On an imaginary exponential functional of Brownian motion

7.1 MB·English

by  D. Gredat

On an imaginary exponential functional of
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1 Brownian motion
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n D Gredat1, I Dornic1, and J M Luck2
a
J 1 Service de Physique de l’Etat Condens´e, IRAMIS/SPEC, CEA Saclay and

6 URA 2464, CNRS, 91191 Gif-sur-Yvette cedex, France
2 Institut de Physique Th´eorique, IPhT, CEA Saclay and URA 2306, CNRS, 91191
]
h Gif-sur-Yvette cedex, France
c
e
m Abstract. We investigatearandomintegralwhichprovidesa naturalexampleofan
- imaginaryexponentialfunctionalofBrownianmotion. Thisfunctionalshowsupinthe
t
a study of the binary annihilation process, within the Doi-Peliti formalism for reaction-
t
s diffusion systems. The main emphasis is put on the complementarity between the
.
t usual Langevin approach and another approach based on the similarity with Kesten
a
m variablesand other one-dimensionaldisorderedsystems. Eventhough neither of these
routesleadstothefullsolutionoftheproblem,wehaveobtainedacollectionofresults
-
d describing various regimes of interest.
n
o
c
[
PACS numbers: 02.50.Cw, 05.40.Jc, 02.50.Fz

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E-mail: [email protected],[email protected],[email protected]
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On an imaginary exponential functional of Brownian motion 2
1. Introduction
Real exponential functionals of Brownian motion have been the subject of much activity
in probability theory [1]. They have found many applications to problems ranging from
finance to physics [2]. To be more specific, if B(t) is Brownian motion (a Wiener process
with B(0) = 0 and B(t)2 = t), the following stochastic variable
h i
t
X(t) = e s+gB(s)ds, (1.1)
−
Z0
where g is a real coupling constant, shows up in one guise or another in various models
of disordered systems [3, 4]. The above random integral can be shown to represent the
solution of the following Langevin equation with multiplicative noise:
dX(t)
= 1 X(t)+gX(t)η(t), (1.2)
dt −
with η(t) dB(t)/dt being a zero-mean Gaussian white noise, normalized as
≡
η(s)η(t) = δ(t s). (1.3)
h i −
Thederivationofthesolutionto(1.2)intheform(1.1), including therequired stochastic
calculus prescription (Stratonovich), will be reviewed in Section 2. As an illustration of
theversatility ofthesituations inwhich therandomvariableX(t)appears, theLangevin
equation (1.2) may serve to study the effect of multiplicative noise on a deterministic
fixed point. Indeed, the linear force 1 X(t) stabilizes X(t) at the fixed point X = 1
−
in the absence of noise. The solution X(t) however keeps on fluctuating forever because
of the multiplicative noise term gX(t)η(t). In the long-time limit, what emerges out of
this interplay between the deterministic restoring force and the fluctuating noise term
is a non-trivial distribution for the random variable
X = ∞e t+gB(t)dt, (1.4)
−
Z0
representing the stationary solution of (1.2). The distribution of X lim X(t) can
t
≡ →∞
be computed in a number of ways, and we shall review in Section 2 two useful methods
to do so. Using either more elaborate path-integral methods or probabilistic identities,
the full-time dependent distribution of X(t) can also be obtained, and features, among
other things, an interesting continuous spectrum of relaxation rates [2].
A seemingly innocuous and somewhat natural generalization of (1.4) consists in
analytically continuing the coupling constant g ig. We thus obtain the following
→
random integral, defining an imaginary exponential functional of Brownian motion:
Z = ∞e t+igB(t)dt, (1.5)
−
Z0
which now lives in the complex plane (actually in the unit disk).
The main goal of the present article is to investigate the distribution of the
functional Z. Throughout the following, instead of the coupling constant g, we rather
make use of the dimensionless diffusion constant
g2
D = . (1.6)
2
On an imaginary exponential functional of Brownian motion 3
As hinted at above, even though a large number of works have been devoted to real
exponential functionals of Brownian motion, much less is known about the distribution
of complex functionals of Brownian motion such as (1.5). We found interesting from a
conceptual viewpoint to tackle this problem, in particular in order to see if any of the
methods which proved successful for the real case would extend to this complex-variable
setting. Besides this, it turns out that complex stochastic processes have surfaced time
and again in different scientific disciplines ranging from signal theory, where processes
involving imaginary exponential functionals of Brownian motion occur in the study of
phase noise [5], to quantum optics [6], in conjunction with the development of phase-
space representations and of the associated formalism of quasi-probabilities, and finally
to reaction-diffusion processes [7, 8] through the Doi-Peliti approach [9]. The latter
topic has constituted our original thrust to embark on the study of (1.5). Let us now
describe how the connection emerges.
