Convergent Multiplicative Processes Repelled from
Zero: Power Laws and Truncated Power Laws
Didier Sornette, Rama Cont
To cite this version:
Didier Sornette, Rama Cont. Convergent Multiplicative Processes Repelled from Zero: Power
Laws and Truncated Power Laws. Journal de Physique I, EDP Sciences, 1997, 7 (3), pp.431-444.
<10.1051/jp1:1997169>. <jpa-00247337>
HAL Id: jpa-00247337
https://hal.archives-ouvertes.fr/jpa-00247337
Submitted on 1 Jan 1997
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Phj.s.
J.
FFance
I
(1997)
7
431-444
Convergent
Multiplicative
Power
and
Laws
Didier
(~)
Parc
Physique
de
Valrose,
06108
PACS.05.40.+j
Cont
MatiAre
de la
2,
Los
1996,
(**),
CondensAe
received
in
final
form
Universit@
Sciences,
des
BP
70,
of
Geophysics
and
Planetary Physics,
USA
November
12
phenomena,
random
phenomena
Nonequilibrium
thermodynamics,
PACS.05.70.Ln
Zero:
France
Fluctuation
Dynamic
PACS.64.60.Ht
from
431
PAGE
(~)
Space Science, and Institute
Angeles, California 90095,
and
September
Rania
Cedex
Nice
(~) Department of Earth
University of California,
2
and
(~~~,*)
Laboratoire
(Received
1996)
Processes
Repelled
Power
Laws
llfuucated
Sornette
1997,
MARCH
and
processes,
accepted
1996,
Brownian
November
20
motion
critical
irreversible
processes
Levy and Solomon have found that random
multiplicative
processes
wi =11 A2. At
P(wi)
lead,
of
boundary
in
the
constraint,
distribution
0)
in the form
to
>
presence
a
a
physically intuitive
of this result
of a power law w/~~+~~. lve provide a simple exact
deri,,ation
based on a random
walk analogy and show the following: ii the result applies to the asymptotic
it ~ ml
should be distinguished from the
limit
distribution
of wt and
central
theorem
which
) (log wi (log will; 2)
the asymptotic
distribution
of the
reduced
variable
is a
statement
on
Abstract.
(with Aj
the
two
sufficient
and
necessary
(corresponding
conditions
for
P(1Ji)
to
be
a
power
la,v
(logAj)
that
are
<
0
become
small.
discuss
that 1Jt not be allo,ved to
lve
too
~ 0) and
a
1Jt
underlying
for
mechanism
models, previously thought unrelated, showing the
several
common
the
variable log wt undergoes a random
laws by multiplicative
the generation of power
processes:
Fokker-Planck
equation. 3) For all these models,
walk repelled from
which we
describe by a
-oo,
of
and
thus
depends on the
distribution
the
is
solution
of
1
obtain
result
that
(A~)
exact
we
~1
random
A. 4) For finite t, the power law is cut-off by a log-normal tail, reflecting the fact that the
information
off the repulsive force to diffusively
the
walk has not the time to
transport
scatter
drift
to
=
far in
tail.
the
Levy et
0) conduit,
R4sum4.
(avec Aj
>
puiss~nce w/
sur
une
loi de
du
~~~~~
analogie
puissance
thAorAme
Solomon
en
Nous
avec
proposons
marche
une
dAcrit la
limite
montrd
ant
prdsence
une
dArivation
aldatoire.
di~tribution
central
dAcrivant
qu'un
contrainte
d'une
Nous
asymptotique
la
convergence
multiplicatif du type
processus
distribution
de bord, h une
simple,
intuitive
obtenons
les
de wt
de la
aux
et
rdsultats
rAduite
et
=
doit
)(log
Al12.. -At
en
rAsultat
ce
suivants:
grands temps
variable
de
exacte
wi
P(wi)
ii
le
Atre
wt
loi de
basAe
rAgime de
distingud
(log wt))
ndcessaires
suflisantes
Gaussienne; 2) les deux
conditions
et
une
pour
que P(wt) soit
puissance sent (log Aj < 0 (correspondant h une ddrive vers zdro) et la contrainte
que wi
contrainte
de maniAre
Cette
soit empAchde de trap s'approcher de zdro.
peut Atre mise en
oeuvre
rAflAchissante
exarninA
variAe, gAnAralisant h une grande classe de modAles le cas d'une barriAre
approximatif,
devenant
dans
traitement
Solomon.
donnons
aussi
exact
Nous
Levy
et
un
par
vers
loi
la loi
de
(*) Author for correspondence
(**) CNRS URA 190
Q
Les
(ditions
de
Physique
(e-mail:
1997
sornettefinaxos.unice.fr)
JOURNAL
432
off
limite
la
3)
Planck.
h
l'6quation (A~)
4) Pour des
=
exploration
une
finie
t
N°3
I
d'6quation de Fokkerl'exposant ~lest
que
d6pend de la spAcificit6 de la
donc
non-universel
1. p est
et
log-normale
finis, la loi de puissance est tronquAe par une
queue
de A
6troite
est
modAles,
ces
de A.
distribution
due
tous
de
solution
la
distribution
la
Pour
PHYSIQUE
DE
de la
log-normale
ou
obtenons
nous
terme
en
gAn6ral
rdsultat
le
exact
a16atoire.
marche
Introduction
1.
