is the semigroup generated by the operator c
α
x
α
− y
α ∂
∂ y
α
, i.e., the drift term in A
w, c
x
. If 1.2.53 holds we say that a diffusion matrix w has invariant harmonics. This condition guarantees that systems of linearly interacting diffusions with given
w -harmonic functions cluster in a universal way, as we explain now.
1.2.7 Harmonic functions and clustering
Let X = X
i
t
t ≥0, i∈
be a family of stochastic processes, indexed by an arbitrary Abelian group , solving the martingale problem for an operator A of the form
A f x : =
i j,α
a j −ix
α j
−x
α i
∂ ∂
x
α i
f x +
i αβ
w
αβ
x
i ∂
2
∂ x
α i
∂ x
β i
f x. 1.2.55 If X has initial condition
X
i
= θ i
∈ , 1.2.56
and if the diffusion matrix w has invariant harmonics, then it turns out that E[ f X
i
t] = f θ
∀t ≥ 0, i ∈ , f w-harmonic. 1.2.57
If w = w
p,2
, then the only w-harmonic functions are the affine functions, and 1.2.57 says nothing more than that the mean of X
i
t is conserved. In general, we express 1.2.57 by saying that the harmonic mean of X
i
t is conserved. It turns out that the long-time behavior of the process X
= X t
t ≥0
= X
i
t
t ≥0, i∈
depends on the migration kernel a. If the symmetrized random walk on , i.e., the random walk that jumps from i to j with rate
a j − i + ai − j
1.2.58 is recurrent and irreversible, then we believe that
X t ⇒ X ∞
as t → ∞,
1.2.59 where X
∞ has the following properties P[X
i
∞ ∈ ∂
w
K ∀i] = 1
P[X
i
t = X
j
t ∀i, j] = 1.
1.2.60 Here
∂
w
K : = {x ∈ K : wx = 0}
1.2.61 is what we call the effective boundary of the domain K . For example, for a p-type
q-tuple model, ∂
w
ˆ K
p
consists of those compositions of the urn in which less than q
colors are present, so that the resampling process has come to a halt. On the other hand, if the symmetrized random walk on is transient, then one can prove that
1.2.60 cannot hold.
For the p-type 2-tuple model we can understand the importance of the sym- metrized random walk as follows. Consider two balls, drawn at a certain time t at
random from two different urns or from one and the same urn. Both of these balls have in the past migrated according to the random walk with kernel a, and have
been subject to the resampling mechanism described in section 1.1.1. If they have been introduced into the urn as a result of the resampling mechnism, then we call
the ball whose color they copied their parent. The probability that our two balls are of the same color now depends on the probability that they descend from a common
ancestor, and this in turn depends on the time their resepective ancestors have spent together in one urn. When we trace back the ‘historical process’, descibing where
ancestors of the two balls lived at previous times, then the difference between their positions follows the symmetrized random walk. If this symmetrized random walk
is recurrent, then ancestors of the two balls have for a long time lived together in one urn, and therefore with high proability descend from a common ancestor. This
implies that with large probability all balls in one urn are of the same color, and also that after a sufficiently long time any two urns at a finite distance of each other
will contain balls of the same color. This explains 1.2.60.
In the article ‘Clustering of Linearly Interacting Diffusions and Universality of their Long-Time Distribution’ [41], contained in Chapter 3, we show that this
picture holds as long as the w-harmonic functions are invariant,
5
and in that case we can even specify the distribution of X
i
∞ explicitly. In fact, for each θ ∈ K there exists a unique probability distribution on ∂
w
K with a given harmonic mean. If we call this distribution Ŵ
θ
d x, then by the fact that the harmonic mean is conserved we see that we must have
P[X
i
∞ ∈ dx] = Ŵ
θ
d x. 1.2.62
The proof that the recurrence of the symmetrized random walk implies X t ⇒
X ∞, where X ∞ satisfies 1.2.60 and 1.2.62, consists of two main ingredi-
ents: a calculation of the covariances CovX
i
t, X
j
t between urns i and j , and a calculation of the expectation of w-harmonic functions.
If we return to the hierarchical group
N
in the limit of large N , then modulo the technical difficulties explained in section 1.2.5, we may expect the following
5
In Chapter 3, we use the terminology ‘the boundary distribution is stable against a linear drift’. Under a weak technical assumption, this is equivalent to saying that the w-harmonic functions are
invariant; see formulas 3.1.42 and 3.1.46.
behavior. If
∞ k
=1
1 c
k
= ∞, 1.2.63
then K
w, k
θ
· ⇒ Ŵ
θ
· σ
k
F
k
w → w
∗
as k → ∞.
1.2.64 Here Ŵ
θ
· is the unique distribution on the effective boundary ∂
w
K with harmonic means θ , and the fixed shape matrix w
∗
is given by w
∗ αβ
x : =
K
Ŵ
x
d yy
α
− x
α
y
β
− x
β
. 1.2.65
Note that the convergence in 1.2.64 is universal in all w with the same invariant w
-harmonic functions and in all c
k
satisfying 1.2.63. One can check that the random walk with the kernel in 1.2.7 is for large N recurrent if and only if
1.2.63 holds. We end this section by giving a short overview of what we have proved for the
p-type q-tuple models in particular. The first question is: Does the p-type q-tuple model have invariant harmonics?
The answer is:
-
p 2
3 4
5 6
?
q 2
3 4
5 6
yes yes
yes yes
yes yes
no no
no yes
no no
yes no
yes
I.e., the p-type 2-tuple and the p-type p-tuple models have invariant harmonics, the others have not.
The second question is: If we have invariant harmonics, then is the fixed shape w
∗
the same as w
p,q
or not? The answer is:
