I seek to bound the total-variation distance between two probability measures $p_1$ and $p_2$. It is extremely easy to build a parameter space where $p_1$ and $p_2$ are the marginals of some joint probability distribution (see below for details) so it's extremely easy to control the Wasserstein-k distance between the two.

However, what I'm really interested is the total-variation distance between the two. Is there some hope for me there ? Obviously, any sort of bound would involve the standard deviation of the distributions, and would only work for co-measurable $p_1$ and $p_2$. Do you know anything of that kind ?

Details: $p_1$ and $p_2$ are the probability distributions of the position of a particle behaving according to stochastic differential equation, and initialized at slightly different positions. Ie, $X_1$ and $X_2$ both respect:

$$ \dot X = - \phi(X) + 2 dW $$

and their initial positions are very close to one another.

It's easy to couple them by considering the processes driven by the same Wiener process dW, but I don't see how to get good total-variation bounds that way