Power laws have a special status due to
implicit (to the physicist) relationship with critical
phenomena, a subtle many-body problem in which self-similarity and power laws emerge from
function.
characteristic
behavior of the partition or
cooperative effects leading to non-analytic
random
multibased on
mechanism
Recently, Levy and Solomon iii have presented a novel
plicative processes:
II)
Atwt,
wi+i
Many
mechanisms
of
absence
the
lead to
can
distributions.
law
power
scale
characteristic
a
and
the
=
ii is
where
of
a
moving
with
by
/wu.
w
value
describes
then
w
Taken
[2-4].
variable
stochastic
a
reference
which
wu
the
wu.
convolution
end,
we
wt
v
easily
can
express
we
make
reappear
below
frame"
"reference
by replacing everywhere
scale wu
the
our
units
in
wt
analysis
other
no
the
~
~~~~~
where
and
distribution
ingredient, expression ii) leads to the log-normal
log
the
logarithm of ii), we can
distribution
of
the
as
express
w
of log A. Using the
cumulant
of t
distributions
expansion and going back to the
leads, for large times t, to
literally with
Indeed, taking
variable
Hilt
distribution
All
form e~~, with r
constant.
normalized to wu, in other words in the
of wt
distribution
At the
probability
with
could be of the
(log A)
=
f/°
e
dA
~t
log AH(A)
~~~~
~~~12~t ~~°~ ~~
and
D
((log A)2)
=
(2)
'
(log A)2.
Expression (2)
can
be
rewritten
pj~~)
~p(wt)ut
@$
j~)
w)~~~'°~~
with
2~jt ~°~ ~~
~~~~~
/J(wt)
Since
can
is
wi
which
the
log-normal
noise.
smaller.
=
random
function
apparent
an
Indeed, it
undistinguishable
is
Ho~.ever,
is
not
To
pointed
was
from
~1(wi)
~
the I
cc
far
form
out
that
shows
exponent
an
[5]
that
/J
for
wt
/wi distribution,
the
in
tail
wt
the
log-normal
<
e<~+~~J~,
providing
»
distribution
slowly varying with the
a
/J(wi)
«
the
and
II f
log-normal
mechanism
e<~+2~fi and
range
I
for
law.
power
a
of wi, this
law With
power
added by Levy
Solomon
iii
is to
constrain
larger than a
remain
wi to
corresponds to put back wi to wo as soon as it would become
intuitively what happens, it is simpler to think in terms of the variables
understand
and
log A, here following iii. Then obviously; the equation ii) defines a
value
logwt
that
notice
ingredient
minimum
z~
for
measured.
is
distribution
The
slowly varying
a
mistaken
be
~~~
walk in
~o
>
0.
and
This
=
~-space
with
steps
(positive
and
negati;e)
distributed
according
to
the
density
CONVERGENT
CONSTRAINED
N°3
MULTIPLICATIVE
PROCESSES
433
lx)
P~
0.4
0.35
-(VI
drift
0.3
>*
~
«
~'
0.25
0.2
z
~
0.15
i
~'~
-@ (X-X0)
D/lvl
o.05
o
0
Fig.
1.
with
a
Steady-state
negative
drift
distribution
7r(1)
defined: P(~i, t)
.
If
ii
e
v
exponential profile of
reflecting barrier.
a
e~H(e~).
=
7
6
5
probability density
the
distribution
The
4
xlx0
position
of the
of
presence
of the
of the
random
walk
random
is
walk
similarly
e~~P(e~~, t).
=
(log A)
=
of the
presence
and
3
2
barrier
0, the
>
has
no
random
walk is
important
and
biased
and
consequence
drifts
we
to
As
+cc.
the
recover
a
the
consequence,
distribution
log-normal
log wo, as
(2) apart from minor and less and less important boundary effects at xo
t increases.
Thus, this regime is without surprise and does not lead to any power law. We can however
definition of the moving
transform
this
in the following one v e (1) < 0 with a suitable
case
But
random
random
drifts to the left.
reference
scale wu
d~~ such that, in this frame, the
the barrier has to stay fixed in the moving frame, corresponding to a moving barrier in the
=
+~
unscaled
.
If
v
variable
e
wt.
(1) < 0, the
following (see Figs.
1
walk
random
and
2):
a
drifts
towards
steady-state
it
the
barrier.
The
qualitative
cc) establishes itself
reflecting barrier.
The
exponential
D Iv with an
~
in
picture is the
which
the
net
drift
walk
becomes
random
by the reflection on the
tail (see below). Its
trapped in an effective cavity of size of order
diffusion
off the
barrier
and
and repeated
reflections
motion
back and forth
incessant
away
land no more
(concentration)
profile
from it lead to the build-up of an exponential probability
Gaussian). The probability density function of the Walker position x is then of the form e~P~
a
ID. As x is the logarithm of the random variable w, then one obtains a power law
with
~1 m iv
to
the
left is
distribution
We
first
for
is
distribution
of the
present
an
the
formulated
generalization
extreme
w
using
exponent,
problem
balanced
is
values
of the
done
of the
form
w~l~+P~
+~
intuitive
Fokker-Planck
approximate
forinulation
derivation
in
a
of the
random
power
walk
law
analogy.
distribution
In
Section
and
its
2.2, the
rigorously and solved exactly in Section 2.5. Sections 2.3 and 2.4 are
calculation
of the
exponent of the power la~.
process (1). The explicit
using a Wiener-Hopf integral equation, showing that it is controlled by
process.
JOURNAL
434
t
PHYSIQUE
DE
N°3
I
o.48
Reflecting
barrier
jt
x
a(t)
<
1.48
<
and
o
<
bit)
I
<
'
[--
l~
drift
-(v(
~
0
0
Xo
200
0
D/)V)
400
t
X
800
600
1000
hi
aj
Fig.
off
al
2.
the
uniformly
taken
the
In
typical
The
in
the
interval
[0;
log
"
and
vii
ofthe
of
Notice
the
ii
a
random
the
random
describes
that
ensures
equation
2.I.
with
walk
the
interval
exact
distributions.
In
two
[6j. In
this
case,
our
we
pj~
where
defined
the
to
u
excursions.
(5)
iii
0
<
to
escape
left.
the
to
This
-cc.
