## An unusually-coupled simple random walk

For motivation see section 3.3 of
A lecture on the averaging process.
Consider continuous-time simple symmetric random walk on the integers;
that is, transition rates \(q(i,i+1) = q(i,i-1) = 1/2\).
Now let \( (X_1(t), X_2(t)), 0 \le t < \infty) \)
be two such processes coupled as follows.

(i) From a state \( (i,j) \) with \( |j - i| \ge 2 \)
they move independently.

(ii) From a state \( (i,i+1) \)
they make each of the following moves at rate \( 1/2 \):
\[ (i,i+1) \to (i-1,i+1); \quad (i,i+1) \to (i,i+2) \]
and they make each of the following moves at rate \( 1/4 \):
\[ (i,i+1) \to (i+1,i+1); \quad (i,i+1) \to (i,i); \quad (i,i+1) \to (i+1,i). \]

(ii) From a state \( (i,i) \) they make each of the following moves at rate \( 1/4 \):
\[ (i,i) \to (i,i+1); \quad (i,i) \to (i+1,i+1); \quad (i,i) \to (i+1,i) \]
\[ (i,i) \to (i,i-1); \quad (i,i) \to (i-1,i-1); \quad(i,i) \to (i-1,i) \]

Intuitively, the two component processes are "independent to first order".
From duality results in the paper above we know
\( \mathbb{P}(X_1(t) = i, X_2(t) = i) \ge (\mathbb{P}(X_1(t) = i))^2 \)
and the problem is to find the order of magnitude of the "second order" quantity
\[ v(t) :=
\sum_i ( \mathbb{P}(X_1(t) = i, X_2(t) = i) - (\mathbb{P}(X_1(t) = i))^2 ) \]
as \(t \to \infty\) .