It is clear that the hierarchical group with its structure of blocks made out of smaller blocks is ideally suited for renormalization theory. For certain 2-type mod-
els it has been shown that the system in 1.2.8 admits a rigorous description in terms of a renormalization transformation, in the limit where the dimension N of
the hierarchical group tends to infinity. Since we expect this result to hold more generally, we formulate it here as a non-rigorous conjecture.
For c ∈ 0, ∞, x ∈ ˆ
K
p
and for any diffusion matrix w on ˆ K
p
, let us write A
w, c
x
for the operator A
w, c
x
f y : =
α
cx
α
− y
α ∂
∂ y
α
f y +
αβ
w
αβ
y
∂
2
∂ y
α
∂ y
β
f y. 1.2.11
We expect that for ‘reasonable’ w this is one point where we are non-rigorous the martingale problem for A
w, c
x
is well-posed and the associated diffusion process has a unique equilibrium and is ergodic. By Z
w, c
x
we denote the solution to the martingale problem for A
w, c
x
with initial condition Z
w, c
x
= x, and by ν
w, c
x
d y we denote the equilibrium distribution associated with A
w, c
x
. For each c
∈ 0, ∞ we define a renormalization transformation F
c
, acting on diffusion matrices w we are vague as to the precise domain of F
c
by the formula F
c
w
αβ
x : =
ˆ K
p
w
αβ
yν
w, c
x
d y. 1.2.12
Conjecture 1.2.1 Assume that X
N
solves 1.2.8 with initial condition X
i
= θ for all i
∈
N
. Then for each k ≥ 0
X
N,k
N
k
t
t ≥0
⇒ Z
F
k
w, c
k +1
θ
t
t ≥0
as N → ∞,
1.2.13 where F
k
w is the k-th iterate of renormalization transformations F
c
applied to w
: F
k
w :
= F
c
k
◦ · · · ◦ F
c
1
w. 1.2.14
Furthermore, for any t 0 X
N,k
N
k
t, . . . , X
N,0
N
k
t ⇒ Z
k
, . . . , Z
as N → ∞,
1.2.15 where Z
k
, . . . , Z
is a Markov chain in this order with transition probabilities P[Z
n −1
∈ dy|Z
n
= x] = ν
F
n −1
w, c
n
x
d y n
= 1, . . . , k. 1.2.16
Note that in order to get a non-trivial limit in 1.2.13, we need to rescale space and time. While we rescale space by going to k-block variables, we rescale time
by a factor N
k
. In the limit N → ∞ the block averages of differently sized blocks
evolve on separate time scales. The average of a large block changes much slower than the average of a smaller block.
If we consider the time evolution of a k-block average, then we may treat its interaction due to migration with the much larger k
+ 1-block that it is part of as if this k
+ 1-block is an infinite reservoir in which the frequencies of colors are fixed. As we saw in section 1.1.4, the process Z
w, c
x
describes the behavior of an urn that is in interaction with such a reservoir.
After a sufficiently long time, the k-blocks reach equilibrium, subject to the value of the k
+1-block with which they interact. The diffusion matrix describing the evolution of this k
+ 1-block can then be found by averaging the diffusion matrix of the k-blocks with respect to this equilibrium distribution. This is how the
renormalization transformation F
c
arises. For the details behind the heuristics, we refer to Chapter 2.
1.2.3 A renormalization transformation
In [10], Dawson and Greven proved a rigorous form of Conjecture 1.2.1 for a class of 2-type models. They considered X
N i
t, indexed by the hierarchical group
N
and taking values in ˆ K
2
= [0, 1], solving an equation of the form compare 1.2.8 d X
N i
t =
∞ k
=1
c
k
N
k −1
X
N,k i
t − X
N i
t dt
+ gX
N i
td B
i
t t ≥ 0, i ∈
N
, 1.2.17
where g is taken from the class
H
of functions g : [0, 1] → [0, ∞ that are
Lipschitz continuous and satisfy gx = 0 ⇔ x ∈ {0, 1}. On
H
and for c ∈ 0, ∞,
the renormalization transformation F
c
:
H
→
H
is defined by F
c
gx : =
[0,1]
gyν
g,c x
d y, 1.2.18
with ν
g,c x
the unique equilibrium distribution of a diffusion whose generator extends A
g,c x
f y : = cx − y
∂ ∂
y
f y + gy
∂
2
∂ y
2
f y. 1.2.19
The equilibrium measure ν
g,c x
occuring in 1.2.18 is in fact known in closed form, and the transformation F
c
is given by the following explicit formula:
F
c
gx =
1
d y
e
−
y x
d z
cz −x
gz 1
d y
1 gy
e
−
y x
d z
cz −x
gz
. 1.2.20
As we see from formula 1.2.20, F
c
g depends in a complicated and non-linear way on g, so it is not obvious how the iterates of F
c
behave. In [1], Baillon, Cl´ement, Greven and Den Hollander studied these iterates. They were able to show the
following.