The
process
+03
=
Fokker-Planck
(I) and D
=
1)
t +
(12)
#
~m
7r(I)P(X
get a physical
equation by its
To
Master
equation
between
coefficient
diffusion
this
Fokker-Planck
the
time
leading
not
+
xt
=
ii)
drift
a
does
walk
ANALYSIS.
approximate
Usually,
1.47
m
is
barrier
described
at
xo
by
=
the
log
wo
Master
[11
PERTURBATIVE
now
/J
expression II) reads
log At variables,
"
PIT,
we
to
Analogy
Tt+i
and
times
defined
variable
large
intermittent
showing the multiple reflections
by the equation (19) with at
according to (17) and bt uniformly
large
at
Kesten
the
[0.48;1.48] leading
interval
ii.
walker
random
evolution
time
Walk
RandoTn
it
trajectory
b)
taken
the
in
The
2.
A
barrier.
becomes
exact
I,
t)di.
(6)
of the
intuition
corresponding
the
in
limit
~v-here
underlying
Fokker-Planck
the
variance
mechanism,
equation.
of
~
iii
and
finite ratio defining the
keeping a constant
corresponds to taking the limit uf very
7r(I)
case, this
narrow
l, t) up to second order
expand P(~
can
steps go
i, t)
"
to
P(~,
while
zero
~~
~l
~S'~~'~~
~
i~~
~
o~P
°~~
j~~~~~
formulation
(I)~
the
are
leading
cumulants
of
H(logA). j(x, t)
is
the
flux
by
J (Xi t)
"
Up IX,
f)
D
°~j ~ ~~
18)
CONSTRAINED
N°3
Expression (7)
description (7)
nothing
is
CONVERGENT
but
the
probability.
of
conservation
generic in the limit of very
for the large t behavior; only
is
important
PROCESSES
MULTIPLICATIVE
narrow
first
its
It
can
distributions:
7r
cumulants
435
be
shown
that
the
details
of
the
results
control
this
are
7r
[6j.
D/ju). In the overdamped approxima~
and D
introduce
characteristic
"length" I
u
a
tion, we can neglect the inertia of the random walker, and the general Langevin equation
-i(( + F + Ffluct reduces to
m(()
not
two
=
=
j
which
equivalent
is
correlation,vith
Fokker-Planck
the
to
variance
This
D.
form
~(t),
+
u
=
(9)
equation (7). ~ is a iloise of zero
mean
exemplifies the competition between drift u
delta
and
and
iv
=
~(t).
diffusion
stationary
The
solution
~@
(7),
of
0, is immediately
=
Pcc(x)
be
to
~e~~~,
A
=
found
(10)
l£
,vith
)~
(II)
lL+
A
and
B
scheme,
two
are
the
/J is
obviously A
is
B
=
0
=
~v-ays to
deal
with
leading
when
The
it.
integration.
Notice
characteristic
length
of the
log-normal form (2)
the
of
constants
in,>erse
t
to
the
~
cc.
trivial
obvious
most
the
In
In
absence
Pm ix)
presence
of the
impose
/m Pm (x)dx
expected
as
solution
is to
one
that,
+.
=
0,
in
of the
which
barrier,
approximation
this
the
barrier,
is
there
indeed
are
t,vo
the
solution
limit
of
equivalent
normalization
~
1,
(12)
/1e~~~~~~°~
i13)
=
~~
where
xo
%
log
wo.
This
leads
to
Pmix)
Alternatively,
flux
the
(and
wi
to
not
=
when
wo
condition
j(xo)
from
"
Let
(12)
correspond
retrieve
we
us
satisfies
the
that
stress
instance)
becomes
it
would
0,
for
(13)
as
smaller
to
kill
which
condition
barrier at xo is reflectwe, namely that
boundary
condition
is indeed of this type
the rule of the
multiplicati,,e
put back
process is that
we
than wo, thus ensuring wt > wo. An absorbing boundary
Substituting (10) in (8) with
the
wlien wt < wo.
process
Reciprocally, (13) obtained
automatically
normalized.
is
condition
the
express
can
we
j(xo)
0.
absorbing
=
j(xo)
the
that
the
correct
0.
=
Brownian
motion
(13) using an analogy with'a
in
(I) < 0 corresponds to the existence of a constant
force
direction.
This force derives from the linearly increasing potential V
iv ix.
vi in the -x
In thermodynamic
equilibrium, a Brownian particle is found at the position x ~v.ith probability
factor e-P'~l~. This is exactly (13) with D
I/fl as it should from
Boltzmann
given by the
definition of the
the
random
thermal
fluctuations.
noise modelling the
Translating in the initial variable wt
distribution
e~, ,ve get the Paretian
There
is
equilibrium
a
faster
with
a
way
to
thermal
get
bath.
this
result
The
bias
=
=
"
Pm (wt
~)(
#
~~
(14)
,
JOURNAL
436
~~~~ ~ ~~~~~
~~ ~~~~
~
PHYSIQUE
DE
N°3
I
ill°g~ll
((log A)~) (log A)~
"
(Is)
give the impression that we have found the exact solution.
the exponential form is correct
but the value of /J given
Fokker-Planck
As already stressed, the
is valid in the limit
by (15) is
of narrow
distributions
thermal
equilibrium,
of step lengths. The
Boltzmann
analogy
assumes
distribution,
distributed
corresponding to a
that
noise is
according to a Gaussian
the
i-edistribution
for the A's. These
hypothesis are not obeyed in general for
log-normal
restrictive
arbitrary nil). The power law
distribution
(14) is sensitive to large deviations not captured
Within the
Fokker~Planck
approximation.
These
As
should
derivations
two
not
below, it turns out
only an approximation.
show
we
that
where
these
approximations do not hold, we
by the equations IS, 6). Let us consider first the
barrier is absent.
As already stated, the
random
the
walk eventually
where
to
case
escapes
probability
around
exploring
with
However,
will
wander
initial
it
its
starting
point,
-cc
one.
maybe to the right and left sides for a while before escaping to -cc.
For a given realization,
thus
the
rightmost
all times.
reached
What is
position
it
we
can
measure
ever
xmax
over
Pmax(Max(0,xmax))? The question has been answered in the
the
distribution
mathematical
literature
using renewal theory [7j, p. 402) and the answer is
2.2.