In the context of interacting particle systems such as reaction-diffusion systems,
there exists a standard set of techniques, usually referred to as the Doi-Peliti
formalism [9] (see [10] for a recent review, and the references therein), which allows
to recast the master equation describing the evolution of these processes in terms of
a field theory whose action involves a pair of conjugate fields. Without entering into
much detail, provided certain technical conditions are met, the theory can in turn be
transformedintoaLangevinequationforasingledensity fieldϕ (t), customarilydubbed
n
so because its noise-average ϕ (t) , coincides with the local mean particle number for
n
h i
the underlying reaction-diffusion process ρ (t) . Here the double brackets denote an
n
hh ii
averageover thedynamicsoftheparticles. Inthecaseofthebinaryannihilationreaction
A+A , (1.7)
→ ∅
where particles A diffuse by hopping with rate κ on a hypercubic lattice in dimension d,
and annihilate pairwise with rate λ when they meet on a given lattice site n, one can
show [7, 8, 9] that the stochastic density field ϕ (t) obeys the following Langevin-Itˆo
n
equation:
dϕ (t)
n = κ( 2ϕ) (t) λϕ (t)2 +ζ (t), (1.8)
n n n
dt ∇ −
with 2 being the lattice Laplacian, such that, e.g., ( 2ϕ) = ϕ + ϕ 2ϕ
n n+1 n 1 n
∇ ∇ − −
in one dimension. It can be expected on physical grounds that the amplitude of the
Gaussian noise term ζ [ϕ ] vanishes when ϕ = 0. Indeed, the first two terms on the
n n n
right-hand side of (1.8) respectively account for the diffusion of the particles and for the
(mean-field) decay rate of the particle density due to pairwise annihilation. If the noise
amplitude vanishes when ϕ = 0, there is no evolution at all in regions where ϕ = 0.
n n
The Doi-Peliti approach also yields the following expression for the correlator of the
noise:
ζ (s)ζ (t) = λϕ (t)2δ δ(t s), (1.9)
m n n m,n
h i − −
which therefore has a negative variance. In other words, we have ζ (t) = i√λϕ (t)η (t),
n n n
where η (t) is a normalized real Gaussian white noise. This explains why (1.8)
n
On an imaginary exponential functional of Brownian motion 4
is often referred to as an imaginary-noise equation. One would obtain the same
Langevin equation using Gardiner’s quasi-probability formalism, where the particle-
numberprobabilitydistributionisrepresented asasuperpositionofPoissondistributions
with weights ϕ [8, 11]. The equivalence between the Doi-Peliti formalism and
n
Gardiner’s Poisson representation method has been demonstrated in general in [12].
There is no contradiction in either formalism, as soon as the auxiliary field ϕ (t)
n
is complex-valued, provided one refrains from erroneously identifying ϕ (t) with the
n
(integer-valued) stochastic variable ρ (t), based on the sole equality between the mean
n
values ϕ (t) = ρ (t) . In fact, the precise relationship between the distribution of
n n
h i hh ii
ϕ (t) and that of ρ (t) is that the ordinary moments of ϕ (t) are equal to the factorial
n n n
moments of ρ (t). This is a particular instance of what is referred to as duality between
n
two stochastic processes in the probabilistic literature [13]. Thus in particular:
ρ (t)(ρ (t) 1) = ϕ (t)2 , (1.10)
n n n
hh − ii h i
and so one can have ϕ (t)2 < 0, while keeping varρ (t) = ρ (t)2 ρ (t) 2 =
n n n n
h i hh ii − hh ii
ϕ (t)2 + ϕ (t) ϕ (t) 2 0. It is also commonly accepted that the complex-
n n n
h i h i − h i ≥
valued nature of the trajectories of the field ϕ (t) is needed in order to account for the
n
importance of fluctuation effects in low spatial dimensions, resulting in a slower decay
for the total density of particles than what the naive law of mass action would predict
(viz. t d/2 vs. 1/t in d < 2 for the reaction (1.7) [10, 14]).
