −R′′
0(0)z1∧z2,
in the isotropic case, where
R0(x) = Z∞
−∞
R(x, s)ds(2.13)
is a function of |x|. This implies that the average Wigner distribution W(1)
ε(z, x,p) =
E{Wε(z, x,p)}converges as ε→0 uniformly on compact sets to the solution of the
advection-diffusion equation in phase space
∂W (1)
∂z +p
k· ∇xW(1) =k2D
2∆pW(1)
(2.14)
with the initial data W(1)(0,x,p) = WI(x,p). Here the diffusion coefficient Dis given by
D=−R′′
0(0)
4.(2.15)
The one-point moments E[Wε(z, x,p)]Nconverge as ε→0 to the functions W(N)(z, x,p)
that satisfy the same equation (2.14) but with the initial data W(N)(0,x,p) = [W0(x,p)]N.
This is similar to the spot dancing phenomenon [11], where all one-point moments are gov-
erned by the same Brownian motion. In particular we have that
W(2)(z , x,p)6=hW(1)(z , x,p)i2
so that the process Wεdoes not converge to a deterministic one, in the strong sense pointwise.
Parabolic approximation and time reversal 7
2.4. Multi-point moment equations. As in the previous Section we may also study
the white noise limit ε→0 of the higher moments of Wε(z, x,p) at different points
W(N)
ε(z, x1,...,xN,p1,...,pN) = E[Wε(z, x1,p1)]r1·...·[Wε(z, xN,pN)]rN.
Here the points (xm,pm) are all distinct, (xn,pn)6= (xm,pm). We may account for mo-
ments that have different powers of Wεat different points by taking different powers rjof
Wε(xj,pj).
We now consider the joint process (Xε(z;xm,pm),Pε(z;xm,pm)), m= 1,...,N. As
ε→0 it converges to the solution of the system of stochastic differential equations
dPm
i=−k
2
N
X
n=1
d
X
j=1
σij Xm−Xn
δdBn
j(z), dXm=−1
kPmdz,(2.16)
with the initial conditions
Xm(0) = xm,Pm(0) = pm.
The d-dimensional Brownian motions Bm,m= 1,...,N have the standard covariance tensor
EBm
i(z1)Bn
j(z2)=δmnδij z1∧z2, i, j = 1, . . . , d, m, n = 1,...,N.
The symmetric tensor σij (x) is determined from
N
X
k=1
σik(x)σjk (x) = −∂2R0(x)
∂xi∂xj.(2.17)
We assume that equation (2.17) has a solution that is differentiable in x, which is compatible
with the fact that the matrix on the right is, by Bochner’s theorem, non-negative definite.
The moments W(N)
εconverge as ε→0 to the solution of the advection-diffusion equation
∂W (N)
∂t +
N
X
m=1
pm
k· ∇xmW(N)=k2D
2
N
X
m=1
∆pmW(N)
(2.18)
−k2
4
N
X
n,m=1
n>m
d
X
i,j=1
∂2R0((xn−xm)/δ)
∂xi∂xj
∂2W(N)
∂pn
i∂pm
j
with the initial data
W(N)(0,x1,,...,xN,p1,...,pN) = [WI(x1,p1)]r1·...·[WI(xN,pN)]rN.
From (2.18) we can calculate moments of functionals of Wεof the form
Wε,φ(z) = ZWε(z , x,p)φ(x,p)dxdp.
For example, as ε→0 we have that
E[Wε,φ(z)]2→ZW(2) (z, x1,p1,x2,p2)φ(x1,p1)φ(x2,p2)dx1dp1dx2dp2.
A convenient way to deal with not only the limit of N-point moments but with the
full limit process W(z , x,p), at all points x,psimultaneously, is provided by the theory of
8G. PAPANICOLAOU, L. RYZHIK AND K. SØLNA
stochastic flows [23]. For this we need to show that Wε(z, x,p) converges weakly (in the
probabilistic sense) as ε→0 to the process W(z, x,p) that satisfies the stochastic partial
differential equation
dWδ=−p
k· ∇xWδ+k2D
2∆pWδdz −k
2∇pWδ·dB(x
δ, z).(2.19)
Here the Gaussian random field B(x, z) has the covariance
E{Bi(x1, z1)Bj(x2, z2)}=−∂2R0((x1−x2))
∂xi∂xjz1∧z2.
We call equation (2.19) the Liouville-Ito equation. It allows us to treat all equations of the
form (2.18) simultaneously and is a convenient tool for simulation and analysis. The dimen-
sionless wave number kcan be scaled out of (2.19) by writing W(z , x,p;k) = W(z, x,p
k; 1)
so that we need only consider (2.19) with k= 1. We will use this scaling in Section 3.1.
Note that unlike the single Brownian motion (2.12) that governs the evolution of one-
point moments, the Brownian field that enters the SPDE (2.19) depends explicitly on the
dimensionless correlation length δin the transverse direction. Therefore the limit process
also depends on δand we denote it by Wδ.