EXACT
have
to
ANALYSIS.
address
In
general
the
general problem
the
case
defined
Pmax((MaXjo, ~max))
with /J
given by
/~~ 7r(I)eP~di /~~ H(A)APdA
'
=
The
the
proof can
sequel.
that
xmax
<
walk
verifies
,valk
starting
be
sketched in a few lines [7] and
Consider the probability
distribution
Starting
x.
xi
<
#
from
the origin, this
together with the
event
at
x
less
is
-xi
II?)
I.
=
o
-m
in
i16j
e~~~~'~~,
~
or
equal
<
xmax
it
will
useful
be
f]~
Pmax(xmax)dxmax,
first
step of the
rightmost position
Summing over all possible
the
that
y.
x
e
if the
occurs
x
because
it
MIX)
function
condition
to
summarize
we
random
of the
y,
random
get the
we
Wiener-Hopf integral equation
M(x)
straightforward
It is
to [7j for
Section
How
is
fiI(x)
questions of uniqueness
the
Hopf integral
in
that
check
to
equations.
2.5
below
this
result
We
which
shall
for
and
Mix
prevents
for
[9,10j
the
our
of the
drift
ensures
in
Section
exact
Let
us
position
tribution
the
briefly
xmax
in
a
broad
mention
ever
the
validity of (16)
2.5 for
wt
reached,
variable.
that
to
for
class of
there
get
x
with
classical
for
/J
given by II?).
methods
for
refer
We
handling
1Viener-
integral
equation
type of i&'iener-Hopf
same
general case.
problem? Intuitively,
to
an
Suppose
is
large
reinjecting
of the
x.
processes.
another
way
we
have
to
use
distribution
a
is
random
by the
constant
not
felt
this
and
and
problem,
which
enabling
(16).
distribution
and
walker
the
arguments
intuitive
barrier,
of the
presence
the
described
barrier
These
exponential
that
the
deviations
it to
the
large
(18)
the
random walk,
amounts
escape
sample again and again the large positive
Indeed, for such a large deviation, the presence
the
v)7rlv)dv.
e~P~
~
to
encounter
addresses
useful
/~
=
are
on
therefore
input of random
a
presence
to
shown
the
power
of
be
rightmost
law
walkers,
say
disat
CONSTRAINED
N°3
the origin. They
length) of these
provides
establish
Let
us
is the
also
solution
order
second
from
and
previous
our
when
is
for
PROCESSES
437
processes.
two
results
7r(I)
is
discussion
MULTIPLICATIVE
directed
towards
The density (number per unit
-cc.
obviously decaying as given by (16) with II?). This
generating power laws, based on the superposition of
(IS, 17)
Gaussian
a
perturbatively from II?)
re-exponentiating, we find
obtained
be
mechanism
the
compare
of (17)
now
flux
right
the
to
multiplicative
convergent
many
uniform
a
walkers
alternative
an
CONVERGENT
for /J.
I.e.
expanding
the
that
approximation
of the
straightforward to check
log-normal
distribution.
It is
nil)
is
eP~
a
as
solution
involved
eP~
=
II?)
of
in
I + /JI +
IS).
is
the
use
)/J~12
This
of the
was
(15)
that
(IS)
+..
can
to
up
expected
Fokker-Planck
equation.
RELATION
2. 3.
and
additive
KESTEN
wiTH
defining
process
Consider
VARIABLES.
affine
random
a
Si+i
the
following
mixture
of
multiplicative
map:
bi + Aist,
=
(19)
being positive
independent
random
variables.
The
stochastic
dynamical process
various
occasions, for instance in the physical modelling of ID
disordered
statistical
of financial
representation
time
The
systems ill) and the
series [12j.
variable Sit) is known in probability theory as a Kesten
variable [13].
Consider as an example the number of fish St in a lake in the t-th year. The population Si+i
in the It + I)st year is related to the population St through (19). The growth rate ii depends
the rate of reproduction
and the depletion
due to fishing as well as
environmental
rate
on
conditions, and is therefore a variable quantity. The quantity bt describes the input due to
restocking from an external
such as a fish hatchery in artificial
from migration
source
cases,
or
from adjoining
reservoirs
natural
This
model
(19)
applied
in
be
the
problems of
to
cases.
can
population dynamics, epidemics,
portfolio growth, and immigration
investment
national
across
borders [8j. The justification of our
that it is the simplest
interest
in (19) relies on the fact
linear
stochastic
equation that can provide an
alternative
modelling strategy for describing
complex time series to the nonlinear
deterministic
Notice
that the multiplicative
process,
maps.
with a At that
take
values
larger
than
dependence on
I,
intermittent
sensitive
can
ensures
an
initial
conditions.
The restocking
bt, or more generally the repulsion from the origin,
term
corresponds to a reinjection of the dynamics. It is noteworthy that these two ingredients,
of sensitive
fundamental
dependence on initial
and reinjection,
also the two
conditions
are
properties of systems exhibiting chaotic
behavior.
with
and b
(19)
b
Sit)
has
been
is
in
recovers
II) (without
distributed
according
0
=
introduced
barrier).
the
to
power
a
For b
/ 0,
it is
well-known
that
for
(log A)
<
0,
law
Plsi)
+~
Si~~~~~,
12°)
II?) [13j already encountered
above (A~)
I. In fact, the
of
large
excursions
(16)
of
the
renewal
theory
positive
uses
probability
density
of a random
walk biased
reconstructed
towards
Figure
shows
the
3
[12j.
