A new coupling technique

Let , be two probability measures on the same measurable space .

We draw jointly the couple of random variables according to the following procedure:

Proposition. is a coupling of .

What is nice is that we are able to couple these random variables whereas their densities are known only up to a multiplicative constant. I wonder if it is interesting to couple in that way: and . Up to some tricks, can we deduce a way of coupling two MH starting from different initial distributions, maybe with delayed coupling? Can we compare it to the coupling of Pierre Jacob et al.?
Moreover, if we look at the problem today, if then, is a coupling of , no?

Obviously, where

We now show that is a coupling of . To do so, it is sufficient to check that for any bounded or non-negative function , .

Indeed, write, using the detailed balance condition in the second line : which completes the proof.

The coupling probability is given by:

Question: we know that . But I can't see how to prove