No multiple collisions for mutually repelling Brownian particles

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No multiple collisions for mutually repelling Brownian particles

Emmanuel Cépa, Dominique Lépingle

To cite this version:

Emmanuel Cépa, Dominique Lépingle. No multiple collisions for mutually repelling Brownian parti-

cles. Séminaire de Probabilités, Springer-Verlag, 2007, 40, pp.241-246. �hal-00014051�

ccsd-00014051, version 1 - 17 Nov 2005

Brownian Particles

Emmanuel C´epa and Dominique L´epingle MAPMO, Universit´e d’Orl´eans,

B.P.6759, 45067 Orl´eans Cedex 2, France

e-mail: [email protected], [email protected]

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 = (B_t¹, B_t², B³_t) does not hit the axis {x1 = x2 = x3} except possibly at time 0. An easy proof is obtained by applying Ito’s formula toRt = [(B_t¹−B²_t)²+ (B_t¹−B_t³)²+ (B²_t −B_t³)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,(F^t)t>0,P) and for N >3 the following system of stochastic differential equations

dX_tⁱ = dB_tⁱ + λ X

16j6=i6N

dt

X_tⁱ−X_t^j, i= 1,2, . . . , N with boundary conditions

X_t¹ 6 X_t² 6· · ·6 X_t^N, 06t <∞, and a random,F⁰-measurable, initial value satisfying

X0¹ 6 X0² 6· · ·6 X0^N.

HereBt= (B_t¹, B_t², . . . , B_t^N) denotes a standardN-dimensional (F^t)-Brownian motion andλis a positive constant. This system has been extensively studied in [5], [7], [2], [1], [3], [6]. For comments on the relationship between this sys- tem and the spectral analysis of Brownian matrices, and also conditioning of Brownian particles, we refer to the introduction and the bibliography in [3].

2 Emmanuel C´epa and Dominique L´epingle Whenλ>1

2, establishing strong existence and uniqueness is not difficult, because particles never collide, as proved in [7]. The general case with arbitrary coupling strength is investigated in [2] and it is proved in [3] that collisions occur a.s. if and only if 0< λ < 1

2. 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 key Bessel process

We consider for anyt>0 St =

N

X

j=1 N

X

k=1

(X_t^j−X_t^k)².

Theorem 1 For anyλ >0, the process S divided by the constant2N is the square of a Bessel process with dimension(N−1)(λN+ 1).

Proof. It is purely computational. Ito’s formula provides for anyj6=k (X_t^j−X_t^k)²= (X0^j−X0^k)² + 2

Z t

0

(X_s^j−X_s^k)d(B_s^j−B_s^k)

+2λ X

16l6=j6N

Z t

0

X_s^j−X_s^k

Xs^j−X_s^l ds+ 2λ X

16m6=k6N

Z t

0

X_s^k−X_s^j X_s^k−X_s^mds +2t .

Adding theN(N−1) equalities we get

St=S⁰ + 2

N

X

j=1 N

X

k=1

Z t

0

(X_s^j−X_s^k)d(B_s^j−B_s^k)

+ 4λ

N

X

j=1 N

X

k=1

X

16l6=j6N

Z t

0

X_s^j−X_s^k

Xs^j−X_s^l ds + 2N(N−1)t .

But N

X

j=1 N

X

k=1

X

16l6=j6N

Z t

0

X_s^j−X_s^k Xs^j−X_s^l ds

=

N

X

j=1 N

X

k=1

X

16l6=j6N

Z t

0

X_s^j−X_s^l Xs^j−X_s^lds +

Z t

0

X_s^l−X_s^k Xs^j−X_s^l ds

= N²(N−1)t −

N

X

l=1 N

X

k=1

X

16j6=l6N

Z t

0

X_s^l−X_s^k X_s^l−Xs^j

ds

= 1

2N²(N−1)t .

For the martingale term, we compute

N

X

j=1

(

N

X

k=1

(X_s^j−X_s^k))²

=

N

X

j=1 N

X

k=1 N

X

l=1

(X_s^j−X_s^k)(X_s^j−X_s^l)

=

N

X

j=1 N

X

k=1 N

X

l=1

(X_s^j−X_s^k)² +

N

X

j=1 N

X

k=1 N

X

l=1

(X_s^j−X_s^k)(X_s^k−X_s^l)

= N

2Ss.

Let B^′ be a linear Brownian motion independent of B. The process C defined by :

Ct = Z t

0

1I{S_s>0} N

X

j=1 N

X

k=1

(X_s^j−X_s^k)dB_s^j qN

2Ss

+ Z t

0

1I{S_s=0}dB_s^′

is a linear Brownian motion and we have St = S⁰ + 2

Z t

0

p2N SsdCs + 2N(N−1)(λN+ 1)t ,

which completes the proof. ⊓⊔

3 Multiple collisions are not allowed

Since multiple collisions do not occur for Brownian particles without interac- tion, 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 time0.