2.5. Statistical stability in the broad beam limit. We will now consider the limit
δ→0 of the process Wδ(z, x,p) when the transverse dimension d≥2. We are particularly
interested in the behavior of functionals of Wδas δ→0. The analysis of one-point moments
in Section 2.3 showed that they do not depend on δand are governed by a standard Brownian
motion. Therefore the process Wδdoes not have a pointwise deterministic limit. However,
we will show that functionals of Wδbecome deterministic in the limit δ→0. We refer to this
phenomenon as statistical stabilization and give conditions for it to happen. Stabilization
plays an important role in time reversal, imaging and other applications, as discussed in the
Introduction.
Theorem 2.1. Assume that φ(p)is a smooth test function of rapid decay, the transverse
correlation function R0(x)has compact support, the initial Wigner distribution WI(x,p)is
uniformly bounded and Lipschitz continuous, and the transverse dimension d≥2. Define
Iδ,φ(z, x) = ZWδ(z, x,p)φ(p)dp.(2.20)
Then
lim
δ→0EI2
δ,φ(z, x)=E2{Iδ,φ(z, x)}(2.21)
where E{Iδ,φ(z, x)}is independent of δ.
The assumption of compact support for R0(x) is not essential but simplifies the proof.
We have already noted that the Wigner distribution Wδitself does not stabilize. However,
(2.21) implies that
lim
δ→0V ar {Iδ,φ}= lim
δ→0EI2
δ,φ(z)−E2{Iδ,φ}= 0.(2.22)
Therefore, any smooth functional of the form (2.20) stabilizes in the limit δ→0, that is,
Iδ,φ ≈E{Iδ,φ},(2.23)
Parabolic approximation and time reversal 9
in mean square, and the expectation of Iδ,φ does not depend on δ. We prove Theorem 2.1
in Appendix B.
In the applications of the asymptotic theory to time reversal we need not only functionals
Iδ,φ of the form (2.20) but also of the form
Jδ(z, x) = ZWδ(z, x,p)dp.(2.24)
We need to show that such functionals are well defined with probability one and to analyze
their behavior as δ→0. This is done in the following theorem.
Theorem 2.2. Under the same hypotheses of Theorem 2.1 and with a non-negative
initial Wigner distribution WI≥0, the functional Jδis bounded, continuous and non-
negative with probability one. In the limit δ→0we have
lim
δ→0EJ2
δ(z, x)=E2{Jδ(z, x)}(2.25)
where E{Jδ(z, x)}does not depend on δ.
The proof of this theorem is given in Appendix B.
What is important in both Theorems 2.1 and 2.2 is that we do integrate over the wave
numbers pbecause there is no pointwise stabilization. In time reversal applications, as in
section 3.1, we actually need Theorem 2.2 when the integration is only over a line segment in
pspace, and the dimension of the latter is d≥2. Its proof follows from the one of Theorem
2.2.
3. Application to time reversal in a random medium. We will now apply these
results to the time reversal problem [7] described in the Introduction. A wave emitted
from the plane z= 0 propagates through the random medium and is recorded on the time
reversal mirror at L. It is then reversed in time and re-emitted into the medium. The back-
propagated signal refocuses approximately at the source, as shown in Figure 1.1. There are
two striking features of this refocusing in random media. One is that it is statistically stable,
that is, it does not depend on the particular realization. The other is super-resolution, that
is, the refocused spot is tighter than in the deterministic case. We discuss these two issues
in this section.
3.1. The time-reversed and back-propagated field. We assume that the wave
source at z= 0 is distributed on a scale σsaround a point x0, that is,
ψθ(z= 0,x;k) = eip0·(x−x0)/θψ0(x−x0
σs
;k),
where ψ0is a rapidly decaying and smooth function of xand k. The width of the source
σscould be large or small compared to the Fresnel number θ, and this affects the statistical
stability of the time-reversed, back-propagated field, as we explain in this Section. The
Green’s function, Gθ(z, x;ξ), solves the parabolic wave equation (2.5) with a point source
at (x, z) = (ξ , 0). Using its symmetry properties and the fact that time reversal t→ −tis
equivalent to ω→ −ωor k→ −k, the back-propagated, time-reversed field on the plane of
the source has the form
ψB
θ(L, x0, ξ;k) =(3.1)
ZZ Gθ(L, x;x0+θξ;k)Gθ(L, x0+η;η;k)eip0·η/θ ψ0(η
σs
;−k)χA(x)dxdη.
The complex field amplitude ψB
θis evaluated at x0+θξ, in the plane z= 0. We scale the
observation point off x0by θbecause we expect that the spot size of the refocused signal
10 G. PAPANICOLAOU, L. RYZHIK AND K. SØLNA
will be comparable to the lateral spread of the initial wave function. We denote with χA
the aperture function of the time reversal mirror. It could be its characteristic function,
occupying the region Ain the plane z=L
χA(x) = 1,x∈A
0,x/∈A,
or a more general aperture function like a Gaussian. The time reversal mirror is located in
the plane z=L.
After changing variables, the back-propagated field is given by
ψB
θ(L, x0, ξ;k) = θdZGθ(L, x;x0+θξ;k)Gθ(L, x;x0+θη;k)eip0·ηψ0(θη
σs
;−k)χA(x)dxdη
=θdZGθ(L, x0+θξ, x;k)Gθ(L, x0+θη , x;k)eip0·ηψ0(θη
σs
;−k)χA(x)dxdη.
