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 the same stochastic differential equation, at the same time t, and initialized at slightly different positions. Ie, $X_1$ and $X_2$ both respect:
$$ \dot X = – \nabla \phi(X) + 2 dW $$
and their initial positions are very close to one another. And $p_i$ are the marginals of the position of the particle at some time $t$.
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
Best Answer
Guillaume, the answer given by Dan contains a response to your question. Indeed, you can use the fact that the TV distance between the distributions of $X_t^{(1)}$ and $X_t^{(2)}$ is smaller than the TV-distance between the distributions of these processes on the path space $C([0,t])$ up to time $t$. Then, you can use the Pinsker inequality in conjunction with the Girsanov formula. More details can be found in https://arxiv.org/pdf/1412.7392v3.pdf (cf, in particular, Eq (17))