The exact asymptotic of the collision time tail distribution for
independent Brownian particles with different drifts.
Zbigniew Pucha la1,2,3 and Tomasz Rolski1,2
November 4, 2018
Abstract
In this note we consider the time of the collision τ for n independent Brownian
motions X1t , . . . , X
t with drifts a1, . . . , an, each starting from x = (x1, . . . , xn), where
x1 < . . . < xn. We show the exact asymptotics of IPx(τ > t) = Ch(x)t
−αe−γt(1 + o(1))
as t → ∞ and identify C, h(x), α, γ in terms of the drifts.
Keywords: Brownian motion with drift, collision time.
AMS 2000 Subject Classification: Primary: 60J65.
1 Mathematical Institute, University of Wroc law, pl. Grunwaldzki 2/4, 50-384 Wroc law, Poland
2This work was partially supported by a Marie Curie Transfer of Knowledge Fellowship of the European
Community’s Sixth Framework Programme: Programme HANAP under contract number MTKD-CT-2004-
13389.
3This work was partially supported by This work was partially supported by KBN Grant N201 049 31/3997
(2007).
http://arxiv.org/abs/0704.0215v1
1 Introduction and results
Let W = {y : y1 < . . . < yn} be the Weyl chamber. Consider Xt = (X1t , . . . ,Xnt ), wherein
coordinates are independent Brownian motions with unit variance parameter, drift vector
a = (a1, . . . , an) and starting point X0 = x ∈ W . In this paper we study the collision time
τ , which is the exit time of Xt from the Weyl chamber, i.e.
τ = inf{t > 0 : Xt /∈ W} .
For identical drifts a1 = . . . = an , say ai ≡ 0, the celebrated Karlin-McGregor formula states
(see [7])
IP(τ > t;X t ∈ dy) = det [pt(xi, yj)] dy , (1.1)
where pt(x, y) =
(x−y)2
2t , which yields the tail distribution of τ :
IPx(τ > t) =
det [pt(xi, yj)] dy .
For the use of Karlin-McGregor formula it is essential that processes X1t , . . . ,X
t are inde-
pendent copies of the same strong Markov, with skip-free realizations process, starting at
t = 0 from x ∈ W . In this case the asymptotic of IPx(τ > t) was first studied by Grabiner
[5] (for the Brownian case) (see also proofs by Doumerc and O’Connell [4] and Pucha la [9])
Later Pucha la & Rolski [10]) showed that this asymptotic is also true for the Poisson and
continuous time random walk case. The above mentioned asymptotics is:
IPx(τ > t) ∼ D∆(x)t−n(n−1)/4, (1.2)
where ∆(x) = det
(j−1)
i,j=1
is the Vandermonde determinant, and
D = (2π)
2 ∆ (y) dy , (1.3)
for t → ∞. Here and below 1/cn =
j=1 j!.
In this note we study the same problem, however for Brownian motions with different
drifts. For this we derive first, in Section 2, a formula for IPx(τ > t) by the change of
measure. It is apparent that possible results must depend on the form of drift vector a. For
example we can analyze all cases for n = 2, because in this case the collision equals to the first
passage to zero of the Brownian process X2t − X1t , for which the density function is known
(see e.g. [3]). Hence
IPx(τ > t) =
2πs3/2
−(x + as)
where x = x2 − x1 and a = a2 − a1. This yields
IPx(τ > t) =
xeaxt−3/2e−ta
2/4 (1 + o(1)) , a1 > a2
2 (1 + o(1)) , a1 = a2
1 − e−ax + o(1) a1 < a2 .
For general n the situation is much more complex and different scenarios are possibles.
For example the drifts can be diverging and then IPx(τ > t) tends to a positive constant,
which the situation was analyzed by Biane et al [2]. Another case is when all drifts are equal,
in which the case the probability IPx(τ > t) is polynomially decaying, as it was found by
Grabiner [5]. However there are various situations when the probabilities are exponentially
decaying with polynomial prefactors. The full characterization depends on a concept of the
stable partition of the drift vector, which the notion is introduced in Section 3. In Section 4
we state the main theorem, which shows all possible exact asymptotics of IPx(τ > t) in from
of Ch(x)t−αe−γt, where formulas for C,α and γ are given in terms of the stable partition of
the drift vector.
2 Formula for IPx(τ > t).
We note our basic probabilistic space with natural history filtration (Ω,F , (Ft), IPx) and
consider on it process Xt as defined in the Introduction. Unless otherwise stated we tacitly
assume that x ∈ W . We start off a lemma on the change of measure for the Brownian
case, which the proof can be found for example in Asmussen [1], Theorem 3.4. Let Mt =
e<α,Xt>/IEe<α,Xt> be a Wald martingale. For a probability measure IPx its restriction to
Ft we denote by IPx|t. Let ĨPx be a probability measure obtained by the change of measure
IPx with the use of martingale Mt, that is defined by a family of measures ĨPx|t = Mt dIPx|t,
t ≥ 0. For the theory we refer e.g. to Section XIII.3 in [1]
Lemma 2.1 If Xt is a Brownian motion with drift a under IPx, then this process is a
Brownian motion with drift a + α under ĨPx.
The sought for formula for the tail distribution of the collision time is given in the next
proposition.
Proposition 2.2
IPx(τ > t) =
= (2π)−n/2e−<a,x>−||x||
e−||y−a
t||2/2 det[exiyj/
t] dy . (2.4)
Proof. We use α = −a to eliminate the drift under ĨPx. Thus IPx(τ > t) = ĨEx[M−1t ; τ > t] .
Now by Karlin-McGregor formula (1.1) we write
IPx(τ > t) = ĨEx[e
<a,Xt>IExe
<−a,Xt>; τ > t]
= e<−a,x>−||a||
e<a,y> det[pt(xi, yj)] dy ,
and next, algebraic manipulations yield (2.4).
In the paper we use the following vector notations. For a vector a ∈ IRn we denote a[i,j] =
(ai, ai+1, . . . , aj) and ā[i,j] = (ai+ai+1+ . . .+aj)/(j−i+1). We also use a(i,j] = (ai+1, . . . , aj)
and a(i,j) = (ai+1, . . . , aj−1). By z
k, where z = (z1, . . . , zm) and k = (k1, . . . , km) we denote
j=1 z
3 Stable partition of a.
Let a ∈ IRn. Our aim is to make a suitable partition
(a1, . . . , aν1)(aν1+1, . . . , aν1+ν2), . . . , (aν1+...+νq−1+1, . . . , aν1+...+νq ) . (3.5)
of a, where νi > 0. For short we denote m1 = ν1,m2 = ν1 + ν2, . . . ,mq = ν1 + . . . + νq = n.
We also set m0 = 0.
We say that sequence a is irreducible if
ā[1;1] > ā[2;n]
ā[1;2] > ā[3;n]
ā[1;n−1] > ā[n;n]
. (3.6)
Suppose we have a partition defined by m1, . . . ,mq . The mean of the i
th sub-vector is
denoted by f i = ā(mi−1;mi]. Furthermore we define a vector f = (f1, . . . , fn) by
fi = f
k, if mk−1 < i ≤ mk .
It is said that partition (3.5) of vector a is stable if
f1 ≤ f2 ≤ . . . ≤ f q (3.7)
and each vector a(mi−1,mi] is irreducible (i = 1, . . . , q). Remark that a stable partition is
defined if we know m = (m1, . . . ,mq) for which (3.7) hold and each a(mi−1,mi] is irreducible
(i = 1, . . . , q). In the sequel, for a given stable partition of a, characters q,f ,m are reserved
for it.
Consider now fm1 , fm2 , . . . , fmq and define a subsequence m
= (m′1, . . . ,m
q′) of m =
(m1,m2, . . . ,mq) as follows. Let q
′ be the number of strict inequalities in f1 ≤ f2 ≤ . . . ≤ f q
plus 1. Furthermore we define inductively by m′0 = 0 and for i = 1, . . . , q
′ − 1
m′i = inf{mj > m′i−1 : mj ∈ m, fmj < fmj+1} .
and finally we set mq′ = n. We also define a subsequence of indices i0, i1, . . . , iq′ inductively
by i0 = 0 and
ik = inf{j > ik−1 : fmj < fmj+1}.
Hence we have
fm′1 < fm
< · · · < fm′
In this case we say that (m′1, . . . ,m
q′) is a strong representation of the stable partition of a
and q′, (m′1, . . . ,m
q′) are characters reserved for it. Set ν
i = m
i −m′i−1, (i = 1, . . . , q′).
Example 1 Suppose that a = (3, 1, 2, 5, 1). Then q = 3 and m1 = 2,m2 = 3,m3 = 5 define
the stable partition (3, 1)(2)(5, 1) with means f1 = 2, f2 = 2, f3 = 3. furthermore q′ = 2,
m′1 = 3,m
2 = 5 and i1 = 2, i2 = 5.
Proposition 3.1 For each vector a, there exists its unique stable partition.
Before we state a proof of Proposition 3.1 we prove few lemmas.
Lemma 3.2 If a = (a1, . . . , an) is irreducible, then
ā[1;1] > fn > ā[2;n]
ā[1;2] > fn > ā[3;n]
ā[1;n−1] > fn > ā[n;n]
. (3.8)
Proof. fn is a nontrivial weighted mean of every pair ā[1;i] and ā[i+1;n].
Lemma 3.3 In a stable partition, for each element a(mi−1;mi]
ā(mi−1;mi−1+k] ≥ fmi .
Proof. The case k ≤ mi −mi−1 follows from Lemma 3.2. Clearly for k = mi −mi−1 we have
equality. Consider now k > mi −mi−1. Than ā(mi−1;mi−1+k] is a weighted mean of fmi and
ā(mi;mi+k−(mi−mi−1)] and the later term is greater or equal than fmi by (3.7) and (3.8).
In the next lemma we consider two vectors a1 ∈ IRn1 and a2 ∈ IRn2 . The corresponding
f -s are fn1 and fn2 respectively. We consider a situation of creating a new vector (a1,a2) =
(a1 . . . , an1+n2) ∈ IRn1+n2 .
Lemma 3.4 Suppose that a1 and a2 are irreducible and fn1 > fn2. Then vector (a1,a2) is
irreducible.
Proof. Recall that (a1,a2) = (a1 . . . , an1+n2) ∈ IRn1+n2 . Suppose 1 ≤ k ≤ n1. By Lemma
3.2 we have ā[1;k−1] > fn1 > ā[k;n1], also ā[1;k−1] > fn1 > fn2 . Hence ā[1;k−1] > ā[k;n1+n2]
becasue ā[k;n1+n2] is a weighted mean of ā[k;n1] and fn2 . Suppose now n1 < k. Then ā[1;k−1]
is a weighted mean of fn1 and ā[n1+1,k] and both by Lemma 3.2 are greater than ā[k;n1+n2],
which completes the proof.
Proof of Proposition 3.1. The existence part is by induction with respect n. For n = 2 we
have two situations
1. if a1 ≤ a2, than q = 2 with m1 = 1, m2 = 2 is a stable partition,
2. if a1 > a2, than q = 1 with m1 = 2 is a stable partition.
Assume that there exists a stable partition with q partition vectors of a vector a ∈ Rn. We
add a new element an+1 at the end of vector a to create new one (a, an+1) = (a1, . . . , an+1).
We have two situations.
1. If an+1 ≥ f q than in a stable partition an+1 is alone in the q + 1 partition vector.
2. If an+1 < f
q, than we proceed inductively as follow. We use Lemma 3.4 with a1 =
a[mq−1;mq] and a2 = (an+1) and let f
q and f q+1 = an+1 are means of these partition
vectors. In result (a(mq−1;mq], an+1) form an irreducible vector, for which we have to
check whether condition (3.7) holds. If yes, then we end with a stable partition, other-
wise we join the q − 1 partition vector with the new q partition vectors and repeat the
procedure. In the worst case we end up with one partition vector.
For the uniqueness proof , suppose that we have two different stable partitions: m11 <
m12 < · · · < m1q1 and m
1 < m
2 < · · · < m2q2 . The means of fs are (f
1)1, . . . , (f1)q1
for the first partition vector and (f2)1, . . . , (f2)q2 for the second respectively. Since parti-
tions are supposely different, there exists i such that m1i 6= m2i . We take the minimal i with
this property and without loss of generality we can assume m2i > m
i . Set k = m
i −m1i . We
have to analaze the following cases.
1. (m2i = m
i+1). We have
[m2i−1+1;k]
> ā2
[k+1;m2i ]
On the other hand (f1)i = ā[m1i−1+1;m
= ā[m2i−1+1;m
> ā[m1i+1;m
= ā[m1i+1;m
(f2)i and this contradics with (f1)i ≤ (f1)i+1.
2. (m2i > m
i+1). We have ā[m2i−1+1;m
i+1;m
and by Lemma 3.3
(f1)i = ā[m2i−1+1;m
> ā[m1i+1;m
≥ (f1)i+1 ,
which is a contradiction.
3. (m2i < m
i+1). We have by Lemma 3.3
(f1)i = ā[m1i−1+1;m
> ā[m1i+1;m
≥ (f1)i+1 ,
which is a contradiction.
The proof is completed.
Remark The stable partition can be obtained by considering the following simple determin-
istic dynamical system. We have n particles starting from x1 < x2 < · · · < xn. The ith
particle has speed ai. Each particle moves with a constant speed on the real line until it
collides with one of its neighboring particle (if it happens). Then both the particles coalesce
and from this time on they move with the proportional speed which is the mean of speed of
colliding particles, and so on. Ultimately the particles will form never colliding groups, which
are the same as in the stable partition of a. Notice that resulted grouping do not depend on
a starting position x.
4 The theorem and examples.
We begin introducing some notations. Suppose that a has a stable partition with character-
istics q, (mi), q
′, (m′i) respectively. In the sequel we will use the following notations:
ml−1<u<v≤ml
(au − av)2
 , (4.9)
+ (n− q) +
 , (4.10)
h(x) = e−<x,a> det
exifjx
(j−m′
l−1−1)1I{m′
l−1<j≤m
. (4.11)
Moreover we define a function
I(a, t)
|z|2e
ml−1<u<v≤ml
(zu−zv)(av−au)
∆(z(m′j−1;m
) d z .
(4.12)
Remark that from Lemma 5.1 it will follow
ml−1<u<v≤ml
(au − av)2 =
ml−1<u<v≤ml
(au − āl)2,
where
āl =
u=ml−1+1
Using this notation we now state a proposition which is useful for calculations in some
cases.
Proposition 4.1
IPx(τ > t) = (2π)
e−γtt−
j=1 (
×e−<x,a> det
exkfjx
(j−mil−1−1)1I{mil−1<j≤mil}
×I(a, t) (1 + o(1)). (4.13)
Remark that formula (4.13) does not give us straightforward asymptotic because inte-
gral I(a, t) depends on t. However in some cases this dependence vanishes and this is why
Proposition 4.1 can be sometimes useful.
The next theorem gives us asymptotic for all cases.
Theorem 4.2 For some C given below, as t → ∞
IPx(τ > t) = Ch(x)t
−αe−γt(1 + o(1)),
γ, α, and h(x) are defined in (4.9),(4.10),(4.11) respectively.
To show C we need few more definitions. Let
H(s1, . . . , sℓ) =
1≤i≤j≤ℓ+1
(si + . . . + sj−1). (4.14)
Define now
C = A1 ×A2 ×A3 , (4.15)
where
A1 = (2π)
−n/2√2πn
· · ·
ξi>0:i/∈{m1,...,mq}
ml−1<u<v≤ml
(ξu+···+ξv−1)(au−av)
ξ(mi−1;mi−1)
i/∈{m1,...,mq}
dξi ,
· · ·
ξi>0:i∈{m1,...,mq}\{ml1 ,...,mlq′ }
· · ·
ξi>−∞:i∈{ml1 ,...,mlq′ }
k,l∈{m1,...,mq}
Sklξkξl
i:{i,i+1,...,i+k}
∈{1,...,q}\{l1,...,lq′ }
ξmi+j
νiνi+k+1
i∈{m1,...,mq}
dξi ,
where Skl = (n− 2)k for k ≤ l and Skl = Slk. In the remaining part of this section we diplay
some special cases.
Example 2 (a1 = a2 = · · · = an) This is no drift case. Here q = n and m1 = 1,m2 =
2, . . . ,mn = n, also q
′ = 1 and m′1 = n. In result fm1 = a1, . . . , fmn = an. Let a be the
common value of the drift. Using Proposition 4.1 we have
IPx(τ > t) = (2π)
−n/2cne
−<x,a> det
exkfjx
2 ∆n(z[1;n]) dz (1 + o(1)).
First we notice that since all the coordinates in vector f are the same, we have
exkfjx
= e<x,f> det
= e<x,a>∆(x).
Furthermore W −f
t = W because y1 < y2 < · · · < yn if and only if y1 + a
t < y2 + a
· · · < yn + a
t. Finally we write
IPx(τ > t) = C h(x) t
−α (1 + o(1)),
where
h(x) = ∆n(x),
C = (2π)−n/2cn
2 ∆(z) dz.
Before we state the next example we prove the following lemma.
Lemma 4.3 If a ∈ W , then {W − at} → IRn as t → ∞.
