Abstract
Consider
n
non-intersecting Brownian motions on
R
, depending on time
t
∈
[
0
,
1
]
, with
m
i
particles forced to leave from
a
i
at time
t
=
0
,
1
≤
i
≤
q
, and
n
j
particles forced to end up at
b
j
at time
t
=
1
,
1
≤
j
≤
p
. For arbitrary
p
and
q
, it is not known if the distribution of the positions of the non-intersecting Brownian particles at a given time
0
<
t
<
1
, is the same as the joint distribution of the eigenvalues of a matrix ensemble. This paper proves the existence, for general
p
and
q
, of a partial differential equation (PDE) satisfied by the log of the probability to find all the particles in a disjoint union of intervals
E
=
∪
i
=
1
r
[
c
2
i
−
1
,
c
2
i
]
⊂
R
at a given time
0
<
t
<
1
. The variables are the coordinates of the starting and ending points of the particles, and the boundary points of the set
E
. The proof of the existence of such a PDE, using Virasoro constraints and the multicomponent KP hierarchy, is based on the method of elimination of the unwanted partials; that this is possible is a miracle! Unfortunately we were unable to find its explicit expression. The case
p
=
q
=
2
will be discussed in the last section.
►
N
non-intersecting Brownian motions leaving from and going to several points. ► We study the probability that at time
0
<
t
<
1
all the particles are in a compact set. ► An integrable deformation of the probability provides multi-component KP tau-functions. ► Virasoro constraints are constructed for the deformed probability. ► Existence of a PDE for the logarithm of the probability is proven.