Clustering: Theorem 3.1.3 Introduction and main results
Let us assume that for each initial condition x ∈ K , the non-interacting equa-
tion 3.1.19 has a unique weak solution X
x
t
t ≥0
, and let us denote the associated semigroup on BK by
S
t
f x : = E[ f X
x
t] x
∈ K , t ≥ 0. 3.1.40
We add a ‘last element’ S
∞
to this semigroup by defining S
∞
f x : = E[ f X
x
∞] =
K
Ŵ
x
d y f y x
∈ K , f ∈ BK , 3.1.41
where Ŵ
x x
∈K
is the boundary distribution associated with w, introduced in 3.1.22.
With this notation, we formulate a condition that will guarantee that the long- time behavior of the non-interacting model is not changed by the introduction of a
linear drift.
Definition 3.1.1 Let w be a diffusion matrix on K such that weak uniqueness holds for 3.1.19, and let Ŵ
x x
∈K
be the associated boundary distribution. We say that Ŵ
x x
∈K
is stable against a linear drift if S
∞
T
θ, t
S
∞
f = T
θ, t
S
∞
f ∀θ ∈ K , t ≥ 0, f ∈ BK .
3.1.42 Since S
∞
S
∞
= S
∞
, we can read equation 3.1.42 as: S
∞
and T
θ, t
commute on functions of the form S
∞
f . For technical reasons, we will restrict ourselves to the case that
S
∞
C
K ⊂
C
K . 3.1.43
This condition guarantees that S
∞
f is a w-harmonic function for all f ∈
C
K , where the space of w-harmonic functions is defined as
H : = { f ∈
D
G : G f = 0},
3.1.44 with G the full generator of the process in 3.1.19 and
D
G its domain. In par- ticular,
C
2
-functions are w-harmonic if and only if they solve the equation
αβ
w
αβ
x
∂
2
∂ x
α
∂ x
β
f x = 0
x ∈ K .
3.1.45 It turns out that condition 3.1.42 is equivalent to
T
θ, t
H ⊂ H
∀θ ∈ K , t ≥ 0. 3.1.46
That is, for each θ the space of w-harmonic functions is invariant under the semi- group T
θ, t
t ≥0
. With these definitions, our main result reads as follows.
Theorem 3.1.5 Let X be a shift-invariant solution to 3.1.2 such that
L
X 0 is spatially ergodic and
E[X
i
0] = θ
i ∈
3.1.47 for some θ
∈ K . Assume that weak uniqueness holds for the non-interacting equation 3.1.19, that the associated boundary distribution is stable against a
linear drift, that S
∞
C
K ⊂
C
K and that H is contained in the bp-closure of
C
2
K ∩ H . If the random walk with kernel a
S
is recurrent, then there exists a K
-valued random variable X ∞ such that
X t ⇒ X ∞
as t → ∞,
3.1.48 where
L
X
i
∞ = Ŵ
θ
i ∈ .
3.1.49 The bp-closure of a set is the smallest set containing it that is closed under bounded
pointwise limits. Note that by Theorem 3.1.3, P[X
i
∞ = X
j
∞ ∀i, j ∈ ] = 1. Thus, the fact that the boundary distribution is stable against a linear drift not only allows us
to conclude that X t converges to a limit X ∞, it also allows us to completely
specify its distribution. This distribution turns out to be universal in all recurrent random walk kernels a
S
and Abelian groups , and in all diffusion matrices w sharing the same boundary distribution Ŵ
x x
∈K
.