Proof. Let a ∈ W . We show that for all y ∈ IRn there exists s > 0, such that for all t > s,
y ∈ {W − at}. Let y ∈ IRn. We note bi = yi+1 − yi and di = ai+1 − ai. Condition a ∈ W
implies di > 0 for all i = 1, 2, . . . , n − 1. We take s = max{−bi, 0}/min{di} and t > s. Set
zi = yi + tai, then we get that z ∈ W , because
zi+1 − zi = yi+1 + tai+1 − yi − tai = bi + tdi > bi + sdi ≥ bi + max{−bi, 0} ≥ 0.
Thus for t > s we have y = z − ta, where z ∈ W , and so y ∈ {W − ta} for all t > s.
Example 3 (a1 < a2 < · · · < an) This is the case of non-colliding drifts. Here q = q′ = n,
m1 = m
1 = 1, . . . ,mn = m
n = n, fm1 = a1, . . . , fmn = an. Using Proposition 4.1 we have
IPx(τ > t) = (2π)
−n/2e−<x,a> det [exkaj ]
|z|2 dz (1 + o(1)).
By Lemma 4.3 we have that
|z|2 dz =
|z|2 dz = (2π)n/2.
Finally we write
IPx(τ > t) = e
−<x,a> det [exkaj ] .
This result was derived earlier by Biane et al [2]
Example 4 Case when q = q′ = 1. This is the case of a one irreducible drift vector. Here
m1 = m
1 = n, f1 = f2 · · · = fn = ā[1;n] =
a1+···+an
. Using Proposition 4.1 we have
IPx(τ > t) = Ch(x)t
−αe−γt(1 + o(1)),
where
0<u<v≤n
(au − av)2
(n− 1)(n + 1)
h(x) = e−<x,a> det
exifjx
C = (2π)−n/2
2πncn
· · ·
ξi>0:i=1,2,...,n−1
ml−1<u<v≤ml
(ξu+···+ξv−1)(au−av)
×H(ξ[1;n−1])
dξi .
We now analyze a remaining situation for n = 3.
Example 5 (a1 > a2 and
a1+a2
< a3). This is the case of two subsequences. Thus q =
2, q′ = 2 and m1 = m
1 = 2, m2 = m
2 = 3. By Theorem 4.2 we have
IPx(τ > t) = Ch(x)e
(a2−a1)2t−
where
(a2 − a1)2
h(x) = e−<x,a>
a1+a2
2 ex1
a1+a2
2 x1 e
a1+a2
2 ex2
a1+a2
2 x2 e
a1+a2
2 ex3
a1+a2
2 x3 e
C = (2π)−3/2
(a1 − a2)2
5 Auxiliary results.
For the proof we need a set of lemmas and propositions, presented in subsections below.
5.1 Useful lemmas.
We need a few technical lemmas, which we state without proofs.
Lemma 5.1 For a ∈ IRm
ā[1;m] − ai
1≤u<v≤m
(au − av)2.
Lemma 5.2 For a,z ∈ IRm
ā[1;m] − ai
(zv − zu)(au − av).
The proof of the following lemma follows easily from Lemmas 5.1 and 5.2.
Lemma 5.3 For a,f ∈ Rn such that f is is a vector obtained from the stable partition of a,
and z ∈ Rn, we have
t + z|2 = |z|2 +
ml−1<u<v≤ml
(au − av)2
ml−1<u<v≤ml
(zv − zu)(au − av)
Lemma 5.4
−1 1 0 . . . 0 0
0 −1 1 . . . 0 0
. . .
0 0 0 . . . −1 1
1 1 1 . . . 1 1
(A−1)TA−1 =
n− 2 2(n − 2)
n− 3 2(n − 3) 3(n − 3)
. . .
1 2 3 . . . n− 1
0 0 0 . . . 0 1
Note that (A−1)TA−1 is symmetric.
By Proposition 2.2 we have
IPx(τ > t) = (2π)
−n/2e−<a,x>−||x||
e−||y−a
t||2/2 det(exiyj/
t) dy .
We now introduce new variable z by
y = f
t + z,
where f = (f1, . . . , fn) is a vector obtained from the stable partition of a.
Finally we rewrite formula (2.4) in new variables by the use of Lemma 5.3:
Lemma 5.5
IPx(τ > t) = (2π)
−n/2e−<a,x>−||x||
2/2te−γt
ml−1<u<v≤ml
(zv−zu)(au−av)
exi(zj/
t+fj)
dz. (5.16)
5.2 Asymptotic behavior of determinant.
The following lemma is an extension of Lemma 2 from Pucha la [9] .
We define functions
gk(z) =
det[z
det[z
for k = (k1, . . . , kn) ∈ Zn and 0 ≤ k1 < · · · < kn Functions g corresponds to Schur functions
gk = sk−(0,1,...,n); see e.g. Macdonald [8], Ch. 1.3.
Lemma 5.6 Let k0 =
exi(zj/
t+fj)
t−k/2Tk , (5.17)
where
z(m′j−1,m
k1+···+kn=k
k1<···<km′
......
<···<k
gk(m′
(z(m′0,m
k1! . . . km′1 !
. . .
gk(m′
(z(m′
q′−1,m
q′−1+1
! . . . km′
exifjx
In particular as t → ∞
exi(zj/
t+fj)
j=1 (
∆(z(m′
j−1,m
× det
exkfjx
(j−mil−1−1)1I{mil−1<j≤mil}
(1 + o(1)).
Proof. By Sn we denote the group of permutations on n-set. We write
exi(zj/
t+fj)
(−1)σe
xifσ(i)e
xizσ(i)/
(−1)σe
xifσ(i)
t−k/2(x1zσ(1) + · · · + xnzσ(n))k/k!
t−k/2
(−1)σe
xifσ(i)(x1zσ(1) + · · · + xnzσ(n))k.
Now the coefficient at t−k/2 is equal to
Tk = Tk(z) =
(−1)σe
xifσ(i)(x1zσ(1) + · · · + xnzσ(n))k
(−1)σe
xifσ(i)
k1+···+kn=k
k1! . . . kn!
(x1zσ(1))
kσ(1) . . . (xnzσ(n))
kσ(n)
k1+···+kn=k
k1! . . . kn!
(−1)σe
xifσ(i)(x1zσ(1))
kσ(1) . . . (xnzσ(n))
kσ(n)
k1+···+kn=k
k1! . . . kn!
det[exifjx
Recall that
f1 = · · · = fm′1 < fm′1+1 = · · · = fm′2 < · · · < fm′q′−1+1 = · · · = fm′q′ .
If ki = kj and fi = fj, then the determinant det
exifjx
is 0. Thus we have non-zero
determinant if ki are different for those i such that fi are equal. Thus index k such that Tk
is non-zero must be at least
kj ≥ k0 =
Moreover we get all nonzero det
exifjx
putting in each subsequence
(k(m′0,m
, . . . ,k(m′
q′−1,m
all possible permutations of strictly ordered numbers from Z+ such that all sum up to k.
Thus we have
k1+···+kn=k
k1<···<km′
......
<···<k
σ1∈Sν′
· · ·
σq′∈Sν′
σ1(k(m′
(m′0,m
k1! . . . km′1 !
. . .
σ1(k(m′
q′−1,m
q′−1+1
! . . . km′
× det
exifjx
σl(kj)1m′
l−1<j≤m
Again we notice that permutations in the determinant influence only by the change of sign.
These signs and sums over the group of permutations form determinants, thus we have
k1+···+kn=k
k1<···<km′
......
<···<k
i,j=1
k1! . . . km′1
. . .
i,j=m′
q′−1+1
q′−1+1
! . . . km′
exifjx
Remark. Using Itzykson–Zuber integral (see e.g. [6]) we can write
exi(zj/
t+fj)
∆(x)∆(z/
t + f)
eTrdiag(x)Udiag(z/
t+f)U∗µ( dU) ,
where µ( dU) is (normalized) Haar measure on the unitary group U(n). Now letting t → ∞,
eTrdiag(x)Udiag(z/
t+f)U∗µ( dU) →
eTr(diag(x)Udiag(f)U
∗)µ( dU)
exifj
∆(x)∆(f)
t + f) = t−
i=1 (
∆(z(m′i−1;m
1≤k<l≤n
(f l − fk)ν′kν′l(1 + o(1)) .
Hence, as t → ∞
exi(zj/
t+fj)
i=1 (
∆(z(m′i−1;m
1≤k<l≤n
(f l − fk)ν′kν′l
exifj
This is a less detailed version of the formula from Lemma 5.6.
