For a 2-type model with diffusion matrix w as in 3.1.16, the class H contains only affine functions x
→ a + bx a, b ∈
R
, and 3.1.51 says no more than that the mean of the components is conserved. Since there is only one distribution
on {0, 1} with a given mean, it is then immediately clear for recurrent a
S
that there is only one possible long-time distribution for the process in 3.1.2. In the
general higher-dimensional case, we need to specify a distribution on the effective boundary ∂
w
K , and for this we need the expectation of sufficiently many harmonic functions. We may describe 3.1.51 by saying that the ‘w-harmonic mean’ of the
components is conserved.
3.1.9 Special models: Corollary 3.1.7
The proof of Theorem 3.1.5 uses a comparison argument, in the spirit of the ‘du- ality comparison argument’ in [6]. In our comparison argument we use objects
related to the special diffusion matrix
w
∗ αβ
x : =
K
Ŵ
x
d yy
α
−x
α
y
β
−x
β
x ∈ K , α, β = 1, . . . , d. 3.1.52
We do not have a duality for the model with w
∗
, but we can find an expression for second moments, which is enough for our purposes. For the special model with
w = λw
∗
λ 0 the proof of Theorem 3.1.5 yields the following corollary.
Corollary 3.1.7 In addition to the assumptions in Theorem 3.1.5, assume that for some λ
∈ 0, ∞ w
x = λw
∗
x x
∈ K . 3.1.53
Then for each t ≥ 0, i, j ∈ , α, β = 1, . . . , d
E[X
α i
t − θ
α
X
β j
t − θ
β
] = w
∗ αβ
θ K
λ t
i, j , 3.1.54
where K
λ t
i, j denotes the probability that two delayed coalescing random walks, each with kernel a, starting in points i and j respectively, and coalescing with rate
2λ, have coalesced before time t.
3.1.10 Examples
We close this introduction by giving two examples of classes of diffusion matrices w
satisfying the assumptions in Theorem 3.1.5. The first example arises when we generalize the 2-type models mentioned in
3.1.16 to p-type models in the following way.
Example 3.1.8 p-type models Assume that K is the p
−1-dimensional simplex S
p −1
, and that x → wx is Lipschitz continuous and satisfies compare 3.1.8
α,β
z
α
w
αβ
xz
β
= 0 ⇔ z ∈ I
⊥ x
x ∈ K .
3.1.55 Then:
a Weak uniqueness holds for the non-interacting equation 3.1.19. The boundary distribution is stable against a linear drift, S
∞
C
K ⊂
C
K , and H is contained in the bp-closure of
C
2
H ∩ H .
b The class of w-harmonic functions consists of all affine functions
x → a +
α
b
α
x
α
a, b
1
, . . . , b
d
∈
R
. 3.1.56
c The associated special diffusion matrix is the Wright-Fisher diffusion matrix
w
∗ αβ
x = x
α
δ
αβ
− x
β
x ∈ K , α, β = 1, . . . , d.
3.1.57 The second example is formed by the class of isotropic diffusion matrices compare
3.1.17.
Example 3.1.9 isotropic models Assume that K has non-empty interior K
◦
, and let ∂ K :
= K \K
◦
denote its topological boundary. Assume that w
αβ
x = δ
αβ
gx x
∈ K , α, β = 1, . . . , d 3.1.58
for some Lipschitz continuous function g : K → [0, ∞ satisfying
gx = 0 ⇔ x ∈ ∂ K .
3.1.59 Then:
a Weak uniqueness holds for the non-interacting equation 3.1.19. The boundary distribution is stable against a linear drift, S
∞
C
K ⊂
C
K , and H is contained in the bp-closure of
C
2
H ∩ H .
b The class of w-harmonic functions is given by
H =
f ∈
C
K ∩
C
2
K
◦
:
α ∂
2
∂ x
α 2
f x = 0 on K
◦
. 3.1.60
c The associated special diffusion matrix is given by
w
∗ αβ
x = δ
αβ
g
∗
x x
∈ K , α, β = 1, . . . , d, 3.1.61
where g
∗
∈
C
K ∩
C
2
K
◦
is the unique solution of −
1 2
α ∂
2
∂ x
α 2
g
∗
x = 1
x ∈ K
◦
g
∗
x = 0
x ∈ ∂ K .
3.1.62
One can find a few more examples of diffusion matrices satisfying the assumptions in Theorem 3.1.5, but it turns out that these are mainly trivial variations on the
two examples mentioned above. The message of Theorem 3.1.5 is that all these examples fall into the same framework. The common property that unites them is
the stability of the boundary distribution against a linear drift.
In fact, we conjecture that this property is a necessary condition for the uni- versality of the long-time distribution. If the boundary distribution is not stable
against a linear drift, it seems likely that still X t converges weakly to some limit as t
→ ∞, although we do not know how to prove this for infinite . But we believe that in this case the law of X
∞ will depend on the choice of the recur- rent kernel a
S
and the Abelian group . However, we have at present very little knowledge about the nature of this dependence.
In conclusion, we have found that the ‘duality comparison argument’ devel- oped in [6] is linked to universality of the long-time distribution of solutions to
3.1.2. A similar relation between comparison arguments and universality has been found for models on the hierarchical group
N
with N large in [1, 10, 21]. There, the system in 3.1.2 is studied by means of a renormalization transforma-
tion acting on diffusion matrices. Under iteration of the transformation, the renor- malized diffusion matrices converge to a limit. In the clustering case a comparison
argument shows that this limit is universal within a large ‘universality class’ of ma- trices. This has been worked out for 2-type models in [1] and for isotropic models
in [21]. The universal limit that is found is exactly the w
∗
in formula 3.1.52. The conclusion we can draw from Theorem 3.1.5 is that the correct ‘universality
classes’ of diffusion matrices one should look at are formed by all diffusion matri- ces w that share the same boundary distribution Ŵ
x x
∈K
. Furthermore, universal behavior can be expected only if this boundary distribution is stable against a linear
drift.
3.2 Proofs of Theorems 3.1.1 and 3.1.2
3.2.1 Proof of Theorem 3.1.1
If ⊂ is finite then
C
2
K is the space of real functions on K
that have a
C
2
-extension to all of
R
d
.
C
2 fin
K consists of all functions that are the lifting
to the larger space K of a function in
C
2
K for some finite
⊂ .
Lemma 3.2.1 The operator A in 3.1.10 with domain
D
A in 3.1.12 is a densely defined linear operator on the Banach space
C
K , and satisfies the max-
imum principle.
