Renormalization theory Overview of the three articles
we expect X
N
to solve the martingale problem for an operator of the form A f x :
=
i j
a j − i
α
x
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, 1.2.5
with domain the
C
2
-functions f that depend on finitely many x
α i
only. Here the dif- fusion matrix w can be the p-type q-tuple diffusion matrix w
p,q
, originating from a q-tuple resampling mechanism, but we also allow for more general w, originating
from a composition-dependent resampling mechanism. We choose the migration kernel a in such a way that the strength of the migra-
tion between two urns depends only on their hierarchical distance. The collection of all urns at hierarchical distance at most k from an urn i
{ j ∈
N
: j − i ≤ k}
1.2.6 we call the k-block around i . We fix constants c
1
, c
2
, . . . ∈ 0, ∞ and for all
k = 1, 2, . . . we let the balls in our urns be subject to the following migration
mechanism: With rate c
k
N
k −1
each ball in an urn i chooses a random urn in the k-block around i possibly itself and migrates to that urn. This means that the
migration kernel a is given by
ai =
∞ k
=i
c
k
N
2k −1
. 1.2.7
To understand why 1.2.7 is the correct formula, note that a ball in urn i decides with rate c
k
N
k −1
to jump to another urn in the k-block around i . If k ≥ i, this
urn is with probability N
−k
the origin. The process X
N
can be represented, on an appropriately chosen probability space equipped with p
−1-dimensional independent Brownian motions B
i i
∈
N
, as a solution to the following system of stochastic differential equations:
d X
N,α i
t =
∞ k
=1
c
k
N
k −1
X
N,k,α i
t − X
N,α i
t dt
+
β
σ
αβ
X
N i
td B
β i
t i
∈
N
, α = 1, . . . , d, t ≥ 0,
1.2.8 where
1 2
γ
σ
αγ
xσ
βγ
x = w
αβ
x 1.2.9
and X
N,k i
t is the k-block average around i : X
N,k,α i
t : = N
−k j :
j−i≤k
X
N,α i
t. 1.2.10
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