6 Proof of the Theorem.
Using (5.17) and formula (5.16) we write
IPx(τ > t) =
= (2π)−n/2e−||x||
2/2te−<x,a>e−γt
|z|2e
ml−1<u<v≤ml
(zu−zv)(av−au)
t−k/2Tk(z) dz ,(6.18)
First we will analyze above expression by taking only the first term in the sum (6.18), and
then we show that it gives the right asymptotic. Thus the first term equals to
(2π)−n/2e−||x||
2/2te−γt
ml−1<u<v≤ml
(zu−zv)(av−au)
×e−<x,a>
∆(z(m′j−1;m
× det
exkfjx
(j−mil−1−1)1I{mil−1<j≤mil }
k0 dz
= (2π)−n/2e−||x||
2/2te−γt
×e−<x,a> det
exkfjx
(j−mil−1−1)1I{mil−1<j≤mil}
I(a, t),
where I(a, t) was introduced in (4.12).
6.1 Asymptotic behavior of integral.
If s = Az, where
−1 1 0 . . . 0 0
0 −1 1 . . . 0 0
. . .
0 0 0 . . . −1 1
1 1 1 . . . 1 1
than zu − zv = sv + sv+1 + · · · + su−1 and
|z|2 = zTz = (A−1s)T (A−1s) = sT (A−1)TA−1s.
Hence by Lemma 5.4 we have
|z|2 =
s2n + s
(n)((A
−1)TA−1)(n)s(n),
where s(n) is obtained from s by deleting the n
th coordinate and A(n) is matrix A without
nth row and nth column.
After substitution s = Az, integral I(a, t) is
I(a, t) =
· · ·
si>(fi−fi+1)
for i=1,...,n−1
((A−1)TA−1)(n)s(n))
ml−1<u<v≤ml
(su+···+sv−1)(au−av)
(6.19)
H(s(m′
k−1;m
) ds(n) dsn
· · ·
si>(fi−fi+1)
for i=1,...,n−1
((A−1)TA−1)(n)s(n))
ml−1<u<v≤ml
(su+···+sv−1)(au−av)
(6.20)
H(s(m′
k−1,m
)) ds(n).
It is important to notice that the second exponent in integral I(a, t) in (6.19) depends only
on those si, where i /∈ {m1, . . . mq}. We also see that if mi−1 < k < mi, then the coefficient
at sk in (6.19) is
mi −mi−1
(mi − k)(k −mi−1)
ami−1+1 + · · · + ai
i−mi−1
ai+1 + · · · + ami
mi − i
and it is strictly negative by the definition of the stable partition. Note also that polynomials
H in integral I(a, t) depends only on sj, where j /∈ {m′1, . . . ,m′q′}.
We now introduce new variables ξ = (ξ1, . . . , ξn−1) by
tsj, for j 6= mi, j = 1, . . . , n− 1, i = 1, . . . , q − 1
sj, for j = mi, j = 1, . . . , n− 1, i = 1, . . . , q − 1.
(6.21)
We define function K by K
k−1,m
k−1,m
Consider now H
s(1;m′1)
. Since m′ is a subsequence of m, we recall that i1 is such that
mi1 = m
1. Similarly are defined i1, . . . , iq′ . We now factorize H
s(1;m′1)
into parts in which
there in none of mi, where is exactly one mi, exactly two and so on. Thus
s(m0;m′1)
s(mk−1;mk)
mk−1<i≤mk
mk<j≤mk+1
(si + · · · + sj−1)
mk−1<i≤mk
mk+1<j≤mk+2
(si + · · · + sj−1)
m0<i≤m1
mi1−1<j≤mi1
(si + · · · + sj−1).
We make analogous factorization for other H(s(mk−1;mk)).
Lemma 6.1 As t → ∞
K(ξ(m′
k−1,m
), t) = t
i=1 (
H(ξ(mi−1;mi))
i:{i,i+1,...,i+k}
∈{1,...,q}\{i1,...,iq′ }
ξmi+j
νiνi+k+1
(1 + o(1))
Proof. After the substitution we get
K(ξ(m′
k−1,m
), t) =
ξ(mk−1;mk)/
mk−1<i≤mk
mk<j≤mk+1
r=i,r 6=mk
t + ξmk
mk−1<i≤mk
mk+1<j≤mk+2
r=i,r /∈{mk ,mk+1}
t + ξmk + ξmk+1
m0<i≤m1
mi1−1<j≤mi1
r=i,r /∈{m1,...,mi1−1}
It is not difficult to see that asymptotic behavior of the above expression is
K(ξ(m′
k−1,m
), t) = t
l=1 (
ξ(mk−1;mk)
(ξmk)
νkνk+1
(ξmk + ξmk+1)
νkνk+2
)ν1νi1
(1 + o(1)).
In result the whole polynomial is asymptotically
K(ξ(m′
k−1;mk)
, t) = t−
i=1 (
ξ(mi−1;mi)
i:{i,i+1,...,i+k}
∈{1,...,q}\{i1,...,iq′ }
ξmi+j
νiνi+k+1
(1 + o(1)).
For substitution (6.21), we have ds(n) = t
−(n−q)/2 dξ. Note that fk+1 = fk for k 6= mi,
and hence the integration on the kth coordinate starts from 0. On the other hand if k = mi
for some i, and k 6= m′j for every j, then we also have fk+1 = fk and therefore the integration
starts from 0. Finally if k = mij for some j, then fk+1 > fk and the integrations starts from
(fk − fk+1)
t. Hence we have after the substitution
I(a, t) = t−(n−q)/2
· · ·
ξj>(fj−fj+1)
for i=1,...,n−1
k,l∈{m1,...,mq}
Sklξkξl)
k,l/∈{m1,...,mq}
Sklξkξl/t+2
k∈{m1,...,mq},l/∈{m1,...,mq}
Sklξkξl/
ml−1<u<v≤ml
(ξu+···+ξv−1)(au−av)
K(ξ(m′
k−1,m
), t) dξ .
So we can clearly see that
k=1K(ξ(m′k−1,m
), t) depends only on ξi’s such that i /∈
{ml1 , . . . ,mlq′} and it can be factorized into a part which depends only on i /∈ {m1, . . . ,mq}
and a part that depends on i ∈ {m1, . . . ,mq} \ {ml1 , . . . ,mlq′}. Thus finally we can write
I(a, t) = t−(n−q)/2
· · ·
ξi>(fi−fi+1)
for i=1,...,n−1
k,l∈{m1,...,mq}
Sklξkξl)
k,l/∈{m1,...,mq}
Sklξkξl/t+2
k∈{m1,...,mq},l/∈{m1,...,mq}
Sklξkξl/
ml−1<u<v≤ml
(ξu+···+ξv−1)(au−av)
i=1 (
H(ξ(mi−1;mi), t)
i:{i,i+1,...,i+k}
∈{1,...,q}\{i1,...,iq′ }
ξmi+j
νiνi+k+1
dξ (1 + o(1))
Hence
I(a, t) = t−(n−q)/2t−
i=1 (
· · ·
ξi>0:i=1,...,n−1
i/∈{m1,...,mq}
ml−1<u<v≤ml
(ξu+···+ξv−1)(au−av)
H(ξ(mi−1;mi))
i/∈{m1,...,mq}
· · ·
ξi>0:i∈{m1,...,mq}\{ml1 ,...,mlq′ }
· · ·
ξi>−∞:i∈{ml1 ,...,mlq′ }
k,l∈{m1,...,mq}
Sklξkξl
i:{i,i+1,...,i+k}
∈{1,...,q}\{l1,...,lq′ }
ξmi+j
νiνi+k+1
i∈{m1,...,mq}
dξi (1 + o(1)) . (6.22)
Concluding we have
I(a, t) = C1t
−(n−q)/2t−
i=1 (
2 )(1 + o(1)),
where C1 depends only on drift vector a.
6.2 Proof of Theorem 4.2.
Following considerations of Section 6.1, notice first that it suffices to take the first term
from the sum (6.18) for asymptotic analysis because next terms consists of positive rank
polynomials of variable z and therefore they will tend to zero faster after substitution (6.21).
For the proof of the main theorem we have to plug the asymptotics (6.22) to integral (6.18).
References
[1] S. Asmussen (2003) Applied Probability and Queues. Second Ed., Springer , New York.
[2] Ph. Biane, Ph. Bougerol and N. O’Connell (2005) Littelmann paths and Brownian paths.
Duke Math. J. 130, 127–167.
[3] A.N. Borodin and P. Salminen (2002) Handbook of Brownian Motion - Facts and For-
mulae. Birkhäuser Verlag, Basel.
[4] Y. Doumerc, N. O’Connell (2005) Exit problems associated with finite reflection groups.
Probability Theory and Related Fields 132, 501 - 538
[5] D.J. Grabiner, Brownian motion in a Weyl chamber, non-colliding particles, and random
matrices. Ann. Inst. H. Poincaré. Probab. Statist.35 (1999), 177-204.