−
Henceforth, along the lines of [7, 8], we focusonto thesingle-site problem associated
with (1.8), neglecting any spatial dependence. This simplification will allow us to better
understand the role of the excursions of the field ϕ(t) in the complex plane. Setting
κ = 0, and absorbing the reaction rate λ into the time scale, one ends up with the
following Langevin-Itˆo equation for a single complex stochastic variable ϕ(t):
dϕ(t)
= ϕ(t)2 +iϕ(t)η(t), (1.11)
dt −
where η(t) is a normalized Gaussian white noise (see (1.3)). The initial condition ϕ(0)
is real and non-negative (e.g., ϕ(0) = ρ if one starts from a Poisson distribution with
0
density ρ for the original particle system). The conjugation symmetry ϕ ϕ¯ ensures
0
→
that the imaginary part of ϕ(t) averages over to zero, so as to maintain the reality
and the non-negativity of ϕ(t) = ρ(t) . It has already been noticed by several
h i hh ii
authors [6, 7, 8] that (1.11) becomes linear in the variable (t) = 1/ϕ(t). One thus
Z
obtains the explicit solution
t
(t) = (0)e t/2 iB(t) + e (t s)/2+i[B(s) B(t)]ds. (1.12)
− − − − −
Z Z
Z0
Rescaling time, and using the scaling property B(at) √aB(t) of Brownian motion,
≡
we obtain by identifying (1.5) and (1.12) the following identity between the stationary
solution lim (t) and the functional Z for g = √2, i.e., D = 1 [8]:
t
Z ≡ →∞Z
= lim (t) 2Z = 2 ∞e t+i√2B(t)dt. (1.13)
Z t→∞Z ≡ |D=1 Z0 −
The above representation of the stationary solution has striking consequences. At
the level of the original process (1.7), the stationary state is rather featureless, as there
On an imaginary exponential functional of Brownian motion 5
just remains either zero or one particle, depending on the parity of the initial condition.
By (1.10), and the corresponding equations for higher-order moments, this implies that
ϕp = 0 for any integer p 2. Owing to (1.13), this reads Z p = 0. The
h i ≥ h − i|D=1
full distribution of Z will however turn out to be highly non-trivial. In particular, in
stark contrast to what intuition backing up (1.8) or (1.11) could let us foresee, in the
stationary state the variable ϕ = 2/Z never reaches the value ϕ = 0, characteristic of
the absorbing state. Indeed, as already announced, Z lies within the unit disk.
The setup of this article is the following. In Section 2 we present a self-contained
investigation of the real exponential functional X (see (1.4)). The emphasis is put
on the complementarity between the usual Langevin approach and another approach
based on the similarity with the random recursions met in the study of Kesten variables
and one-dimensional disordered systems. The main section of the paper (Section 3)
is devoted to a detailed study of the imaginary exponential functional Z (see (1.5)).
We present numerical illustrations of the distribution of Z and investigate many facets
of the problem by analytical means, including the relationship between the Langevin
and Kesten approaches, the moments ZkZ¯l , the weak-disorder and strong-disorder
h i
regimes, and the asymptotic behavior of the distribution near the unit circle. Section 4
contains a brief summary of our findings.
2. A warming up: the real functional
This section is to a large extent intended as a warming up. It is devoted to a self-
contained study of the real exponential functional (see (1.4))
X = ∞e t+gB(t)dt. (2.1)
−
Z0
The positive random variable thus defined is one of the exponential functionals of
Brownian motion which have been investigated in probability theory, chiefly by Yor
and his collaborators (see [1] for a review). It also appears in the physics literature, in
the context of one-dimensional disordered systems [2, 3, 4]. The coupling constant g
measuring the strength of noise, or, equivalently, the diffusion constant D (see (1.6)), is
the sole parameter entering the definition of X.
2.1. Langevin approach
A first approach to study the random variable X consists in using Langevin equations.
At this point it is useful to recall some elements of stochastic calculus [11, 15, 16].