It is now convenient to introduce the Wigner distribution
Wθ(z, x0,p;k) = Zθdeip·y
(2π)dGθ(z, x0−yθ/2,x;k)Gθ(z , x0+yθ/2,x;k)χA(x)dxdy,(3.2)
and express the back-propagated field as
ψB
θ(L, x0, ξ;k) = Zeip·(ξ−η)Wθ(L, x0+θ(ξ+η)
2,p;k)eip0·ηψ0(θη
σs
;−k)dpdη.(3.3)
The Wigner distribution is scaled here differently from (2.8) because of the way we have
scaled the source function.
In the high frequency limit θ→0, Wθ(z, x,p;k) tends to W(z, x,p;k), which solves the
random Liouville equation (2.11). Then, in the white noise limit, it solves the Liouville-Ito
equation (2.19). The mean of Wsolves (2.14), in the high-frequency and white noise limit,
with initial data
W(0,x,p;k) = χA(x)
(2π)d.(3.4)
Let
β=σs
θ
(3.5)
be the ratio of the width of the source to the Fresnel number and assume that it remains
fixed as θ→0. In this limit, the time-reversed and back-propagated field is given by
ψB(L, x0, ξ;k) = Zeip·(ξ−η)W(L, x0,p
k)eip0·ηψ0(η/β;−k)dpdη(3.6)
=Zeip·ξW(L, x0,p
k)βdˆ
ψ0(β(p−p0); −k)dp.
Here we have used the scaling W(z, x,p;k) = W(z, x,p
k; 1) in (2.19) and we have dropped
the last argument k= 1.
Parabolic approximation and time reversal 11
3.2. Statistical stability. From the form (3.6) of the back-propagated and time-
reversed field we see that when β=O(1) (or small), which means that σsis comparable
to the Fresnel number θ(or smaller), we can apply the results of Section 2.5 and conclude
that it is statistically stable or self-averaging in the broad beam limit δ→0. Theorems 2.1
and 2.2 are exactly what is needed for this. The fact that the initial function (3.4) may
be discontinuous at the boundary of the set Ais not a problem. This is because, we may
approximate the function χAfrom above and below by two smooth positive functions, to
which we may apply Theorems 2.1 and 2.2, and then use the maximum principle to deduce
the decorrelation property when the initial data is χA. We have, therefore,
ψB(L, x0, ξ;k)≈ hψB(L, x0, ξ;k)i
in the sense of convergence in probability or in mean square, in the broad beam limit δ→0,
for each fixed frequency ω=kc0. Statistical stability of time reversal does not depend on
having a broad-band signal if the source is localized in space. This is true in the regime of
parameters reflected by the scaling θ≪ε≪δconsidered here, which is a high frequency
regime encountered in optical or infrared applications like ladar. The numerical experiments
in [7] and [8] are closer to the regime of ultrasound experiments [16] and in underwater sound
propagation, which is different from the high frequency regime analyzed here.
For distributed sources the parameter βis large and we cannot apply Theorems 2.1 and
2.2 to (3.6). It is necessary for statistical stability in this case to have broad-band signals.
For βlarge the time reversed and back propagated signal in the time domain has the form
ψB(L, x0, ξ, t)(3.7)
= (2π)dei(p0·ξ−k0c0t)ψ0(ξ/β)ZW(L, x0,p0
k0+k)e−ikc0tˆg(−c0k)c0dk
2π
= (2π)dei(p0·ξ−ω0t)ψ0(ξ/β)ZW(L, x0,c0p0
ω0+ω)e−iωt ˆg(−ω)dω
2π
with ˆg(c0k) the Fourier transform of the initial pulse relative to the central frequency ω0=
c0k0. This means that we have replaced the actual wave number kby k0+k, or ωby ω0+ω,
with the new ω, the baseband frequency, bounded by Ω, the bandwidth, |ω| ≤ Ω< ω0.
The integration is over the bandwidth [−Ω,Ω]. This integral is well defined with probability
one and is self-averaging in the broad beam limit δ→0 by Theorem 2.2 and the remark
following it. We will compute its average in Section 3.4.
3.3. The effective aperture of the array. From the explicit expression for the
Green’s function of (2.14), with k= 1,
U(z, x,p;x0,p0) = Zdwdr
(2π)2dexp iw·(x−x0) + ir·(p−p0)−izw·p0
×exp −Dz
2r2+zr·w+w2z2
3,
and with the time reversal mirror a distance Lfrom the source and x0= 0, it follows from
(3.6) that
hψB(z, ξ;k)i=(3.8)
Zdpdydw
(2π)2deip·ξβdψ0(β(p−p0); −k)χA(y) exp −iw·y−izw·p
k−Dz3w2
6
12 G. PAPANICOLAOU, L. RYZHIK AND K. SØLNA
The high-frequency, white-noise limit of the self-averaging time-reversed and back-propagated
field is therefore given by a convolution
hψB(L, ξ;k)i=ψβ
0(·,−k)∗ W(·)(ξ)(3.9)
with
W(η) = W(η;L, k) = kd
(2πL)dˆχA(ηk/L)e−η2/(2σ2
M),(3.10)
the point spread function, and
ψβ
0(η, −k) = eip0·ηψ0(η/β)ˆg(−kc0)(3.11)
with ψ0(η/β) the spatial source distribution function and ˆgthe Fourier transform of the
pulse shape function g(t). This notation is consistent with (3.7), with the time factor
e−ik0c0tomitted, along with the horizontal phase eikz which cancels in time reversal. We
have also introduced the refocused spot size with multipathing
σ2
M=3
DLk2=L2
k2a2
e
(3.12)
and the effective aperture ae=ae(L),
ae=rDL3
3,(3.13)
which we now interpret.