[6] C. Itzykson and J.-B. Zuber (1980) The planar approximation II. J. Math. Phys. 21,
411–421 .
[7] S. Karlin and J. McGregor (1959) Coincidence probabilities. Pacific J. Math. 1141–1164.
[8] I.G. Macdonald, Symetric Functions and Hall Polynomials. Clarendon Press (1979),
Oxford.
[9] Z. Pucha la (2005) A proof of Grabiner theorem on non-colliding particles. Probability
and Mathematical Statistics 25, 129–132.
[10] Z. Pucha la and T. Rolski (2005) The exact asymptotics of the time to collison. Electronic
Journal of Probability 10, 1359–1380.
	Introduction and results
	Formula for IPbold0mu mumu xxxxxx(>t).
	Stable partition of bold0mu mumu aaaaaa.
	The theorem and examples.
	Auxiliary results.
	Useful lemmas.
	Asymptotic behavior of determinant. 
	Proof of the Theorem.
	Asymptotic behavior of integral.
	Proof of Theorem ??.

  In this note we consider the time of the collision $\tau$ for $n$ independent
Brownian motions $X^1_t,...,X_t^n$ with drifts $a_1,...,a_n$, each starting
from $x=(x_1,...,x_n)$, where $x_1<...<x_n$. We show the exact asymptotics of
$P_x(\tau>t) = C h(x)t^{-\alpha}e^{-\gamma t}(1 + o(1))$ as $t\to\infty$ and
identify $C,h(x),\alpha,\gamma$ in terms of the drifts.

Introduction and results
Let W = {y : y1 < . . . < yn} be the Weyl chamber. Consider Xt = (X1t , . . . ,Xnt ), wherein
coordinates are independent Brownian motions with unit variance parameter, drift vector
a = (a1, . . . , an) and starting point X0 = x ∈ W . In this paper we study the collision time
τ , which is the exit time of Xt from the Weyl chamber, i.e.
τ = inf{t > 0 : Xt /∈ W} .
For identical drifts a1 = . . . = an , say ai ≡ 0, the celebrated Karlin-McGregor formula states
(see [7])
IP(τ > t;X t ∈ dy) = det [pt(xi, yj)] dy , (1.1)
where pt(x, y) =
(x−y)2
2t , which yields the tail distribution of τ :
IPx(τ > t) =
det [pt(xi, yj)] dy .
For the use of Karlin-McGregor formula it is essential that processes X1t , . . . ,X
t are inde-
pendent copies of the same strong Markov, with skip-free realizations process, starting at
t = 0 from x ∈ W . In this case the asymptotic of IPx(τ > t) was first studied by Grabiner
[5] (for the Brownian case) (see also proofs by Doumerc and O’Connell [4] and Pucha la [9])
Later Pucha la & Rolski [10]) showed that this asymptotic is also true for the Poisson and
continuous time random walk case. The above mentioned asymptotics is:
IPx(τ > t) ∼ D∆(x)t−n(n−1)/4, (1.2)
where ∆(x) = det
(j−1)
i,j=1
is the Vandermonde determinant, and
D = (2π)
2 ∆ (y) dy , (1.3)
for t → ∞. Here and below 1/cn =
j=1 j!.
In this note we study the same problem, however for Brownian motions with different
drifts. For this we derive first, in Section 2, a formula for IPx(τ > t) by the change of
measure. It is apparent that possible results must depend on the form of drift vector a. For
example we can analyze all cases for n = 2, because in this case the collision equals to the first
passage to zero of the Brownian process X2t − X1t , for which the density function is known
(see e.g. [3]). Hence
IPx(τ > t) =
2πs3/2
−(x + as)
where x = x2 − x1 and a = a2 − a1. This yields
IPx(τ > t) =
xeaxt−3/2e−ta
2/4 (1 + o(1)) , a1 > a2
2 (1 + o(1)) , a1 = a2
1 − e−ax + o(1) a1 < a2 .
For general n the situation is much more complex and different scenarios are possibles.
For example the drifts can be diverging and then IPx(τ > t) tends to a positive constant,
which the situation was analyzed by Biane et al [2]. Another case is when all drifts are equal,
in which the case the probability IPx(τ > t) is polynomially decaying, as it was found by
Grabiner [5]. However there are various situations when the probabilities are exponentially
decaying with polynomial prefactors. The full characterization depends on a concept of the
stable partition of the drift vector, which the notion is introduced in Section 3. In Section 4
we state the main theorem, which shows all possible exact asymptotics of IPx(τ > t) in from
of Ch(x)t−αe−γt, where formulas for C,α and γ are given in terms of the stable partition of
the drift vector.
2 Formula for IPx(τ > t).
We note our basic probabilistic space with natural history filtration (Ω,F , (Ft), IPx) and
consider on it process Xt as defined in the Introduction. Unless otherwise stated we tacitly
assume that x ∈ W . We start off a lemma on the change of measure for the Brownian
case, which the proof can be found for example in Asmussen [1], Theorem 3.4. Let Mt =
e<α,Xt>/IEe<α,Xt> be a Wald martingale. For a probability measure IPx its restriction to
Ft we denote by IPx|t. Let ĨPx be a probability measure obtained by the change of measure
IPx with the use of martingale Mt, that is defined by a family of measures ĨPx|t = Mt dIPx|t,
t ≥ 0. For the theory we refer e.g. to Section XIII.3 in [1]
Lemma 2.1 If Xt is a Brownian motion with drift a under IPx, then this process is a
Brownian motion with drift a + α under ĨPx.
The sought for formula for the tail distribution of the collision time is given in the next
proposition.
Proposition 2.2
IPx(τ > t) =
= (2π)−n/2e−<a,x>−||x||
e−||y−a
t||2/2 det[exiyj/
t] dy . (2.4)
Proof. We use α = −a to eliminate the drift under ĨPx. Thus IPx(τ > t) = ĨEx[M−1t ; τ > t] .
Now by Karlin-McGregor formula (1.1) we write
IPx(τ > t) = ĨEx[e
<a,Xt>IExe
<−a,Xt>; τ > t]
= e<−a,x>−||a||
e<a,y> det[pt(xi, yj)] dy ,
and next, algebraic manipulations yield (2.4).
In the paper we use the following vector notations. For a vector a ∈ IRn we denote a[i,j] =
(ai, ai+1, . . . , aj) and ā[i,j] = (ai+ai+1+ . . .+aj)/(j−i+1). We also use a(i,j] = (ai+1, . . . , aj)
and a(i,j) = (ai+1, . . . , aj−1). By z
k, where z = (z1, . . . , zm) and k = (k1, . . . , km) we denote
j=1 z
3 Stable partition of a.
Let a ∈ IRn. Our aim is to make a suitable partition
(a1, . . . , aν1)(aν1+1, . . . , aν1+ν2), . . . , (aν1+...+νq−1+1, . . . , aν1+...+νq ) . (3.5)
of a, where νi > 0. For short we denote m1 = ν1,m2 = ν1 + ν2, . . . ,mq = ν1 + . . . + νq = n.
We also set m0 = 0.
We say that sequence a is irreducible if
ā[1;1] > ā[2;n]
ā[1;2] > ā[3;n]
ā[1;n−1] > ā[n;n]
. (3.6)
Suppose we have a partition defined by m1, . . . ,mq . The mean of the i
th sub-vector is
denoted by f i = ā(mi−1;mi]. Furthermore we define a vector f = (f1, . . . , fn) by
fi = f
k, if mk−1 < i ≤ mk .
It is said that partition (3.5) of vector a is stable if
f1 ≤ f2 ≤ . . . ≤ f q (3.7)
and each vector a(mi−1,mi] is irreducible (i = 1, . . . , q). Remark that a stable partition is
defined if we know m = (m1, . . . ,mq) for which (3.7) hold and each a(mi−1,mi] is irreducible
(i = 1, . . . , q). In the sequel, for a given stable partition of a, characters q,f ,m are reserved
for it.
Consider now fm1 , fm2 , . . . , fmq and define a subsequence m
= (m′1, . . . ,m
q′) of m =
(m1,m2, . . . ,mq) as follows. Let q
′ be the number of strict inequalities in f1 ≤ f2 ≤ . . . ≤ f q
plus 1. Furthermore we define inductively by m′0 = 0 and for i = 1, . . . , q
′ − 1
m′i = inf{mj > m′i−1 : mj ∈ m, fmj < fmj+1} .
and finally we set mq′ = n. We also define a subsequence of indices i0, i1, . . . , iq′ inductively
by i0 = 0 and
ik = inf{j > ik−1 : fmj < fmj+1}.