-cc
of the Kesten
variable St for At and bi uniformly sampled in the interval [0.48;1.48] and in [0, Ii
constructed
respectively. This corresponds to the theoretical
We have also
value /t m 1.47.
the probability density function of the
Si+i St of the Kesten variable for the same
variations
with
values.
same
This
barrier
by the
determined
/J
derivation
of
We
(20)
with
observe
condition
iii)
again
=
the
a
power
result
law
tail
for
the
positive
and
negative
variations,
with
the
exponent.
is
and
not
the
by
chance
Kesten
and
variable
now
we
are
show
deeply
that
related.
the
reflective
with the
multiplicative
process
First, notice that for (log A) < 0 in absence
PHYSIQUE
DE
JOURNAL
438
Probability density
for
N°3
I
variable
Kesten
o
o
~
o
o
o
i
o
I
1.5
Reconstructed
3.
of
theoretical
of
b(t); St
and
zero
logarithm of
natural
logarithm of St, for 0.48
prediction /J m 1.47 from ii?) is
function
a
the
shrink
would
thus
acts
to
The
zero.
similarly
quantitativelv
term
bit)
can
~~ii
)
~~
"
status
to
as
the
overdalnped
used
one
derive
to
Langevin
the
=
~v.here
u
~v-e
see
we
=
repulsion
force.
The
the
of the
bi
<
I,
of
as
an
Kesten
uniformly
variable
St
sampled.
as
The
thought
wo.
~ ~~
To
see
more
~'
~~~~
~'
difference
equation
variable
repulsion from
quantitatively, we
effective
this
x
and
e
~ill
logs,
as
lead
@
to
It has
results
expression
the
same
correct
up
(21) gives
the
lit)
(log A) and
b(t)e~~
)U) +
~(t),
122)
the sum of its
and a purely fluctuating part. We thus
mean
(log A)~. Compared to (9),
(~2)
(A)2 ci (log(A)~)
(A~)
additional
This
term b(t)e~~, corresponding to a repulsion from the x < 0 region.
replaces the reflective barrier, which can itself in turn be modelled by a
concentrated
corresponding
Fokker-Planck
equation is
have
(A)
the
finite
<
equation:
j
get
the
Fokker-Planck
Introducing again
cumulant.
second
of~v.riting
approximation
the
the
density
0
,>erified.
be
barrier
form
~Ve make
and
1.48
<
previous
the
to
probability
the
ii
<
5
4
S(t)
log
Fig.
3.5
3
2.5
2
I
,vritten
1
t
D
°~l'~~
=
as
e
bitje~~Pix, tj
=
iv
+
bitje~~)°~j~'~i
+
D°~ (~l'~i
j23j
CONSTRAINED
N°3
It
also
and
+cc
~
x
presents
stationary
~v-ell-defined
a
~
z
previous
the
CONVERGENT
-cc.
results
(13)
first
the
In
with
xo
the
can
we
b(t)e-~
terms
determined
439
easily obtain in
be neglected and
can
asymptotic matching with
that
solution
the
case,
now
PROCESSES
IIULTIPLICATIVE
from
regions
~v.e
the
recover
solution
drop all the terms except those in factor of the exponentials
For x ~
can
-cc.
-cc,
we
diverge and get Fix) ~ e~. Back in the wi variable, Pm (St) is a constant for St ~ 0
ill, IS) for St ~ +cc. Beyond
and decays algebraically- as given by (14) with the exponent
these
approximations. we can solve exactly expression (21) or equi,>alentlj- (19) and we
recover
ii?). This is presented in Section 2.5 below. Again~ notice that ill, 15) is equal to the solution
of ii?) up to second
order in the
cumulant
distribution
of log A.
expansion of the
branching processes
(19)
generalization
of
It is interesting to note that the Kesten
is
process
a
either die
[14]. Consider the simplest example of a branching process in which a branch
can
with probability p2
with probability po or give two
branches
1- po. Suppose in addition
that,
branch
Then, the number of branches Si+i at generation
time step, a new
nucleates.
at each
number of
I and At
t + I is given by- equation (19) with bi
~~)~~, where ji+i is the
branches.
from
branches
The distribution nil) is simply deduced
out of the St which give two
~~'+~
of
binouiial
distribution
nauiely
pl'+~
the
ji+i,
pl'~~~+~ For
e
(~(j~ )pl'+~ p(~
~~)jj(j~~~~j,
at
~
x
which
"
=
"
~,
large St, nil)
to
goes
for
zero
processes
the
sense
is
approximately a
large St We thus
Gaussian
with
pinpoint
here
and
the
Kesten
model:
that
the
number
of
children
increases,
in
contrast
generation
amplitude. This
of the
is
standard
the
key
equal
deviation
difference
bet,veen
branching models, large generations
fluctuates
less
at a given generation
(19)
exhibiting
the
equation
to
same
in
fundamental
the
a
for
reason
robustness
the
of the
@,
to
I.
it
e.
standarj branching
self-averaging
are
and
less
as
of
in
size
fluctuation
relative
existence
the
law
power
a
only for the special
to
contrast
a
power
for
population
(for
population
dies
off,
while
critical
the
po < p2 the
case
po > p2,
po
p2
general
branching
prolifates exponentially). The same
conclusion
carries
out directly for
more
becomes
models.
Note finally that it can be shown that the branching model previously defined
itself a
formed
from
single
if
the
number
of
branches
is
equivalent to a Kesten
a
one
process
distributed
according to a power law with the special exponent /t
I, ensuring
random
variable
fluctuations
~v-ith the size of the generations.
the scaling of the
distribution
branching
in
models
in
law is
which
found
=
=
GENERALIZATION
2.4.
AT
ORIGIN.
THE
TO
The
fi)
log(
)).