If the time reversal mirror is the whole plane z=L, then χA≡1 and
ψB(L, ξ;k)=ψβ
0(ξ, −k).
In this case the back-propagated field is the source field reversed in time, both in the random
and in the deterministic case. The point spread function Wdetermines the resolution of
the refocused signal for a time reversal mirror of finite aperture. Multipathing in a random
medium gives rise to the Gaussian factor (3.12) whose variance is σ2
M. We can give an
interpretation of this variance, or spot size, as follows. For a square time reversal mirror of
size a, the Fourier transform of χAis the sinc function so that
W(η1, η2;L, k) = 1
πL 2
sin(η1ka
2L) sin( η2ka
2L)e−(η2
1+η2
2)/(2σ2
M)
For a deterministic medium (D= 0) the Rayleigh resolution is the distance ηFto the first
zero of the sine, the first Fresnel zone in either direction,
ηF=2πL
ka =λL
a.
In general, if χAis supported by a region of size awe may define the Fresnel resolution, or
the Fresnel spot size, by
σF=L
ka .
Parabolic approximation and time reversal 13
For weak multipathing we have σM≫σFand
W(η;L, k)∼k
2πL d
ˆχA(ηk/L),
which is the diffractive point spread function whose integral over η∈Rdis one. If, however,
we have strong multipathing,σM≪σF, then we may approximate ˆχA(ηk/L) by ˆχA(0) =
adin (3.10), and the point spread function becomes
W(η;L, k)∼ka
2πL d
e−|η|2/(2σ2
M).
By writing the variance (spot size) σ2
Min the form (3.12) we can interpret aeas an effective
aperture of the time reversal mirror. We can rewrite the point spread function in terms of
a normalized Gaussian as
W(η;L, k)∼σM
√2πσFde−|η|2/(2σ2
M)
(2πσ2
M)d/2
with the factor in front of the normalized Gaussian also equal to
a
√2πaed
.
This means that when there is strong multipathing the integral of the point spread function
over Rdis not equal to one but to this ratio, which can be much smaller than one if ae≫a.
Multipathing produces a tighter point spread function but there is also loss of energy, as of
course we should expect.
A more direct interpretation for the effective aperture can be given if the time reversal
mirror has a Gaussian aperture function
χA(η) = e−|η|2/(2a2).
The point spread function Whas now the form
W(η;L, k) = ka
√2πL d
e−|η|2/(2σ2
g),
with
σg=L
kag
and the effective aperture aggiven by
ag=ra2+DL3
3=pa2+a2
e.
Clearly, ag≈aewhen there is strong multipathing and ae≫a. Written with a normalized
Gaussian the point spread function for a Gaussian aperture has the form
W(η) = a
agde−|η|2/(2σ2
g)
(2πσ2
g)d/2.
14 G. PAPANICOLAOU, L. RYZHIK AND K. SØLNA
L
σs
a
|p0|L
k0
Fig. 3.1.A directed field propagates from a distributed source of size σstoward the time reversal
mirror of size a. The time-reversed, back-propagated field depends on the location of the mirror relative to
the direction of the propagating beam.
3.4. Broad-band time reversal for distributed sources. For a distributed source,
its support σsis large compared to the Fresnel number θso the ratio β=σs/θ is large. In
this case we can compute the average of (3.7) the same way as we did in (3.8) and we find
that
hΨB(L, x0, ξ, t)i(3.14)
= (2π)dei(p0·ξ−k0c0t)ψ0(ξ/β)ZhW(L, x0,p0
k0+k)ie−ikc0tˆg(−c0k)c0dk
2π
=ei(p0·ξ−k0c0t)ψ0(ξ/β)Zdydwc0dk
(2π)d+1 χA(y)ei(Lwp0
k0+k−w·y−kc0t)e−DL3w2
6ˆg(−c0k).
The yintegral on the right gives the Fourier transform of the aperture function χA(y) so
with ω0=c0k0and a change of variable from kto ω=c0kwe have
hΨB(L, x0, ξ, t)i(3.15)
=ei(p0·ξ−ω0t)ψ0(ξ/β)Zdω
2πe−iωt ˆg(−ω)χA∗ e−x2/(2a2
e)
(2πa2
e)d/2!(Lc0p0
ω0+ω)
Here the star denotes convolution with respect to the spatial variables x, and aeis the
effective aperture defined by (3.13).
When multipathing is weak we can ignore the Gaussian factor in the convolution and
we have
hΨB(L, x0, ξ, t)i(3.16)
=ei(p0·ξ−ω0t)ψ0(ξ/β)Zdω
2πe−iωt ˆg(−ω)χALc0p0
ω0+ω
In the opposite case, when there is strong multipathing and the effective aperture is much
larger than the physical one, ae≫a, we have
hΨB(L, x0, ξ, t)i(3.17)
Parabolic approximation and time reversal 15
=ei(p0·ξ−ω0t)ψ0(ξ/β)a
√2πaedZdω
2πe−iωt ˆg(−ω)e−1
2(Lc0p0
ae(ω0+ω))2
To interpret these results we note first that a distributed source function of the form
(3.11) can be considered as a phased array emitting an inhomogeneous plane wave, a beam,
in the direction (k, p0), within the paraxial or parabolic approximation. The ratio |p0|/k
is the tangent of the angle the direction vector makes with the zaxis, and L|p0|/k is the
transverse distance of the beam center to the center of the phased array (see Figure 3.1). If
for each ωthe beam displacement vector Lc0p0/(ω0+ω) is inside the set Aoccupied by the
time reversal array, then we recover at the source the full pulse in (3.16), time-reversed,
hΨB(L, x0, ξ, t)i=ei(p0·ξ−ω0t)ψ0(ξ/β)g(−t).
If, however, for some frequencies the transverse displacement vector is outside the time
reversal array, these frequencies will be nulled in the integration and a distorted time pulse
will be received at the source. Depending on the position of the time reversal mirror relative
to the beam, high or low frequencies may be nulled.
In a strongly multipathing medium the situation is quite different because the expression
(3.17), or more generally (3.15), holds now. Even if the beam from the phased array does not
intercept the time reversal mirror at all, we will still get a time reversed signal at the source
but with a much diminished amplitude. If the beam falls entirely within the time reversal
mirror then the time reversed pulse will be a distorted form of g(−t), with its amplitude
reduced by the factor (a/ae)d. An interesting and important application of the time reversal
of a beam in a random medium is the possibility of estimating the effective aperture ae
by pointing the beam in different directions toward the time reversal mirror, measuring the
time reversed signal that back propagates to the source, that is, to the phased array, and
inferring aeby fitting the measurements to (3.15).
4. Summary and conclusions. We have analyzed and explained two important phe-
nomena associated with time reversal in a random medium:
•Super-resolution of the back-propagated signal due to multipathing
•Self-averaging that gives a statistically stable refocusing
Our analysis is based on a specific asymptotic limit (see Section 2.1) where the longitudinal
distance of propagation is much larger than the size of the time-reversal mirror, which in
turn is much larger than the correlation length of the medium, fluctuations in the index of
refraction are weak, and the wave length is short compared to the correlation length. This
asymptotic regime is more relevant to optical or infrared time reversal than it is to sonar or
ultrasound. We have related the self-averaging properties of the back-propagated signal to
those of functionals of the Wigner distribution. Self-averaging of these functionals implies
the statistical stability of the time-reversed and back-propagated signal in the frequency
domain, provided that the source function is not too broad compared to the Fresnel number
(2.4). Time reversal refocusing of waves emitted from a distributed source is self-averaging
only in the time domain.
We apply our theoretical results about stochastic Wigner distributions to time reversal
and discuss in detail super-resolution and statistical stability in section 3.
Appendix A. The white noise limit and the parabolic approximation.
We collect here some comments on the scaling analysis of section 2.1 and refer to [1, 27,
33] for additional comments and results on scaling and asymptotics in the high-frequency
and white-noise regime.
The dimensionless parameters δ, ε, γ introduced by (2.2) in Section 2.1, along with the
Fresnel number θdefined by (2.4), lead to the scaled parabolic wave equation (2.5). If we do
16 G. PAPANICOLAOU, L. RYZHIK AND K. SØLNA
not make the parabolic approximation and keep the ψzz term we have the scaled Helmholtz
equation, with the phase eikz removed,
ε2θ2
δ2ψzz + 2ikθψz+θ2∆xψ+k2δ
ε1/2µ(x
δ,z
ε)ψ= 0.(A.1)
Here, as in (2.5), we relate the strength of the fluctuations σto εand δby (2.6). Is the
parabolic approximation valid in the ordering (2.7), θ≪ε≪δ≪1, that we have analyzed?
The answer is yes but not before both θand εlimits have been taken, in which case the
scaled Wigner distribution (2.8) converges to the Liouville-Ito process that is defined by the
stochastic partial differential equation (2.19).
It is in the white noise limit ε→0, with Fresnel number θand δfixed, that the parabolic
approximation is valid for (A.1), as was pointed out in [1]. This is easily seen if the random
fluctuations µare differentiable in z. The parabolic approximation is clearly not valid in the
high frequency limit θ→0, before the white noise limit ε→0 is also taken. In the white
noise limit, the wave function ψ(z, x) satisfies an Ito-Schr¨odinger equation
2ikθdzψ+θ2∆xψdz +ik3δ2
4θR0(0)ψdz +k2δψdzB(x
δ, z) = 0.(A.2)
Here R0is the integrated covariance of the fluctuations µgiven by (2.15) and (2.13), and
the Brownian field B(x, z) has covariance
hB(x, z1)B(y, z2)i=R0(x−y)z1∧z2.