Hence we have
fm′1 < fm
< · · · < fm′
In this case we say that (m′1, . . . ,m
q′) is a strong representation of the stable partition of a
and q′, (m′1, . . . ,m
q′) are characters reserved for it. Set ν
i = m
i −m′i−1, (i = 1, . . . , q′).
Example 1 Suppose that a = (3, 1, 2, 5, 1). Then q = 3 and m1 = 2,m2 = 3,m3 = 5 define
the stable partition (3, 1)(2)(5, 1) with means f1 = 2, f2 = 2, f3 = 3. furthermore q′ = 2,
m′1 = 3,m
2 = 5 and i1 = 2, i2 = 5.
Proposition 3.1 For each vector a, there exists its unique stable partition.
Before we state a proof of Proposition 3.1 we prove few lemmas.
Lemma 3.2 If a = (a1, . . . , an) is irreducible, then
ā[1;1] > fn > ā[2;n]
ā[1;2] > fn > ā[3;n]
ā[1;n−1] > fn > ā[n;n]
. (3.8)
Proof. fn is a nontrivial weighted mean of every pair ā[1;i] and ā[i+1;n].
Lemma 3.3 In a stable partition, for each element a(mi−1;mi]
ā(mi−1;mi−1+k] ≥ fmi .
Proof. The case k ≤ mi −mi−1 follows from Lemma 3.2. Clearly for k = mi −mi−1 we have
equality. Consider now k > mi −mi−1. Than ā(mi−1;mi−1+k] is a weighted mean of fmi and
ā(mi;mi+k−(mi−mi−1)] and the later term is greater or equal than fmi by (3.7) and (3.8).
In the next lemma we consider two vectors a1 ∈ IRn1 and a2 ∈ IRn2 . The corresponding
f -s are fn1 and fn2 respectively. We consider a situation of creating a new vector (a1,a2) =
(a1 . . . , an1+n2) ∈ IRn1+n2 .
Lemma 3.4 Suppose that a1 and a2 are irreducible and fn1 > fn2. Then vector (a1,a2) is
irreducible.
Proof. Recall that (a1,a2) = (a1 . . . , an1+n2) ∈ IRn1+n2 . Suppose 1 ≤ k ≤ n1. By Lemma
3.2 we have ā[1;k−1] > fn1 > ā[k;n1], also ā[1;k−1] > fn1 > fn2 . Hence ā[1;k−1] > ā[k;n1+n2]
becasue ā[k;n1+n2] is a weighted mean of ā[k;n1] and fn2 . Suppose now n1 < k. Then ā[1;k−1]
is a weighted mean of fn1 and ā[n1+1,k] and both by Lemma 3.2 are greater than ā[k;n1+n2],
which completes the proof.
Proof of Proposition 3.1. The existence part is by induction with respect n. For n = 2 we
have two situations
1. if a1 ≤ a2, than q = 2 with m1 = 1, m2 = 2 is a stable partition,
2. if a1 > a2, than q = 1 with m1 = 2 is a stable partition.
Assume that there exists a stable partition with q partition vectors of a vector a ∈ Rn. We
add a new element an+1 at the end of vector a to create new one (a, an+1) = (a1, . . . , an+1).
We have two situations.
1. If an+1 ≥ f q than in a stable partition an+1 is alone in the q + 1 partition vector.
2. If an+1 < f
q, than we proceed inductively as follow. We use Lemma 3.4 with a1 =
a[mq−1;mq] and a2 = (an+1) and let f
q and f q+1 = an+1 are means of these partition
vectors. In result (a(mq−1;mq], an+1) form an irreducible vector, for which we have to
check whether condition (3.7) holds. If yes, then we end with a stable partition, other-
wise we join the q − 1 partition vector with the new q partition vectors and repeat the
procedure. In the worst case we end up with one partition vector.
For the uniqueness proof , suppose that we have two different stable partitions: m11 <
m12 < · · · < m1q1 and m
1 < m
2 < · · · < m2q2 . The means of fs are (f
1)1, . . . , (f1)q1
for the first partition vector and (f2)1, . . . , (f2)q2 for the second respectively. Since parti-
tions are supposely different, there exists i such that m1i 6= m2i . We take the minimal i with
this property and without loss of generality we can assume m2i > m
i . Set k = m
i −m1i . We
have to analaze the following cases.
1. (m2i = m
i+1). We have
[m2i−1+1;k]
> ā2
[k+1;m2i ]
On the other hand (f1)i = ā[m1i−1+1;m
= ā[m2i−1+1;m
> ā[m1i+1;m
= ā[m1i+1;m
(f2)i and this contradics with (f1)i ≤ (f1)i+1.
2. (m2i > m
i+1). We have ā[m2i−1+1;m
i+1;m
and by Lemma 3.3
(f1)i = ā[m2i−1+1;m
> ā[m1i+1;m
≥ (f1)i+1 ,
which is a contradiction.
3. (m2i < m
i+1). We have by Lemma 3.3
(f1)i = ā[m1i−1+1;m
> ā[m1i+1;m
≥ (f1)i+1 ,
which is a contradiction.
The proof is completed.
Remark The stable partition can be obtained by considering the following simple determin-
istic dynamical system. We have n particles starting from x1 < x2 < · · · < xn. The ith
particle has speed ai. Each particle moves with a constant speed on the real line until it
collides with one of its neighboring particle (if it happens). Then both the particles coalesce
and from this time on they move with the proportional speed which is the mean of speed of
colliding particles, and so on. Ultimately the particles will form never colliding groups, which
are the same as in the stable partition of a. Notice that resulted grouping do not depend on
a starting position x.
4 The theorem and examples.
We begin introducing some notations. Suppose that a has a stable partition with character-
istics q, (mi), q
′, (m′i) respectively. In the sequel we will use the following notations:
ml−1<u<v≤ml
(au − av)2
 , (4.9)
+ (n− q) +
 , (4.10)
h(x) = e−<x,a> det
exifjx
(j−m′
l−1−1)1I{m′
l−1<j≤m
. (4.11)
Moreover we define a function
I(a, t)
|z|2e
ml−1<u<v≤ml
(zu−zv)(av−au)
∆(z(m′j−1;m
) d z .
(4.12)
Remark that from Lemma 5.1 it will follow
ml−1<u<v≤ml
(au − av)2 =
ml−1<u<v≤ml
(au − āl)2,
where
āl =
u=ml−1+1
Using this notation we now state a proposition which is useful for calculations in some
cases.
Proposition 4.1
IPx(τ > t) = (2π)
e−γtt−
j=1 (
×e−<x,a> det
exkfjx
(j−mil−1−1)1I{mil−1<j≤mil}
×I(a, t) (1 + o(1)). (4.13)
Remark that formula (4.13) does not give us straightforward asymptotic because inte-
gral I(a, t) depends on t. However in some cases this dependence vanishes and this is why
Proposition 4.1 can be sometimes useful.
The next theorem gives us asymptotic for all cases.
Theorem 4.2 For some C given below, as t → ∞
IPx(τ > t) = Ch(x)t
−αe−γt(1 + o(1)),
γ, α, and h(x) are defined in (4.9),(4.10),(4.11) respectively.
To show C we need few more definitions. Let
H(s1, . . . , sℓ) =
1≤i≤j≤ℓ+1
(si + . . . + sj−1). (4.14)
Define now
C = A1 ×A2 ×A3 , (4.15)
where
A1 = (2π)
−n/2√2πn
· · ·
ξi>0:i/∈{m1,...,mq}
ml−1<u<v≤ml
(ξu+···+ξv−1)(au−av)
ξ(mi−1;mi−1)
i/∈{m1,...,mq}
dξi ,
· · ·
ξi>0:i∈{m1,...,mq}\{ml1 ,...,mlq′ }
· · ·
ξi>−∞:i∈{ml1 ,...,mlq′ }
k,l∈{m1,...,mq}
Sklξkξl
i:{i,i+1,...,i+k}
∈{1,...,q}\{l1,...,lq′ }
ξmi+j
νiνi+k+1
i∈{m1,...,mq}
dξi ,
where Skl = (n− 2)k for k ≤ l and Skl = Slk. In the remaining part of this section we diplay
some special cases.
Example 2 (a1 = a2 = · · · = an) This is no drift case. Here q = n and m1 = 1,m2 =
2, . . . ,mn = n, also q
′ = 1 and m′1 = n. In result fm1 = a1, . . . , fmn = an. Let a be the
common value of the drift. Using Proposition 4.1 we have
IPx(τ > t) = (2π)
−n/2cne
−<x,a> det
exkfjx
2 ∆n(z[1;n]) dz (1 + o(1)).
First we notice that since all the coordinates in vector f are the same, we have
exkfjx
= e<x,f> det
= e<x,a>∆(x).