A Langevin equation of the form
dX(t)
= a(X(t))+b(X(t))η(t) (2.2)
dt
is ambiguous as soon as it is non-linear, in the sense that the noise η(t) multiplies a
non-trivial function b(X(t)) of the position X(t). This ambiguity due to the usage of
a continuous-time formalism can be lifted in many ways. The two most useful and
well-known prescriptions are the following (see [11, 16] for a detailed exposition):
On an imaginary exponential functional of Brownian motion 6
Stratonovich prescription. The Langevin-Stratonovich differential equation
•
dX(t)
[S] = a (X(t))+b(X(t))η(t) (2.3)
S
dt
can be essentially thought of as an ordinary differential equation. It is amenable
to non-linear changes of variable according to the usual rules of differential and
integral calculus. The main disadvantage is that X(t) and η(t) at the same time t
are not independent.
Itˆo prescription. The Langevin-Itˆo differential equation
•
dX(t)
[I] = a (X(t))+b(X(t))η(t) (2.4)
I
dt
has the advantage that the process X(t) and the noise η(t) at the same time t are
independent, sothatonehase.g. a (X) = 0inthestationarystate. Equation(2.4)
I
h i
alsoprovidesanaturaldiscretizationoftheprocessX(t). Consideringdiscretetimes
t = nε so that X X(t ), we obtain the recursion
n n
≡
X = X +a (X )ε+b(X )ζ , (2.5)
n+1 n I n n n+1
where ζ is a Gaussian random variable, independent of X , such that ζ = 0
n+1 n n+1
h i
and ζ2 = ε. This discrete scheme can be efficiently used in a numerical
h n+1i
simulation. The main disadvantage of the Itˆo prescription is that care must be
exercised when making non-linear changes of variable.
Both Langevin equations (2.3) and (2.4) describe the same stochastic process
X(t) if their drift terms are related to each other by the correspondence formula
{ }
1 d 1 db(X)
a (X) a (X) = b(X)2 = b(X) . (2.6)
I S
− 4 dX 2 dX
The corresponding time-dependent probability density P(x,t) = δ(X(t) x) obeys
h − i
the Fokker-Planck equation
∂P ∂ 1 ∂2
= a (x)P + b(x)2P
I
∂t − ∂x 2 ∂x2
(cid:16) (cid:17) (cid:16) (cid:17)
∂ 1 ∂ ∂
= a (x)P + b(x) b(x)P . (2.7)
S
− ∂x 2 ∂x ∂x
(cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
The stationary density of the process, f (x) lim P(x,t), reads
X t
≡ →∞
N x a (y) N x a (y)
I I S S
f (x) = exp 2 dy = exp 2 dy , (2.8)
X b(x)2 Zx0 b(y)2 ! b(x) Zx0 b(y)2 !
where x is an arbitrary initial point, and N and N are normalization constants.
0 I S
It is now time to return to our functional X (see (1.4)). The Langevin-Stratonovich
equation (1.2), i.e.,
dX(t)
[S] = 1 X(t)+gX(t)η(t), (2.9)
dt −
is equivalent to the Langevin-Itˆo equation
dX(t)
[I] = 1+(D 1)X(t)+gX(t)η(t). (2.10)
dt −
On an imaginary exponential functional of Brownian motion 7
Indeed b(X) = gX and a (X) = 1 X yield a (X) = 1+(D 1)X (see (1.6)).
S I
− −
The Langevin-Stratonovich equation (2.9) can be readily integrated. We thus
obtain the explicit stochastic representation of the process as
t
X(t) = X(0)e t+gB(t) + e (t s)+g(B(t) B(s))ds. (2.11)
− − − −
Z0
In the t limit, the above expression loses the memory of its initial condition X(0).
→ ∞
Moreover, using the stationarity of Brownian motion, the integral can be recast as an
integral over τ = t s, which identifies with (2.1) or (1.4). We have thus shown that
−
the exponential functional X represents the stationary solution of (2.9) or (2.10), i.e.,
X lim X(t).
t
≡ →∞
The Itˆo prescription allows us to directly read off the stationary mean value of X
from (2.10):
1
X = . (2.12)
h i 1 D
−
This expression only makes sense for D < 1. It is in agreement with the behavior of the
time-dependent mean value X(t) . Equation (2.10) indeed yields
h i
d X(t)
h i = 1+(D 1) X(t) , (2.13)
dt − h i
whose solution, i.e.,
1 1
X(t) = + X(0) e(D 1)t, (2.14)
−
h i 1 D − 1 D
− (cid:18) − (cid:19)
relaxes exponentially fast to (2.12) for D < 1, whereas it diverges for D > 1. The
result (2.14) can be recovered by averaging (2.11) over the Brownian motion B(s) .