A~
f(wi, Iii, bi,
model (I) is
for
More
wt
a
thus
wo.
generally,
it generated,
by
<
Ill
the
the
new
ef<~t,l't.~',
factor
reasonable
to
0 for
~
special
The
we
case
Kesten
to
us
)> j
propose
PROCESS
(~4)
t ~ 11
the
case
#
=
"
a
process
in
which
at
each
where
f(wi
Iii: bi,
step t, after the
time
value ii ~i (or Aiwi + bt in the case of Kesten
lJ reflecting the
imposed on the
constraints
consider
REPULSION
WITH
following generalization
the
))) ~ cc for wi ~ 0.
and f(wi, Iii, bt,
~
cc
))
f(wt, lit, bt, II
0 for wt > wo and f(~i, Iii, bi,
))
log(I
model (19) is the special case f(wi, Iii, bt,
consider
can
lead
~f(w~,(A<,b<;.
t+I
~i
MULTIPLICATIVE
OF
considerations
~
where
CLASS
BROAD
A
above
The
variables)
is
+
variable
readjusted
dynamical process.
t only through
on
already holds for the
)) depends
It is
the
which
dynamical variables ii land in special cases bi), a condition
two
the
shall
consider
Fokker-Planck
approximation,
examples above.
In the following
case
we
)) is actually a function of the product Aiwi, which is the value generated
where f(wi, Iii, bt,
represented by f(Aiwi) is applied. We
constraint
by the process at step t and to which the
shall
turn
In the
To
back
to
general case (24) in Section 2.5.
approximation, f(Aiwi) defines an effective repulsive
repulsive mechanism, it is enough to consider the restricted
the
Fokker-Planck
illustrate
the
stochastic
force.
where
f(wi)
case
JOURNAL
440
PHYSIQUE
DE
N°3
I
v
V(x)
-(vi
o
Fig.
leads
to
only
is
force
equation
the
of
F(xt)
the
of
f(wt) acting
"
Fix) decays
region:
random
the
coefficient.
In
random
walker.
to
this
at x
zero
the
is
~
~~~
~~
and
cc
translation
~
~~~~~~~ ~~
The
establishes
a
random
the
in
the
in
noise
ii
term
analogy, ~v.e thus
Fokker-Planck
corresponding
~~~~
~
~
part
random
the
walk
~~
~~~~
repulsion of the diffusive process in the negative
f(wt) ~ cc for
walk analogy of the
condition
0.
~
wi
the
on
freezing
to
diffusion
is
~~~~
x
corresponds
This
wt.
definition
the
to
the
have
function
a
leading
a
of
form
steady-state
of the
distribution
x
potential whose gradient gives the force felt by the random walker.
exponential profile of its density probabilitv, corresponding to a power law
wi-variable.
Generic
4.
This
x
m
,
P(~) for large
according
times is given by Pm (~)
e~P~,
law,
with
given
again
exponent
power
/t
), showing
approximately by ill; 15). The shape of the potential defined by v + Fix)
fundamental
mechanism, is depicted in Figure 4. As we have already not@d, the bound
the
leading
reflecting
barrier is a special case of this general situation, corresponding to a
to
a
wo
concentrated
repulsive force at ~o.
from the overdamped Langevin
The expression (24) for the general model can be "derived"
Fokker-Planck
equation equivalent to the
equation (25) :
With
and
properties,
these
as
tail of
distributed
wi is
consequence
a
the
and
~
to
large
+~
a
=
)
=
Let
take
us
discrete
the
exponentiate
and
to
of
version
(26)
Fl~)
~i+i
as
wi+i
that
=
wt
+
xi
"
126)
F(~t)
Iv + ~t,
replace with
~i
=
eF<i°gW<J>iwi,
log
wi
127j
ii % e~l~l+~t
Since F(~) ~ 0 for large wi, we
multiplicative
recover
a
pure
becomes
large
for
Aiwi for the tail. The condition
that Fix)
negative x
very
decrease
cannot
"
obtain
wi+i
where
~lt).
)U) +
to
zero
as
it
gets
multiplied by
a
diverging
number
When it
model
ensures
goes
to
zero.
EXACT
2.5.
of
a
zero
limiting
for large
DERIVATION
oF
THE
distribution
for
wt
w
and
going
to
TAIL
oF
PoivER
THE
obeying (24),
infinity for ~
for
~
a
LAW
large
0, is
class
ensured
DisTRiBuTioN.
of
by
The
existence
f(w, (A, b;.. )) decaying
the
competition
to
between
CONSTRAINED
N°3
the
to
w
establish
~
result
e
u
~v.here
(A, b,.. )
the
represents
(28)
expression
The
the
It is
~
P(w),
method
we
determine
e~f<"',1',~;.
I.h.s.
lJ
used
its
also
suppose
large
class of
a
in
what
smooth
mathematical
problem
11,10]. Assuming the
shape, which must obey
in
(28)
used
have
r.h.s.
shall
for
~# Aw,
variables
and
441
interesting
an
can
We
it.
satisfied
is
by the
stochastic
of
set
that
means
cc,
conditions.
instance
w
from
which
w
above
rigorously, for
distribution
asymptotic
this
of the
existence
0 for
PROCESSES
MULTIPLICATIVE
sharp repulsion
the
and
zero
0f(w, (A, b,.. )) /0~
already satisfying the
that
functions
to
of
convergence
follows
CONVERGENT
the
define
to
same
random
the
distribution.
We
process.
can
thus
write
+m
Pv (vi
Introducing
V
=
+m
Hi>)
d>
-
log
u,
~
e
log
+m
dW Pm iwi~iU
and
w
P(1/)
log
e
=
A,
we
>WI
-
dj
iii
illi>iPw
~
get
/~~ di H(I)P~ iv
I).
(29)
-m
f(~, (A, b,...)), showing that V ~ ~ for
Taking the logarithm of (28), we have V
~
fix, (A,b,...)) ~ 0 for large x. We can write
large ~ > 0, since we have assumed
that
P(V)dV
P~(x)dx leading to P(V)
~~~)~))~)J~jj~~~ ~ P~(V) for x ~ cc. We thus
announced
the Wiener-Hopf integral equation (18) iieiiing the
results (16) with ii?)
recover
ii?).
the power law
distribution
(14) for wi with /t given by
and
therefore
This
derivation
explains the origin of the generality of these results to a large class of conrepelled from the origin.
vergent multiplicative
processes
=
=
=
Discussion
3.