Proposition 1.2.2 Let g
∗
∈
H
be given by g
∗
x : = x1 − x.
1.2.21 Then g
∗
is a fixed shape under F
c
, c ∈ 0, ∞:
F
c
λ g
∗
= c
λ + c
λ g
∗
∀λ 0. 1.2.22
Moreover, for k = 1, 2, . . . assume that c
k
∈ 0, ∞ satisfy
∞ k
=1
1 c
k
= ∞, 1.2.23
and define σ
n
: =
n k
=1
1 c
k
F
n
g : = F
c
k
◦ · · · ◦ F
c
1
g. 1.2.24
Then for all g ∈
H
, one has lim
n →∞
sup
x ∈[0,1]
σ
n
F
n
gx − g
∗
x =
0. 1.2.25
Proposition 1.2.2 shows that the renormalization transformations F
c
have a unique fixed shape g
∗
that attracts all diffusion functions g ∈
H
after appropriate scal- ing. This implies that the system in 1.2.17 exhibits universal behavior on large
space and time scales. As we already saw in section 1.2.2, the renormalized diffu- sion functions F
k
g describe the behavior of k-block averages on their appropriate time scales N
k
. Thus, formula 1.2.25 shows that large block averages k → ∞
evolve according to the universal diffusion function g
∗
, independent of the dif- fusion function g of the individual components X
N i
t of the system. The scale factors σ
n
are not important here, because they can always be absorbed in a redef- inition of the time scale. The same is true for the factor cc
+ λ in 1.2.22, so the fact that our renormalization transformation has a fixed shape instead of a fixed
point is not important. It turns out that g
∗
is the Wright-Fisher diffusion function see 1.1.30, which arises in a natural way as the diffusion limit of the 2-type
2-tuple model.
Condition 1.2.23 is necessary for the universality observed in 1.2.25. It is known that condition 1.2.23 corresponds to clustering behavior in the model
Theorem 3 in [10]. This means that the components X
N i
t of the system, after a long time, spend most of their time near the boundary of [0, 1]. In fact, according
to Conjecture 1.2.1,
lim
N →∞
P[X
N
N
k
t ∈ dy|X
N,k
N
k
t = x] = K
k x
d y, 1.2.26
where K
g,k x
d y =
· · · ν
F
k −1
g,c
k
x
d z
1
ν
F
k −2
g,c
k −1
z
1
d z
2
· · · ν
F g,c
1
z
k −1
d y. 1.2.27
Thus, the probability measure K
g,k x
· describes the conditional distribution of the urns in a k-block, given that the k-block average is x. It has been shown in [1] that
K
g,k θ
· ⇒ 1 − θδ + θδ
1
as k → ∞,
1.2.28 if and only if 1.2.23 holds. Therefore 1.2.23 corresponds to the situation where
after a long time each urn with probability close to θ contains almost only balls of color 1, and with probability close to 1
− θ almost only balls of color 2.
1.2.4 Higher-dimensional generalizations
The results in Proposition 1.2.2 leave one with a number of questions. Notably, one would like to understand better the origin of the observed universality. What
is so special about the Wright-Fisher diffusion function that all other diffusion functions in the class
H
are attracted to it? In October 1995 my supervisor and me took this question as our motivation to study higher-dimensional equivalents of the
transformation F
c
. For simplicity, we first restricted ourselves to isotropic diffusions. Thus, we
considered a transformation of the form F
c
gx : =
K
gyν
g,c x
d y, 1.2.29
where K ⊂
R
d
is some compact and convex domain, and ν
g,c x
is the unique equi- librium distribution of a diffusion whose generator extends
A
g,c x
f y : =
α
cx
α
− y
α ∂
∂ y
α
f y + gy
α ∂
2
∂ y
α 2
f y. 1.2.30
This Ansatz immediately raised a number of questions.