This Ito-Schr¨odinger equation is the result of the central limit theorem applied to (A.1). Let
Bε(x, z) = 1
√εZz
0
µ(x,s
ε)ds.
Then, as ε→0 this process converges weakly, under suitable hypotheses, to the Brown-
ian field B(x, z) with the above covariance. The extra term in (A.2) is the Stratonovich
correction.
The white noise limit for stochastic partial differential equations is analyzed in [10] and a
rigrous theory of the Ito-Schr¨odinger equation is given in [11]. The ergodic theory of the Ito-
Schroedinger equation is explored in [17]. Wave propagation in the parabolic approximation
with white-noise fluctuations is considered in detail in [18, 30].
The scaled Wigner distribution for the process ψ, defined by (2.8), satisfies the stochastic
transport equation
dzWθ(z, x,p) + p
k· ∇xWθ(z, x,p)dz(A.3)
=k2δ2
4θ2Zdq
(2π)dˆ
R0(q)Wθ(z, x,p+θq
δ)−Wθ(z, x,p)dz
+ikδ
2θZdq
(2π)deiq·x/δ Wθ(z, x,p−θq
2δ)−Wθ(z, x,p+θq
2δ)dzˆ
B(q, z),
which is derived from (A.2) equation using the Ito calculus. The Wigner process Wθcon-
verges in the limit θ→0 to the Liouville-Ito process defined by the stochastic partial
differential equation (2.19).
Parabolic approximation and time reversal 17
Appendix B. Decorrelation of the Wigner process.
B.1. Proof of Theorem 2.1. We give here the proof of Theorems 2.1 and 2.2. We
consider Theorem 2.1 first. It will follow from the Lebesgue dominated convergence theorem
if we show that for p16=p2:
E{Wδ(z, x,p1)Wδ(z, x,p2)} − E{Wδ(z , x,p1)}E{Wδ(z, x,p2)} → 0(B.1)
as δ→0 because the function Wδis uniformly bounded and E{Wδ(z, x,p1)}does not
depend on δ. Furthermore, the correlation function at the same spatial point but for two
different values of the wave vector, U(2)
δ(z, x,p1,p2) = E{Wδ(z , x,p1)Wδ(z, x,p2)}is the
solution of (2.18) with N= 2 and the initial data
W(2)
δ(0,x1,p1,x2,p2) = WI(x1,p1)WI(x2,p2),
evaluated at x1=x2=x. Therefore U(2)
δmay be represented as
U(2)
δ(z, x1,p1,x2,p2) = EWI(X1
δ(z),P1
δ(z))WI(X2
δ(z),P2
δ(z)).
The processes X1,2
δand P1,2
δsatisfy the system of SDE’s (2.16) which may be more explicitly
written as
dP1
δ=−σ(0)dB1(z) + 1
2σX1
δ−X2
δ
δdB2(z)
(B.2)
dP2
δ=−σ(0)dB2(z) + 1
2σX2
δ−X1
δ
δdB1(z)
dX1
δ=−P1
δdz, dX2
δ=−P2
δdz
with the initial conditions X1,2
δ(0) = x,Pm
δ(0) = pm,m= 1,2. Here σ2(0) = D, the
diffusion coefficient (2.15) and the coupling matrix σ(x) is given by (2.17). Recall that
Wδ(z, x,p, k) = Wδ(z, x,p/k; 1) and we need only consider the case k= 1.
It is convenient to introduce the processes X1,2and P1,2that are solutions of (2.16)
with no coupling:
dPm=−σ(0)dBm(z), dXm=−Pmdz,(B.3)
X1,2(0) = x,Pm(0) = pm, m = 1,2
and define the deviations of the solutions of the coupled system of SDE’s (B.2) from those
of (B.3): Zm
δ=Xm
δ−Xm,Sm
δ=Pm
δ−Pm. Then we have
dS1
δ=−1
2σX1
δ−X2
δ
δdB2(z), dS2
δ=−1
2σX2
δ−X1
δ
δdB1(z)(B.4)
dZ1
δ=−S1
δdz, dZ2
δ=−S2
δdz
with the initial data Sm
δ(0) = Zm(0) = 0. Define
V(X1,X2,P1,P2,Z1
δ,Z2
δ,S1
δ,S2
δ)(B.5)
=WI(X1+Z1
δ,P1+S1
δ)WI(X2+Z2
δ,P2+S2
δ)−WI(X1,P1)WI(X2,P2)
then we have with the above notation
E{Wδ(z, x,p1)Wδ(z, x,p2)} − E{Wδ(z, x,p1)}E{Wδ(z, x,p2)}(B.6)
=EV(X1(z),X2(z),P1(z),P2(z),Z1
δ(z),Z2
δ(z),S1
δ(z),S2
δ(z))
≤CE |Z1
δ(z)|+|Z2
δ(z)|+|S1
δ(z)|+|S2
δ(z)|
18 G. PAPANICOLAOU, L. RYZHIK AND K. SØLNA
since WIis a Lipschitz function.
