## 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 $$t \to \infty$$ .