Furthermore W −f
t = W because y1 < y2 < · · · < yn if and only if y1 + a
t < y2 + a
· · · < yn + a
t. Finally we write
IPx(τ > t) = C h(x) t
−α (1 + o(1)),
where
h(x) = ∆n(x),
C = (2π)−n/2cn
2 ∆(z) dz.
Before we state the next example we prove the following lemma.
Lemma 4.3 If a ∈ W , then {W − at} → IRn as t → ∞.
Proof. Let a ∈ W . We show that for all y ∈ IRn there exists s > 0, such that for all t > s,
y ∈ {W − at}. Let y ∈ IRn. We note bi = yi+1 − yi and di = ai+1 − ai. Condition a ∈ W
implies di > 0 for all i = 1, 2, . . . , n − 1. We take s = max{−bi, 0}/min{di} and t > s. Set
zi = yi + tai, then we get that z ∈ W , because
zi+1 − zi = yi+1 + tai+1 − yi − tai = bi + tdi > bi + sdi ≥ bi + max{−bi, 0} ≥ 0.
Thus for t > s we have y = z − ta, where z ∈ W , and so y ∈ {W − ta} for all t > s.
Example 3 (a1 < a2 < · · · < an) This is the case of non-colliding drifts. Here q = q′ = n,
m1 = m
1 = 1, . . . ,mn = m
n = n, fm1 = a1, . . . , fmn = an. Using Proposition 4.1 we have
IPx(τ > t) = (2π)
−n/2e−<x,a> det [exkaj ]
|z|2 dz (1 + o(1)).
By Lemma 4.3 we have that
|z|2 dz =
|z|2 dz = (2π)n/2.
Finally we write
IPx(τ > t) = e
−<x,a> det [exkaj ] .
This result was derived earlier by Biane et al [2]
Example 4 Case when q = q′ = 1. This is the case of a one irreducible drift vector. Here
m1 = m
1 = n, f1 = f2 · · · = fn = ā[1;n] =
a1+···+an
. Using Proposition 4.1 we have
IPx(τ > t) = Ch(x)t
−αe−γt(1 + o(1)),
where
0<u<v≤n
(au − av)2
(n− 1)(n + 1)
h(x) = e−<x,a> det
exifjx
C = (2π)−n/2
2πncn
· · ·
ξi>0:i=1,2,...,n−1
ml−1<u<v≤ml
(ξu+···+ξv−1)(au−av)
×H(ξ[1;n−1])
dξi .
We now analyze a remaining situation for n = 3.
Example 5 (a1 > a2 and
a1+a2
< a3). This is the case of two subsequences. Thus q =
2, q′ = 2 and m1 = m
1 = 2, m2 = m
2 = 3. By Theorem 4.2 we have
IPx(τ > t) = Ch(x)e
(a2−a1)2t−
where
(a2 − a1)2
h(x) = e−<x,a>
a1+a2
2 ex1
a1+a2
2 x1 e
a1+a2
2 ex2
a1+a2
2 x2 e
a1+a2
2 ex3
a1+a2
2 x3 e
C = (2π)−3/2
(a1 − a2)2
5 Auxiliary results.
For the proof we need a set of lemmas and propositions, presented in subsections below.
5.1 Useful lemmas.
We need a few technical lemmas, which we state without proofs.
Lemma 5.1 For a ∈ IRm
ā[1;m] − ai
1≤u<v≤m
(au − av)2.
Lemma 5.2 For a,z ∈ IRm
ā[1;m] − ai
(zv − zu)(au − av).
The proof of the following lemma follows easily from Lemmas 5.1 and 5.2.
Lemma 5.3 For a,f ∈ Rn such that f is is a vector obtained from the stable partition of a,
and z ∈ Rn, we have
t + z|2 = |z|2 +
ml−1<u<v≤ml
(au − av)2
ml−1<u<v≤ml
(zv − zu)(au − av)
Lemma 5.4
−1 1 0 . . . 0 0
0 −1 1 . . . 0 0
. . .
0 0 0 . . . −1 1
1 1 1 . . . 1 1
(A−1)TA−1 =
n− 2 2(n − 2)
n− 3 2(n − 3) 3(n − 3)
. . .
1 2 3 . . . n− 1
0 0 0 . . . 0 1
Note that (A−1)TA−1 is symmetric.
By Proposition 2.2 we have
IPx(τ > t) = (2π)
−n/2e−<a,x>−||x||
e−||y−a
t||2/2 det(exiyj/
t) dy .
We now introduce new variable z by
y = f
t + z,
where f = (f1, . . . , fn) is a vector obtained from the stable partition of a.
Finally we rewrite formula (2.4) in new variables by the use of Lemma 5.3:
Lemma 5.5
IPx(τ > t) = (2π)
−n/2e−<a,x>−||x||
2/2te−γt
ml−1<u<v≤ml
(zv−zu)(au−av)
exi(zj/
t+fj)
dz. (5.16)
5.2 Asymptotic behavior of determinant.
The following lemma is an extension of Lemma 2 from Pucha la [9] .
We define functions
gk(z) =
det[z
det[z
for k = (k1, . . . , kn) ∈ Zn and 0 ≤ k1 < · · · < kn Functions g corresponds to Schur functions
gk = sk−(0,1,...,n); see e.g. Macdonald [8], Ch. 1.3.
Lemma 5.6 Let k0 =
exi(zj/
t+fj)
t−k/2Tk , (5.17)
where
z(m′j−1,m
k1+···+kn=k
k1<···<km′
......
<···<k
gk(m′
(z(m′0,m
k1! . . . km′1 !
. . .
gk(m′
(z(m′
q′−1,m
q′−1+1
! . . . km′
exifjx
In particular as t → ∞
exi(zj/
t+fj)
j=1 (
∆(z(m′
j−1,m
× det
exkfjx
(j−mil−1−1)1I{mil−1<j≤mil}
(1 + o(1)).
Proof. By Sn we denote the group of permutations on n-set. We write
exi(zj/
t+fj)
(−1)σe
xifσ(i)e
xizσ(i)/
(−1)σe
xifσ(i)
t−k/2(x1zσ(1) + · · · + xnzσ(n))k/k!
t−k/2
(−1)σe
xifσ(i)(x1zσ(1) + · · · + xnzσ(n))k.
Now the coefficient at t−k/2 is equal to
Tk = Tk(z) =
(−1)σe
xifσ(i)(x1zσ(1) + · · · + xnzσ(n))k
(−1)σe
xifσ(i)
k1+···+kn=k
k1! . . . kn!
(x1zσ(1))
kσ(1) . . . (xnzσ(n))
kσ(n)
k1+···+kn=k
k1! . . . kn!
(−1)σe
xifσ(i)(x1zσ(1))
kσ(1) . . . (xnzσ(n))
kσ(n)
k1+···+kn=k
k1! . . . kn!
det[exifjx
Recall that
f1 = · · · = fm′1 < fm′1+1 = · · · = fm′2 < · · · < fm′q′−1+1 = · · · = fm′q′ .
If ki = kj and fi = fj, then the determinant det
exifjx
is 0. Thus we have non-zero
determinant if ki are different for those i such that fi are equal. Thus index k such that Tk
is non-zero must be at least
kj ≥ k0 =
Moreover we get all nonzero det
exifjx
putting in each subsequence
(k(m′0,m
, . . . ,k(m′
q′−1,m
all possible permutations of strictly ordered numbers from Z+ such that all sum up to k.
Thus we have
k1+···+kn=k
k1<···<km′
......
<···<k
σ1∈Sν′
· · ·
σq′∈Sν′
σ1(k(m′
(m′0,m
k1! . . . km′1 !
. . .
σ1(k(m′
q′−1,m
q′−1+1
! . . . km′
× det
exifjx
σl(kj)1m′
l−1<j≤m
Again we notice that permutations in the determinant influence only by the change of sign.
These signs and sums over the group of permutations form determinants, thus we have
k1+···+kn=k
k1<···<km′
......
<···<k
i,j=1
k1! . . . km′1
. . .
i,j=m′
q′−1+1
q′−1+1
! . . . km′
exifjx
Remark. Using Itzykson–Zuber integral (see e.g. [6]) we can write
exi(zj/
t+fj)
∆(x)∆(z/
t + f)
eTrdiag(x)Udiag(z/
t+f)U∗µ( dU) ,
where µ( dU) is (normalized) Haar measure on the unitary group U(n). Now letting t → ∞,
eTrdiag(x)Udiag(z/
t+f)U∗µ( dU) →
eTr(diag(x)Udiag(f)U
∗)µ( dU)
exifj
∆(x)∆(f)
t + f) = t−
i=1 (
∆(z(m′i−1;m
1≤k<l≤n
(f l − fk)ν′kν′l(1 + o(1)) .