Let us assume for simplicity that the correlation function R(x) has compact support
inside the set |x| ≤ M. Then the coupling term in (B.2) is non-zero only when |X1
δ−X2
δ| ≤
Mδ. We introduce the processes Qδ=P1
δ−P2
δand Yδ=X1
δ−X2
δthat govern (B.4). They
satisfy the SDE’s
dQδ=−σ(0) −1
2σYδ
δd˜
B, dYδ=−Qδdz,(B.7)
Qδ(0) = p1−p2,Yδ(0) = 0
with ˜
B=B1−B2being a Brownian motion.
In order to prove the theorem we show that the coupling term σ(·) in (B.2) introduces
only lower order correction terms, that is, Sm
δand Zm
δare small. We show first that after a
small ‘time’, τ, the points Xm
δare driven apart since Qδ(0) = P1
δ(0) −P2
δ(0) 6= 0. Then we
show that after the points have separated the probability that they come close, so that the
coupling term σ(·) becomes non-zero, is small. This “non-recurrence” condition requires that
the spatial dimension d≥2. It follows that to leading order the points Xm
δare uncorrelated
when d≥2 and that the coupling term introduces only lower order corrections. A similar
argument for d= 1 would require an estimate on the time that points that are originally
separated in the spatial variable spend near each other, where the coupling term in (B.2) is
not zero.
We need the following two Lemmas. The first one shows that particles that start at
the same point xwith different initial directions p1and p2, get separated with a large
probability:
Lemma B.1. Let Yδ,Qδsolve (B.7) with Yδ(0) = 0,Qδ(0) = q6= 0. Then for any
ε > 0there exists τ0(ε)>0that depends only on q=p1−p2but not on δso that we have
P|Yδ(τ)| ≥ |q|τ
2≥1−εfor all τ≤τ0(ε).
The second lemma shows that after the particles are separated, the probability that
they come close to each other is small:
Lemma B.2. Given any fixed r > 0and z > 0, if Yδ,Qδsolve (B.7) with |Yδ(0)| ≥ r,
Qδ(0) = q6= 0, then P(inf0≤s≤z|Yδ(s)| ≤ Mδ)→0as δ→0.
We prove Theorem 2.1 before proving Lemmas B.1 and B.2:
Proof. Let zand q=p1−p2be fixed and defined as above. Given ε > 0, then for any
τ < τ0(ε) (with τ0as defined in Lemma B.1), Lemma B.2 and the Markov property of the
Brownian motion imply that
PSm
δ(z) = Sm
δ(τ)|Yδ(τ)| ≥ τ|q|
2≥1−ε
and
PZm
δ(z) = Zm
δ(τ) + (z−τ)Sm
δ(τ)|Yδ(τ)| ≥ τ|q|
2≥1−ε
for δ < δ0(τ, ε). Furthermore,
E|Z1
δ(τ)|+|Z2
δ(τ)|+|S1
δ(τ)|+|S2
δ(τ)||Yδ(τ)| ≥ τ|q|
2
(B.8)
≤E|Z1
δ(τ)|+|Z2
δ(τ)|+|S1
δ(τ)|+|S2
δ(τ)|/(1 −ε)≤Cτ
Parabolic approximation and time reversal 19
because the function σis uniformly bounded. Therefore we have
EV(X1,X2,P1,P2,Z1
δ,Z2
δ,S1
δ,S2
δ)
=EV(X1,X2,P1,P2,Z1
δ,Z2
δ,S1
δ,S2
δ)|Yδ(τ)| ≥ τ|q|
2P|Yδ(τ)| ≥ τ|q|
2
(B.9)
+EV(X1,X2,P1,P2,Z1
δ,Z2
δ,S1
δ,S2
δ)|Yδ(τ)| ≤ τ|q|
2P|Yδ(τ)| ≤ τ|q|
2=I+II.
The second term above is small because the probability for Yδ(τ) to be very small is bounded
by Lemma B.1. More precisely, given ε > 0 and τ < τ0(ε), Lemma B.1 implies that
II ≤Cε.(B.10)
The first term in (B.9) corresponds to the more likely scenario that Yδat time τhas
left the ball of radius τ|q|/2. We estimate it as follows. The probability that Yδre-enters
the ball of radius M δ is small according to Lemma B.2. Moreover, if Yδstays outside this
ball, the difference variables Zmand Smare bounded in terms of their values at time τ. The
latter are small if τis small. More precisely, using (B.8) we choose τso small that
E|Z1
δ(τ)|+|Z2
δ(τ)|+|S1
δ(τ)|+|S2
δ(τ)||Yδ(τ)| ≥ τ|q|
2≤ε.
Then we obtain
I≤EV(X1,X2,P1,P2,Z1
δ,Z2
δ,S1
δ,S2
δ)|Yδ(τ)| ≥ τ|q|
2
≤EV(X1,X2,P1,P2,Z1
δ,Z2
δ,S1
δ,S2
δ)|Yδ(τ)| ≥ τ|q|
2and inf
τ≤s≤z|Yδ(s)| ≤ Mδ
×Pinf
τ≤s≤z|Yδ(s)| ≤ Mδ|Yδ(τ)| ≥ τ|q|
2
+E|Z1
δ(z)|+|Z2
δ(z)|+|S1
δ(z)|+|S2
δ(z)||Yδ(τ)| ≥ τ|q|
2and inf
τ≤s≤z|Yδ(s)| ≥ Mδ
×Pinf
τ≤s≤z|Yδ(s)| ≥ Mδ|Yδ(τ)| ≥ τ|q|
2=I1+I2.
