No Multiple Collisions for Mutually Repelling
Brownian Particles
Emmanuel Cépa1 and Dominique Lépingle2
MAPMO, Université d’Orléans
B.P.6759, 45067 Orléans Cedex 2, France
1
e-mail: Emmanuel.Cepa@univ-orleans.fr
2
e-mail: dlepingl@univ-orleans.fr
Summary. Although Brownian particles with small mutual electrostatic repulsion
may collide, multiple collisions at positive time are always forbidden.
1 Introduction
A three-dimensional Brownian motion Bt = (Bt1 , Bt2 , Bt3 ) does not hit the
axis {x1 = x2 = x3 } except possibly at time 0. An easy proof is obtained
by applying Ito’s formula to Rt = [(Bt1 − Bt2 )2 + (Bt1 − Bt3 )2 + (Bt2 − Bt3 )2 ]
and remarking that up to the multiplicative constant 3 the process R is the
square of a two-dimensional Bessel process for which {0} is a polar state. This
remark will be our guiding line in the sequel.
We consider a filtered probability space (Ω, F, (Ft )t0 , P) and for N 3
the following system of stochastic differential equations
dt
dXti = dBti + λ
, i = 1, 2, . . . , N
i
Xt − Xtj
1j=iN
with boundary conditions
Xt1 Xt2 · · · XtN ,
0 t < ∞,
and a random, F0 -measurable, initial value satisfying
X01 X02 · · · X0N .
Here Bt = (Bt1 , Bt2 , . . . , BtN ) denotes a standard N -dimensional (Ft )-Brownian
motion and λ is a positive constant. This system has been extensively studied
in [5], [7], [2], [1], [3], [6]. For comments on the relationship between this
system and the spectral analysis of Brownian matrices, and also conditioning of
Brownian particles, we refer to the introduction and the bibliography in [3].
When λ 12 , establishing strong existence and uniqueness is not difficult,
because particles never collide, as proved in [7]. The general case with arbitrary
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E. Cépa and D. Lépingle
coupling strength is investigated in [2] and it is proved in [3] that collisions
occur a.s. if and only if 0 < λ < 12 . As for multiple collisions (three or more
particles at the same location), it has been stated without proof in [9] and [4]
that they are impossible. The proof we give below, with a Bessel process
unexpectedly coming in, is just an exercise on Ito’s formula.
2 A remarkable identity in law
We consider for any t 0
N N
St =
(Xtj − Xtk )2 .
j=1 k=1
Theorem 1. For any λ > 0, the process S divided by the constant 2N is the
square of a Bessel process with dimension (N − 1)(λN + 1).
Proof. It is purely computational. Ito’s formula provides for any j = k
t
j
j
k 2
k 2
(Xt − Xt ) = (X0 − X0 ) + 2
(Xsj − Xsk ) d(Bsj − Bsk )
0
t Xj − Xk
s
s
ds
+ 2λ
j
l
X
−
X
0
s
s
1l=jN
t k
Xs − Xsj
+ 2λ
ds + 2 t .
k
m
0 Xs − Xs
1m=kN
Adding the N (N − 1) equalities we get
St = S0 + 2
N N j=1 k=1
+ 4λ
N N
t
(Xsj − Xsk ) d(Bsj − Bsk )
0
j=1 k=1 1l=jN
But
N N
t
t
0
Xsj − Xsk
Xsj − Xsk
j=1 k=1 1l=jN 0
N N
=
ds
Xsj − Xsl
t j
Xs − Xsl
j=1 k=1 1j=lN
= N 2 (N − 1)t −
ds +
Xsj − Xsl
N N
0
l=1 k=1 1l=jN
=
ds + 2N (N − 1)t .
Xsj − Xsl
1 2
N (N − 1)t .
2
0
0
t
t
Xsl − Xsk
Xsj − Xsl
Xsl − Xsk
Xsl − Xsj
ds
ds
No Multiple Collisions
243
For the martingale term, we compute
N
N
j=1
=
=
2
(Xsj − Xsk )
k=1
N N
N (Xsj
j=1 k=1 l=1
N N
N (Xsj
j=1 k=1 l=1
− Xsk )(Xsj − Xsl )
− Xsk )2 +
N N
N (Xsj − Xsk )(Xsk − Xsl )
j=1 k=1 l=1
N
Ss .
=
2
Let B be a linear Brownian motion independent of B. The process C
defined by:
N N
Ct =
0
t
1I{Ss >0}
j=1 k=1
(Xsj − Xsk ) dBsj
+
N
2 Ss
0
t
1I{Ss =0} dBs
is a linear Brownian motion and we have
t
St = S0 + 2
2N Ss dCs + 2N (N − 1)(λN + 1)t ,
0
which completes the proof.
3 Multiple collisions are not allowed
Since multiple collisions do not occur for Brownian particles without interaction, we can guess they do not either in case of mutual repulsion. Here is the
proof.
Theorem 2. For any λ > 0, multiple collisions cannot occur after time 0.