{ }
Equation (2.8) leads to the following result for the distribution of the functional X:
1
f (x) = x (1+1/D)e 1/(Dx). (2.15)
X D1/DΓ(1/D) − −
This expression [1, 3, 4] falls off exponentially fast as x 0. It exhibits a fat tail
→
at large x, as it decreases as a power law, with the continuously variable exponent
(1+1/D). Asaconsequence, themoment Xs onlyconvergesfors < 1/D (see(2.30)).
− h i
This provides another way to explain the divergence of (2.12) at D = 1.
Finally, at par with the qualitative discussion given in the Introduction, one can
interpret the emergence of a fat tail at large values of X and the possible divergence of
the mean value X as consequences of the exchange of stability between the two fixed
h i
points of the dynamical system (2.9), i.e., the unstable deterministic one at X = 1 and
a fluctuating one sitting formally at X = .
∞
Figure 1 shows a plot of the density f for three values of D.
X
2.2. Kesten approach
An alternative approach to study the random variable X consists in using the similarity
between the random integral (2.1) and random sums of products referred to as Kesten
variables. Letusstartwithareminder. AKestenvariable[17,18,19,20,21]isdefinedas
Z = 1+ξ +ξ ξ +ξ ξ ξ + , (2.16)
1 1 2 1 2 3
···
On an imaginary exponential functional of Brownian motion 8
0.8
D=1/2
0.6
D=1
D=2
)
x 0.4
(
x
f
0.2
0
0 1 2 3 4 5
x
Figure 1. Plot of the probability density fX of the realfunctional X (see (2.15)), for
three values of the diffusion constant D.
where the ξ are i.i.d. positive random variables with probability density f (ξ). If the
n ξ
latter distribution is such that lnξ < 0, the sum (2.16) is almost surely convergent,
h i
and it represents the stationary solution of the random recursion
Z = 1+ξ Z , (2.17)
n+1 n+1 n
where ξ is independent of Z and has density f (ξ). In other words, we have the
n+1 n ξ
identity among random variables
Z 1+ξZ , (2.18)
′
≡
where Z is a copy of Z, and ξ is independent of Z . The density f (z) of the Kesten
′ ′ Z
variable Z obeys the integral equation
dξ z 1
f (z) = ∞ f (ξ)f − , (2.19)
Z ξ Z
Z0 ξ ξ !
which cannot be solved in closed form in general.
Let ξ = a and ξ = b be the smallest and largest values of ξ (lower and upper
min max
bounds of the support of f ), and similarly Z = A and Z = B the smallest and
ξ min max
largest values of Z. The condition lnξ < 0 implies a < 1. We have then A = 1/(1 a).
h i −
If a < b < 1, we have B = 1/(1 b) and f has finite support. In the more interesting
Z
−
situation where a < 1 < b, the support of f extends to infinity. It is known that the
Z
distribution f generically exhibits a fat tail, i.e., a power-law fall-off, of the form
Z
f (z) z (1+α), (2.20)
Z −
∼
where the exponent α > 0 is given by the condition [17, 19, 20]
ξα = 1. (2.21)
h i
The density f (z) has been derived explicitly [18, 20] in the case where f (ξ) is a power
Z ξ
law on an interval of the form [0,b] or [a, ].