The term I1goes to zero as δ→0 by Lemma B.2. However, if the conditions in I2hold,
then
Sm
δ(z) = Sm
δ(τ),Zm
δ(z) = Zm
δ(τ)−1
k(z−τ)Sm
δ(τ).
Therefore the term I2may be bounded with the help of (B.8) by
I2≤E|Z1
δ(z)|+|Z2
δ(z)|+|S1
δ(z)|+|S2
δ(z)||Yδ(τ)| ≥ τ|q|
2and inf
τ≤s≤z|Yδ(s)| ≥ Mδ
≤CE |Z1
δ(τ)|+|Z2
δ(τ)|+|S1
δ(τ)|+|S2
δ(τ)||Yδ(τ)| ≥ τ|q|
2≤Cτ.
Putting together (B.9), (B.10) and the above bounds on I1and I2, we obtain
E|Z1
δ(z)|+|Z2
δ(z)|+|S1
δ(z)|+|S2
δ(z)|≤Cε
for δ < ¯
δand Theorem 2.1 follows from (B.6).
20 G. PAPANICOLAOU, L. RYZHIK AND K. SØLNA
B.2. Proof of Lemmas B.1 and B.2. We first prove Lemma B.1.
Proof. We write
Qδ(z) = q−Zz
0σ(0) −1
2σ(Y(s)/δ)d˜
B(s)≡q+˜
Qδ(z)
so that
Yδ(t) = −qt−Zt
0
˜
Qδ(s)ds.
Then we have
Psup
0≤s≤τ|˜
Qδ(s)|> r≤Cτ/r2
(B.11)
and hence
P(|Yδ(τ) + τq|> rτ)≤Psup
0≤s≤τ|˜
Qδ(s)|> r≤Cτ/r2.
We let r=|q|/2 in the above formula and obtain
P|Yδ(τ)|<τ|q|
2≤C
|q|2τ,
and the conclusion of Lemma B.1 follows.
Finally, we prove Lemma B.2.
Proof. Let τδbe the first time Yδ(z) enters the ball of radius Mδ:
τδ= inf {z:|Yδ(z)| ≤ M δ},
with Yδ(0) = Y06= 0. For 0 < α < 1 let ∆z=δ1−α,n=⌈z/∆z⌉,Ji= (i∆z, (i+ 1)∆z) and
p < 1. Note that until the time τδthe process (Yδ,Qδ) coincides with the process (Y,Q)
governed by (B.7) without the coupling term σ(Yδ/δ). We find
P(τδ< z)≤
n−1
X
i=0 P(|Y(i∆z)|< Mδp) + Pinf
s∈Ji|Y(s)|< Mδ |Y(i∆z)| ≥ M δp.
The process Y(s) is Gaussian with mean Y0and variance O(s2). Therefore, there is a ¯
δ > 0
such that for δ < ¯
δ
P(|Y(i∆z)|< Mδp)≤Cδdp .
If we assume
p < 1−α(B.12)
then also
P(τδ< z)≤nC δdp +Psup
0<s<∆z|Y(s)−Y0| ≥ M[δp−δ]
≤C(δdp+α−1+δα−1Psup
0<s<∆z|B(s)| ≥ M[δp−δ]/∆z)
≤C(δdp+α−1+δα−1EB(∆z)2r∆z2r
(δp−δ)2r)
≤Chδdp+α−1+δα−1−rp+3r(1−α)/2i.
Parabolic approximation and time reversal 21
Note that with p < 1−αand rlarge enough, there is a q > 0 so that
P(τδ< z)≤Cδq
if d≥2 and Lemma B.2 follows.
B.3. Proof of Theorem 2.2. We need to show first that
Jδ(z, x) = ZWδ(z , x,p)dp(B.13)
is finite with probability one. The stochastic flow (Xδ(t, x,p),Pδ(t, x,p) is continuous in
(t, x,p) with probability one, so Wδ(z , x,p) = WI(Xδ(t, x,p),Pδ(t, x,p)) is bounded and
continuous. It is, moreover, non-negative if WI≥0. We know that
ZE{Wδ(z, x,p)}dp
is finite and independent of δ, and the order of integration and expectation can be inter-
changed by Tonelli’s theorem. This theorem implies in addition that Jδ(z, x) is finite with
probability one.
We can now consider
E{J2
δ(z, x)}=ZE{Wδ(z, x,p1)Wδ(z, x,p2)}dp1dp2.
The integrand is bounded by an integrable function uniformly in δbecause
E{Wδ(z, x,p1)Wδ(z, x,p2)} ≤ E1/2{W2
δ(z, x,p1)}E1/2{Wδ(z, x,p2)},
the right side does not depend on δ, and is integrable. Therefore by the Lebesgue dominated
convergence theorem and the results of the previous Section we have that
lim
δ→0E{J2
δ(z, x)}=E2{Jδ(z, x)}
and the right side does not depend on δ. This completes the proof of Theorem 2.2.
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