Hence, as t → ∞
exi(zj/
t+fj)
i=1 (
∆(z(m′i−1;m
1≤k<l≤n
(f l − fk)ν′kν′l
exifj
This is a less detailed version of the formula from Lemma 5.6.
6 Proof of the Theorem.
Using (5.17) and formula (5.16) we write
IPx(τ > t) =
= (2π)−n/2e−||x||
2/2te−<x,a>e−γt
|z|2e
ml−1<u<v≤ml
(zu−zv)(av−au)
t−k/2Tk(z) dz ,(6.18)
First we will analyze above expression by taking only the first term in the sum (6.18), and
then we show that it gives the right asymptotic. Thus the first term equals to
(2π)−n/2e−||x||
2/2te−γt
ml−1<u<v≤ml
(zu−zv)(av−au)
×e−<x,a>
∆(z(m′j−1;m
× det
exkfjx
(j−mil−1−1)1I{mil−1<j≤mil }
k0 dz
= (2π)−n/2e−||x||
2/2te−γt
×e−<x,a> det
exkfjx
(j−mil−1−1)1I{mil−1<j≤mil}
I(a, t),
where I(a, t) was introduced in (4.12).
6.1 Asymptotic behavior of integral.
If s = Az, where
−1 1 0 . . . 0 0
0 −1 1 . . . 0 0
. . .
0 0 0 . . . −1 1
1 1 1 . . . 1 1
than zu − zv = sv + sv+1 + · · · + su−1 and
|z|2 = zTz = (A−1s)T (A−1s) = sT (A−1)TA−1s.
Hence by Lemma 5.4 we have
|z|2 =
s2n + s
(n)((A
−1)TA−1)(n)s(n),
where s(n) is obtained from s by deleting the n
th coordinate and A(n) is matrix A without
nth row and nth column.
After substitution s = Az, integral I(a, t) is
I(a, t) =
· · ·
si>(fi−fi+1)
for i=1,...,n−1
((A−1)TA−1)(n)s(n))
ml−1<u<v≤ml
(su+···+sv−1)(au−av)
(6.19)
H(s(m′
k−1;m
) ds(n) dsn
· · ·
si>(fi−fi+1)
for i=1,...,n−1
((A−1)TA−1)(n)s(n))
ml−1<u<v≤ml
(su+···+sv−1)(au−av)
(6.20)
H(s(m′
k−1,m
)) ds(n).
It is important to notice that the second exponent in integral I(a, t) in (6.19) depends only
on those si, where i /∈ {m1, . . . mq}. We also see that if mi−1 < k < mi, then the coefficient
at sk in (6.19) is
mi −mi−1
(mi − k)(k −mi−1)
ami−1+1 + · · · + ai
i−mi−1
ai+1 + · · · + ami
mi − i
and it is strictly negative by the definition of the stable partition. Note also that polynomials
H in integral I(a, t) depends only on sj, where j /∈ {m′1, . . . ,m′q′}.
We now introduce new variables ξ = (ξ1, . . . , ξn−1) by
tsj, for j 6= mi, j = 1, . . . , n− 1, i = 1, . . . , q − 1
sj, for j = mi, j = 1, . . . , n− 1, i = 1, . . . , q − 1.
(6.21)
We define function K by K
k−1,m
k−1,m
Consider now H
s(1;m′1)
. Since m′ is a subsequence of m, we recall that i1 is such that
mi1 = m
1. Similarly are defined i1, . . . , iq′ . We now factorize H
s(1;m′1)
into parts in which
there in none of mi, where is exactly one mi, exactly two and so on. Thus
s(m0;m′1)
s(mk−1;mk)
mk−1<i≤mk
mk<j≤mk+1
(si + · · · + sj−1)
mk−1<i≤mk
mk+1<j≤mk+2
(si + · · · + sj−1)
m0<i≤m1
mi1−1<j≤mi1
(si + · · · + sj−1).
We make analogous factorization for other H(s(mk−1;mk)).
Lemma 6.1 As t → ∞
K(ξ(m′
k−1,m
), t) = t
i=1 (
H(ξ(mi−1;mi))
i:{i,i+1,...,i+k}
∈{1,...,q}\{i1,...,iq′ }
ξmi+j
νiνi+k+1
(1 + o(1))
Proof. After the substitution we get
K(ξ(m′
k−1,m
), t) =
ξ(mk−1;mk)/
mk−1<i≤mk
mk<j≤mk+1
r=i,r 6=mk
t + ξmk
mk−1<i≤mk
mk+1<j≤mk+2
r=i,r /∈{mk ,mk+1}
t + ξmk + ξmk+1
m0<i≤m1
mi1−1<j≤mi1
r=i,r /∈{m1,...,mi1−1}
It is not difficult to see that asymptotic behavior of the above expression is
K(ξ(m′
k−1,m
), t) = t
l=1 (
ξ(mk−1;mk)
(ξmk)
νkνk+1
(ξmk + ξmk+1)
νkνk+2
)ν1νi1
(1 + o(1)).
In result the whole polynomial is asymptotically
K(ξ(m′
k−1;mk)
, t) = t−
i=1 (
ξ(mi−1;mi)
i:{i,i+1,...,i+k}
∈{1,...,q}\{i1,...,iq′ }
ξmi+j
νiνi+k+1
(1 + o(1)).
For substitution (6.21), we have ds(n) = t
−(n−q)/2 dξ. Note that fk+1 = fk for k 6= mi,
and hence the integration on the kth coordinate starts from 0. On the other hand if k = mi
for some i, and k 6= m′j for every j, then we also have fk+1 = fk and therefore the integration
starts from 0. Finally if k = mij for some j, then fk+1 > fk and the integrations starts from
(fk − fk+1)
t. Hence we have after the substitution
I(a, t) = t−(n−q)/2
· · ·
ξj>(fj−fj+1)
for i=1,...,n−1
k,l∈{m1,...,mq}
Sklξkξl)
k,l/∈{m1,...,mq}
Sklξkξl/t+2
k∈{m1,...,mq},l/∈{m1,...,mq}
Sklξkξl/
ml−1<u<v≤ml
(ξu+···+ξv−1)(au−av)
K(ξ(m′
k−1,m
), t) dξ .
So we can clearly see that
k=1K(ξ(m′k−1,m
), t) depends only on ξi’s such that i /∈
{ml1 , . . . ,mlq′} and it can be factorized into a part which depends only on i /∈ {m1, . . . ,mq}
and a part that depends on i ∈ {m1, . . . ,mq} \ {ml1 , . . . ,mlq′}. Thus finally we can write
I(a, t) = t−(n−q)/2
· · ·
ξi>(fi−fi+1)
for i=1,...,n−1
k,l∈{m1,...,mq}
Sklξkξl)
k,l/∈{m1,...,mq}
Sklξkξl/t+2
k∈{m1,...,mq},l/∈{m1,...,mq}
Sklξkξl/
ml−1<u<v≤ml
(ξu+···+ξv−1)(au−av)
i=1 (
H(ξ(mi−1;mi), t)
i:{i,i+1,...,i+k}
∈{1,...,q}\{i1,...,iq′ }
ξmi+j
νiνi+k+1
dξ (1 + o(1))
Hence
I(a, t) = t−(n−q)/2t−
i=1 (
· · ·
ξi>0:i=1,...,n−1
i/∈{m1,...,mq}
ml−1<u<v≤ml
(ξu+···+ξv−1)(au−av)
H(ξ(mi−1;mi))
i/∈{m1,...,mq}
· · ·
ξi>0:i∈{m1,...,mq}\{ml1 ,...,mlq′ }
· · ·
ξi>−∞:i∈{ml1 ,...,mlq′ }
k,l∈{m1,...,mq}
Sklξkξl
i:{i,i+1,...,i+k}
∈{1,...,q}\{l1,...,lq′ }
ξmi+j
νiνi+k+1
i∈{m1,...,mq}
dξi (1 + o(1)) . (6.22)
Concluding we have
I(a, t) = C1t
−(n−q)/2t−
i=1 (
2 )(1 + o(1)),
where C1 depends only on drift vector a.
6.2 Proof of Theorem 4.2.
Following considerations of Section 6.1, notice first that it suffices to take the first term
from the sum (6.18) for asymptotic analysis because next terms consists of positive rank
polynomials of variable z and therefore they will tend to zero faster after substitution (6.21).
For the proof of the main theorem we have to plug the asymptotics (6.22) to integral (6.18).
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	Introduction and results
	Formula for IPbold0mu mumu xxxxxx(>t).
	Stable partition of bold0mu mumu aaaaaa.
	The theorem and examples.
	Auxiliary results.
	Useful lemmas.
	Asymptotic behavior of determinant. 
	Proof of the Theorem.
	Asymptotic behavior of integral.
	Proof of Theorem ??.