∞
On an imaginary exponential functional of Brownian motion 9
The exponential functional X, defined in (2.1) as an integral of an exponential,
can be viewed as a continuous analogue of the Kesten variable Z, defined in (2.16)
as a sum of products. Similarly, the Langevin equation (2.9) is the continuous time
analogue of the recursion (2.17). This correspondence, already referred to in [2, 3], can
be understood quantitatively in the following way. Let us introduce a small time step ε
and split the integral in (2.1) as X = X(1) +X(2), where
ε
X(1) = e t+gB(t)dt, X(2) = ∞e t+gB(t)dt. (2.22)
− −
Z0 Zε
To first order in ε, we have X(1) = ε. Furthermore, setting t = ε + s, we have
B(t) B(ε) + B(s), so that X(2) = e ε+ζX , where X is a copy of the variable X
− ′ ′
≡
and ζ = gB(ε) is a Gaussian variable, independent of X , such that ζ = 0 and
′
h i
ζ2 = g2ε = 2Dε. Putting everything together, the exponential functional X appears
h i
(in the ε 0 limit) to obey the identity
→
X ε+ξX , (2.23)
′
≡
with
ξ = e ε+ζ, (2.24)
−
where ζ is Gaussian, such that ζ = 0 and ζ2 = 2Dε. Up to an unimportant global
h i h i
factor ε, the exponential functional X therefore identifies (in the ε 0 limit) with the
→
Kesten variable Z generated by the random input variables ξ given by (2.24). This is
precisely the Kesten variable investigated in [21], where the main emphasis is already
put on the ε 0 limit, and where the distribution (2.15) is derived in this limit.
→
The above correspondence directly yields the fall-off exponent of the density of X.
We have indeed esζ = es2 ζ2 /2 = eDs2ε, so that
h i
h i
ξs = es(Ds 1)ε. (2.25)
−
h i
The condition (2.21) thus predicts α = 1/D, in agreement with (2.15).
Thefull distributionofX canactuallybederived fromtheidentity (2.23). Consider
indeed the moment function
M(s) = Xs . (2.26)
h i
Equations (2.23) and (2.25) can be respectively expanded in the ε 0 limit to yield
→
M(s) = ξs M(s)+s ξs 1 M(s 1)ε+ (2.27)
−
h i h i − ···
and
ξs = 1+s(Ds 1)ε+ , (2.28)
h i − ···
where the dots stand for terms of higher order in ε. We are left (in the ε 0 limit)
→
with the functional equation
M(s 1) = (1 Ds)M(s), (2.29)
− −
whose normalized solution reads
Γ( s+1/D)
M(s) = − (Res < 1/D). (2.30)
Γ(1/D)Ds
On an imaginary exponential functional of Brownian motion 10
Setting s = 1, we recover the expression (2.12) for X , provided D < 1. The density
h i
of X is given by the inverse Mellin transform
ds
f (x) = x s 1M(s). (2.31)
X − −
2πi
Z
The expression (2.15) is recovered by summing the contributions of the poles of the
integrand at s = 1/D +n for n = 0,1,... The leftmost pole at s = 1/D is responsible
for the power-law tail with exponent (1+1/D).
−
3. The imaginary functional
We now turn to the main object of this paper, namely the distribution of the random
variable Z defined by the integral (1.5), i.e.,
Z = ∞e t+igB(t)dt. (3.1)
−
Z0
From a formal viewpoint, the complex variable Z can be viewed as the analytical
continuation as g ig of its real counterpart X, investigated in Section 2. This
→
continuation amounts to changing the sign of the diffusion constant (D D). It
→ −
is however worth emphasizing that the study of Z is far more difficult than that of X,
as most of the usual tools which are fit to investigate real random variables cease to
work in the case of a complex random variable.
Setting Z = X +iY, we are equivalently interested in the joint distribution f(x,y)
of the two correlated real random variables
X = ∞e t cos(gB(t))dt, Y = ∞e t sin(gB(t))dt. (3.2)
− −
Z0 Z0
Let us start with a few general facts. The complex random variable Z lives inside the
unit disk, since
Z ∞e tdt = 1. (3.3)
−
| | ≤
Z0
Let us anticipate that its distribution has a smooth density f(x,y), whose support is
the whole unit disk. Before we turn to more specific features, it is illustrative to first
evaluate the mean Z . Using the identity
h i
eigB(t) = e (g2/2) B(t)2 = e Dt, (3.4)
− h i −
h i
we obtain at once
Z = ∞e t eigB(t) dt = ∞e (D+1)tdt, (3.5)
− −
h i h i
Z0 Z0
i.e.,
1
Z = , (3.6)
h i D +1
so that
1
X = , Y = 0. (3.7)
h i D +1 h i