3. I.
NATURE
from
the
holds
this
have
taken.
for
distribution
to
and
To
sum
distributions
law
power
for the
asymptotic regime, namely
different
addresses
question
a
after
unconstrained
thus
processes
Gaussian
the
to
does
not
contain
which
law.
any
multiplicative
multiplicative variable
convergent
up,
This
(log wt))
problem
SoLuTioN.
THE
in the
true
been
fi (log wt
our
oF
origin lead
describes
Notice
useful
that
an
infinite
that
than
the
this
number
of
answered
Ideally,
products
log-normal
itself.
stochastic
by the
of the
convergence
reduced
variable
repelled
processes
wi
reduced
tends
to
variable
zero
for
information.
of the power law
distribution
and its
presented an intuitive
approximate
derivation
Our main
result
Fokker-Planck
formulation
random
walk analogy.
in a
exponent, using the
of a
solution
of the
calculation
is the explicit
exponent of the power law distribution,
as
a
values of the
Wiener-Hopf integral equation, showing that it is controlled by extreme
process.
extend
problem to a large class of systems where the
the
initial
We have also been able to
from
We
of a
mechanism
feature is the
repelling the variable
existence
zero.
away
common
well-known
with the
have in particular
connection
Kesten
drawn a
to produce
po~v.er
process
presented in this paper are of importance for the description of
law
distributions.
The
results
self-similar
dynamics.
intermittent
systems in Nature showing complex
many
We
3. 2.
is the
p is a
if the
have
Fokker-Pljnck
approximation of the random walk analogy, /t
cavity trapping the random walk. In this approximation,
distribution
of log A. In particular,
of the
function of, and only of, the first two
cumulants
and
with no
variance
variables
drift )u) < 2D, /J < 2 corresponding to
no
mean
even
THE
inverse
EXPONENT
/t.
of the size of the
In the
effective
JOURNAL
442
,vhen
smaller
()u)
I
/J <
coefficient
is, the
/J
l&iithin
It is
to
wilder
/J
rather
large
the
as
in
fluctuations
wt
shown
have
we
typical
of the
inverse
ADDITIOXAL
value
the
of the
for
the
"
expression
PosiTivE
3.4.
the
rather
subtle
phenomenon
is a
largest excursion against the flow of
characterized
class of models
a large
negative domain (in the x-variable).
is
special
a
l~o/Cl
l
of
case
(17)
~~~~
by
should
and
p is cont,rolled by wo in general. This is only true
the
The general result is that /J is gi,>en
average.
of the
of log A.
cumulants
distribution
first
two
/J.
that
We
the relationship relating /L to the
FIXING /L.
recover
reflecting barrier problem by specifying [ii the value C of the average
wolf, leading to
straightforwardly using (14), we get (wi)
average
the
this
that
diffusion
Recall
the
~
Notice
a
large
a
CONSTRAINT
wo in
(wi). Calculating
implying that
here of fixing
to
small
there
that
value
particle in random motion with drift. This holds true
by a negative drift and a sufficiently fast repulsion fi.om
from the origin (in the w-variable).
I.e.
minimum
lead
in
quantified by
a
3.3.
N°3
I
fluctuations.
the
are
large
intuitive:
fluctuations
formulatiun,
exact
an
identifies
which
D).
thus
<
and
D
PHYSIQUE
DE
DRIFT
THE
iN
PRESENCE
OF
UPPER
AN
no
additional
an
by (17),
I.e.
plicative process where the drift is re,>ersed (log A) > 0, corresponding to
of a barrier wo limiting wt to be
smatter
tial growth of wi in the
presence
reasoning holds and a parallel derivation yields
Pm(wt)
at
a
Consider
BOUND.
interpreted as
constraint,
by
minimum
be
mean
with
an
purely
a
average
thari it.
multi-
exponen-
The
same
§ wf~~,
#
(31)
w~
with
again given by (17).
0
>
/J
distribution
This
describes
that,
shows
when
Only
relevant.
the
values
the
0 <
speaking of general power law
regime with negative drift and
This
=
ii
where
is
(log it)
equation II) thus
(log at),
>
r
e~<~+~J(I
lower
The
then
and
ii obey exactly
due
to
its
average
input
large
(which
of
lit
exists
< 0.
transforms
conditions
for
for /t >
the
bound
relevant.
is
=
e~P, leading
to
/t
it(i
+
one.
ii:
We
also
St
~v-ith
with
rate
define
ii
e~~(i,
and
=
average
an
r,
we
define
iie~.
If
where
it
#
analysis to apply. The conclusion is that,
growth rate of wi becomes
that of the input,
exhibits
exponentially as (Si)
e~~ and its value
probability density function P(St)
with /t
previous
our
the
grows
power
order
"
fast,
I)
of
(t+i
into
exponentially
governed by
fluctuations
solution
the
growing
a
variable
stochastic
Notice
MO
is
However, in the case of Kesten
variables (21), if St is growing exponentially
(log At) > 0, and if the input flo~v. bi is also increasing with a larger
rate
bi
<
wt
ob,>iously no more a power law of
For
zero.
/t < I; Pm (wi) decays as a power
(while remaining safely normalized). This
for large values, this regime is not
distribution
if /t > 1, the
distribution
with vii
is increasing
the tail, rather a power law for the values close to
however
law,
bounded by MO and diverging at zero
that,
la,v
fi
+~
+~
fl@§
in
=
the
second
order
cumulant
approximation.
BEHAVIOR.
For t large but finite, the exponential (16) with (17) is trundecays typically like a Gaussian for x > li.