Proof. i) For 3 r N and 1 q N − r + 1, let
I = {q, q + 1, . . . , q + r − 1}
j
StI =
(Xt − Xtk )2
j∈I k∈I
τ I = inf{t > 0 : StI = 0} .
244
E. Cépa and D. Lépingle
ii) We first consider the initial condition X0 . From [2], Lemma 3.5, we
know that for any 1 i < j N and any t < ∞, we have a.s.
t
du
< ∞.
j
i
0 Xu − Xu
Therefore for any u > 0 there exists 0 < v < u such that Xv1 < Xv2
< · · · < XvN a.s. In order to prove P(τ I = ∞) = 1, we may thus assume
X01 < X02 < · · · < X0N a.s., which implies for any I that S0I > 0 and so
τ I > 0 a.s.
iii)
√ know ([8], XI, section 1) that {0} is polar for the Bessel process
√ We
St / 2N , which means that τ I = ∞ a.s. for I = {1, 2, . . . , N }. We will
prove the same result for any I by backward induction on r = card(I). Assume there are no s-multiple collisions for any s > r. Then
t
(Xsj − Xsk ) dBsj
StI = S0I + 4
0
j∈I k∈I
+ 4λ
j∈I k∈I l∈I
/
t
0
Xsj − Xsk
Xsj − Xsl
ds + 2r(r − 1)(λr + 1)t .
We set for n ∈ N∗ , τnI = inf{t > 0 : StI 1/n}. For any t 0,
I
log St∧τ
I
n
=
I
t∧τn
Xsj − Xsk
dBsj
SsI
0
j∈I k∈I
t∧τnI (X j − X k ) 1
1
s
s
+ 2λ
−
ds
j
l
SsI
Xsk − Xsl
X
−
X
0
s
s
j∈I k∈I l∈I
/
t∧τnI
ds
+ 2r[(r − 1)(λr + 1) − 2]
> −∞ .
I
S
0
s
log S0I
+4
From the induction hypothesis we deduce that for j, k ∈ I and l ∈
/ I, a.s.
on {τ I < ∞}, (Xτj I − Xτl I )(XτkI − Xτl I ) > 0 and so
1
(Xsj − Xsk )
1
− k
ds
SsI
Xs − Xsl
Xsj − Xsl
0
I
t∧τ
(Xsj − Xsk )2
ds
= −
> −∞ .
j
I
l
Ss
(Xs − Xs )(Xsk − Xsl )
0
t∧τ I
The martingale (Mn , Ft∧τnI )n1 defined by
Mn = 4
j∈I k∈I
0
I
t∧τn
Xsj − Xsk
dBsj
SsI
No Multiple Collisions
has associated increasing process An
=
I
t∧τn
8r
0
245
ds
. It follows that
SsI
1
[(r − 1)(λr + 1) − 2]An either tends to a finite limit or to +∞ as n
4
I
I
tends to +∞. Then for any t 0, log St∧τ
= ∞) = 1,
I > −∞ and so P(τ
which completes the proof.
Mn +
4 Brownian particles on the circle
We now turn to the popular model of interacting Brownian particles on the
circle ([9], [3]). Consider the system of stochastic differential equations
λ Xti − Xtj
i
i
dt , i = 1, 2, . . . , N
dXt = dBt +
cot
2
2
1j=iN
with the boundary conditions
Xt1 Xt2 · · · XtN Xt1 + 2π ,
0 t < ∞.
As expected we can prove there are no multiple collisions for the particles
j
Ztj = ei Xt that live on the unit circle. The proof is more involved and will
be deduced by approximation from the previous one.
Theorem 3. Multiple collisions for the particles on the circle do not occur
after time 0 for any λ > 0.
Sketch of the proof. For the sake of simplicity, we only deal with the
N -collisions. Let
N N
Xtj − Xtk
2
sin
Rt =
2
j=1
k=1
σn = inf{t > 0 : Rt 1
}.
n
We apply Ito’s formula to log Rt and get
log Rt∧σn = log R0 +
N j=1
t∧σn
Hsj dBsj +
0
t∧σn
Ls ds
0
for some continuous processes H j and L. We divide each integral into an
integral over {Rs 12 } and an integral over {Rs < 12 }. The first type integrals
do not pose any problem. When Rs < 12 , we replace Xsj with
Ysj = Xsj
or Ysj = Xsj − 2π
246
E. Cépa and D. Lépingle
in such a way that for any j, k we have |Ysj −Ysk | < π/3. The processes H j and
L have the same expressions in terms of X or Y . With this change of variables
we may approximate sin x by x, cos x by 1 and replace the trigonometric
functions by approximations of the linear ones which we have met in the
previous sections. We obtain that
t∧σn
1
Ds ds
log Rt∧σn = log R0 + Mn + [(N − 1)(λN + 1) − 2]An +
4
0
where Mn is a martingale with associated increasing process An and D is a.s.
a locally integrable process. Details are left to the reader as well as the case
of an arbitrary subset I like those in Section 3.
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