Proof. i) For 36r6N and 16q6N−r+ 1, let I = {q, q+ 1, . . . , q+r−1} S_t^I = X

j∈I

X

k∈I

(X_t^j−X_t^k)² τ^I = inf{t >0 :S_t^I = 0}.

ii) We first consider the initial condition X⁰. From [2], Lemma 3.5, we know that for any 16i < j6N and anyt <∞, we have a.s.

4 Emmanuel C´epa and Dominique L´epingle

Z t

0

du

Xu^j−X_uⁱ < ∞.

Therefore for anyu > 0 there exists 0< v < u such that X_v¹ < X_v² <

· · · < X_v^N a.s. In order to prove P(τ^I = ∞) = 1, we may thus assume X0¹ < X0² < · · · < X0^N a.s., which implies for any I that S0^I > 0 and so τ^I >0 a.s.

iii) We know ([8], XI, section 1) that {0} is polar for the Bessel process

√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). As- sume there are nos-multiple collisions for anys > r. Then

S_t^I =S0^I + 4X

j∈I

X

k∈I

Z t

0

(X_s^j−X_s^k)dB_s^j + 4λX

j∈I

X

k∈I

X

l /∈I

Z t

0

X_s^j−X_s^k

Xs^j−X_s^l ds + 2r(r−1)(λr+ 1)t . We set forn∈N^∗,τ_n^I = inf{t >0 :S^I_t 6 1

n}. For anyt>0,

logS_t∧τ^I I

n = logS0^I + 4X

j∈I

X

k∈I

Z t∧τ_n^I

0

X_s^j−X_s^k S^I_s dB_s^j + 2λX

j∈I

X

k∈I

X

l /∈I

Z t∧τ_n^I

0

(X_s^j−X_s^k) S_s^I

1

Xs^j−X_s^l − 1 X_s^k−X_s^l

ds

+ 2r[(r−1)(λr+ 1)−2]

Z t∧τ_n^I

0

ds S^I_s

>−∞.

>From the induction hypothesis we deduce that forj, k∈Iandl /∈I, a.s.

on{τ^I <∞}, (X_τ^jI−X_τ^lI)(X_τ^kI−X_τ^lI) > 0 and so Z t∧τ^I

0

(X_s^j−X_s^k) Ss

1

Xs^j−X_s^l − 1 X_s^k−X_s^l

ds

= − Z t∧τ^I

0

(X_s^j−X_s^k)² Ss

ds

(Xs^j−X_s^l)(X_s^k−X_s^l)

−∞.

The martingale (Mn,Ft∧τ_n^I)n>1 defined by

Mn = 4X

j∈I

X

k∈I

Z t∧τ_n^I

0

X_s^j−X_s^k S_s^I dB_s^j

has associated increasing process An = 8r Z t∧τ_n^I

0

ds

S_s^I. It follows that Mn + 1

4[(r−1)(λr+ 1)−2]An either tends to a finite limit or to +∞as ntends to +∞. Then for anyt >0, logS_t∧τ^I I > −∞and soP(τ^I =∞) = 1, which

completes the proof. ⊓⊔

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

dX_tⁱ = dB_tⁱ + λ 2

X

16j6=i6N

cot(X_tⁱ−X_t^j

2 )dt , i= 1,2, . . . , N with the boundary conditions

X_t¹ 6 X_t² 6· · ·6 X_t^N 6X_t¹ + 2π , 06t <∞.

As expected we can prove there are no multiple collisions for the particles Z_t^j = e^{i Xtj} 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 anyλ >0.

Sketch of the proof. For the sake of simplicity, we only deal with the N- collisions. Let

Rt=

N

X

j=1 N

X

k=1

sin²(X_t^j−X_t^k

2 )

σn = inf{t >0 : Rt6_n¹}. We apply Ito’s formula to logRt and get

logRt∧σ_n = log R⁰ +

N

X

j=1

Z t∧σn

0

H_s^jdB_s^j + Z t∧σn

0

Lsds

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. WhenRs<¹₂, we replaceX_s^j with

Y_s^j = X_s^j orY_s^j = X_s^j − 2π

in such a way that for anyj, kwe have|Y_s^j−Y_s^k|< π/3. The processesH^jand Lhave the same expressions in terms ofXorY. With this change of variables

6 Emmanuel C´epa and Dominique L´epingle

we may approximate sinx by x, cosx by 1 and replace the trigonometric functions by approximations of the linear ones which we have met in the previous sections. We obtain that

log Rt∧σ_n = logR⁰ +Mn + 1

4[(N−1)(λN+ 1)−2]An + Z t∧σn

0

Dsds

whereMnis a martingale with associated increasing processAn andDis a.s.

a locally integrable process. Details are left to the reader as well as the case of an arbitrary subsetI like those in Section 3. ⊓⊔

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