Translated
in the wt variable, the
e/%
law
distribution
(14)
extends
and
transforms
into an
approximately
power
up to ~i
log-normal law for large values. Refining these results for finite t using the theory of renewal
is an
mathematical
problem left for the future.
interesting
processes
3.$.
cated
TRANSIENT
and
+~
CONSTRAINED
N°3
NON-STATIONARY
3.6.
time, for
in
time
of
T
the
evolution
distribution
u(t),D(t)
varying
or
with
diffusion
It).
u
and
establishment
the
from
according
x
Fix)
has
reach
to
It is
important
back
and
barrier
the
time
not
time
not
the
are
to
3.7.
STATUS
multiplicative
from
exponential
constituting
of
processes
the
factor
an
the
in
such
stationary
not
characteristic
t*(~)
that
adiabatically
but
wi
with
sucll
that
barrier
energy
>
that
«
T,
following
exponent
an
t*(x)
the
>
the
T,
parameters
be
cut-off
diffusion
from
stems
additive.
a
single
the
fact
system is
This
holds
that
for
the
of
presence
understand
to
variable,
law in the wi
power
x-variable.
that
the
parameter,
the
In
that
a
the
multiplicative
when
true
like
additive
e~P~.
distribution
we
faster
tail in the
would
propose
is to
distribution
by
described
[I]
Solomon
evolve
of the
the
in
profile
the~exponential
T,
of the
modification
Boltzmann
equilibrium
an
of the
exponential
Boltzmann
E is
regime t*(x)
parameters
Levy and
distribution
macro-state
a
while
,,-eakly
Q(Ei + E2)
can
the
in
the
analysis
the
what
fluctuations
Boltzmann
freedom
is
their
~
x
off the
separate work. In particular,
cut-off of the
exponential
an
PROBLEM.
THE
the
case,
defined
oF
a
lead to
exponential of
an
is to
latter
for
"large"
For
bounce
to
this
In
since
The
which
processes
corresponding
then
stress
to
forth.
itself
establish
to
"scattering time" off the barrier.
non-stationarity
effects is left to
~v-hat
ill, IS).
and
x
distribution
law
power
a
equations
to
(I)
process
of time,
function
443
again the physical phenomenon at the origin of
of the exponential profile: the repeated
of the diffusing particle
encounters
For large x, the repeated
take a large time, the time to diffuse
encounters
barrier.
to
predict
thus
We
D
has
process
with
or
multiplicative
the
become
=
~o
the
xo(t)
PROCESSES
x2 ID. For "small"
must be compared ~v-ith t* ix)
Fix, t) keeps an exponential tail with an exponent
already changed.
have
MULTIPLICATIVE
~vhen
PRocEssEs.
u(t), D(t)
if
instance
CONVERGENT
processes.
recall
the
(fl~~)
that
the
number
Q of
microstates
in the
number
of degrees
partitionned
system
a
In
temperature
nutshell,
(14)
law
power
can
be
The only
of the resulting
sub-sj,stems.
solution
functional
equation
Q(Ei)Q(E2) is the exponential.
No such principle applies in the multiplicative
Furthermore, the
Boltzmann
reasoning
case.
hypotheses and provides at best
that we have used in Section 2.I is valid only under
restrictive
approximation for the general case. We have sho,vn that the correct
exponent /t is in fact
an
of the drifting random
controlled by extreme
walk against the main "flo,v" and not
excursions
by its average behavior. This rule~ out the analogy proposed by Levy and Solomon.
interacti,>e
into
"
Acknowledgments
D.S.
acknowledges
Publication
is
California,
Los
no.
stimulating
4642
of the
work
and U.
Frisch.
This
correspondence
Solomon
with S.
of Geophysics and
Planetary Physics~ University of
Institute
Angeles.
References
[Ii Levy
M.
and
Solomon
S.,
Po,ver
(1996) 595-601; Spontaneous
filod. Phys. C 7 (1996) 745-751.
C 7
[2]
Gnedenko
variables
B.V.
and
laws
are
logarithmic
scaling
Kolmogorov A.N, Limit
emergence
Boltzmann
laws,
generic
stochastic
in
distributions
(Addison Wesley, Reading MA, 1954).
for
sum
of
Int.
J.
Mod.
systems,
independent
Phys.
Int.
J.
random
JOURNAL
444
[3]
[4]
Aitcheson J. and Brown J.A.C., The log-normal
London, England, 1957).
multiplicative
Redner S., Random
processes:
(1990)
Montroll
[6]
Risken
E.W.
H.,
(Berlin;
Feller
(Calnbridge University Press,
distribution
An
tutorial,
elementary
Am.
Phys.
J.
58
D.
and
P.M.
and
Nat.
Acad.
methods
Sci.
of
USA
solution
79
and
(1982)
3380-3383.
applications,
2nd
ed.
Springer-Verlag, 1989).
introduction
and
Proc.
equation:
Fokker-Planck
York:
W., An
Sornette
Shlesinger M.F.,
and
The
New
(John Wiley
[8j
N°3
I
267-273.
[5]
[7]
PHYSIQUE
DE
sons,
Knopoff L.,
theory
probability
to
New York,
Linear
applications,
and its
Vol
II, second
edition
1971).
dynamics
stochastic
with
nonlinear
fractal
properties,
submitted.
[9j
Morse
Feshbach
H.,
Methods
of
theoretical
physics (McGraw Hill,
New
York,
1953).
[10j Frisch H., J. Quant. Spectrosc. Radiat. Transfer 39 (1988)149.
of a
Th.M. and
Petritis D., On the
distribution
Nieuwenhuizen
[11j de Calan C., Luck J.-M.,
variable
random
occurring in 1D disordered
systems, J. Phys. A 18 (1985) 501-523.
behavior of solutions
[12j de Haan L., Resnick S.1., Rootz4n H. and de Vries C.G., Extremal
Stoch. Proc. 32
difference
stochastic
equation with applications to ARCH
to a
processes,
(1989)
[13j
[14j
213-224.
Kesten
H.,
Random
trices,
Acta
Math.
Harris
T.E., The
difference
equations
(1973) 207-248.
theory of branching
and
renewal
theory for products of
131
processes
(Springer, Berlin, 1963).
random
ma-