I would be interested in references on the topic of distances between probability measures that are singular with one another and not reduced to trivial ones. For example from here we know that total variation distance is not suitable for such a problem. In the end I am looking for a structure that would be fine enough to be able to address mutually singular Wiener measures in a non trivial way (for example it is well known that the Wiener measures associate with the two process driven by a BM are mutually singulars : $X^1_t =\sigma_1.W_t$ and $X^2_t =\sigma_2.W_t$, with $X^i_0=0, i=1,2$ as soon as $\sigma_1\not=\sigma_2$ so finding a meaningful distance between those measures is a matter of interest for me). Best regards
Edit following the advice of Ilya here i what I could set up using the Wasserstein representation of Kantorovitch distance.
So let's formalize the set up : Let $(X^1_t)_{t\geq 0},(X^2_t)_{t\geq 0}$ be two processes defined by as $X_0^i =1,X^i_t=e^{\mu.t + \sigma_i.W^i_t$}, i=1,2$ with respect to 2 Brownian motions $W^1_t$ and $W^2_t$. The relation between the 2 process $W^1_t, W^2_t$ is left unspecified and we do not impose that $(W^1_t, W^2_t)$ is a 2 dimensional BM at the moment.
Let's note $\mathcal{L}_i$ the marginal law's of $(X^i_t)_{t\geq 0},i=1,2$, $X^i$ seen as a random variables in $C_1(\mathbb{R^+},\mathbb{R^+})$ the space of continuous function s.t. $f(0)=1$, and $\forall t\geq 0, f(t)\geq 0$, the measures being over sets of the Borel sigma-field resulting from topology of the sup norm over compacts of this functional space.
Now the space $\mathcal{J}$ of all the joint laws over state space $C_1(\mathbb{R^+},\mathbb{R^+})^2$ with marginals equals to $\mathcal{L}_1$ and $\mathcal{L}_2$ We define the distance between those 2 measure (which are btw mutually singulars) as :
$$d_{KW}(\mathcal{L}_1,\mathcal{L}_2)=inf_{J\in \mathcal{J}}(\mathbb{E^J}[d(X^1,X^2)])$$
So now using Ilya's ideas and specifying the $d$ in the preceding infimum, as $d(x,y)=sup_{s\geq0}(|x(s)-y(s)|) $, using a particular $J_W\in \mathcal{J}$ which is the one where $W^1_t= W^2_t$ I can rewrite using Burkholder, Davies, Gundy inequality (maybe that's too much) that :
$d_{KW}(\mathcal{L}_1,\mathcal{L}_2)\leq \mathbb{E^{J_W}}[|X^1_s-X^2_s|^*])$ (here the star is for the sup) $\mathbb{E^{J_W}}[|X^1_s-X^2_s|^*])\leq C_1\mathbb{E^{J_W}}[<X^1_.-X^2_.>_\infty^{1/2}])=F^{J_W}$
I have not finished the calculations yet but even though, I bet that thanks to the negative drift it should be possible for $F^{J_W}$ to be bounded and that this bound tends to $0$ as $\sigma_1 - \sigma_2\to 0$ ( I think this was Ilya's original idea). Anyway this would not prove that $d_{KW}(\mathcal{L}_1,\mathcal{L}_2)$ is not trivially equal to 0 for a better choice of $J \in\mathcal{J}$ this is why I'm still a bit unsatisfied with this example. Any thoughts ?
Wassertstein/Kantorovich metric is quite natural to use in case TV is too conservative, you can also check out this classic reference on other distances.
For example, think of two processes with exponential decay, e.g. GBMs with very strong negative drift (same for both) and slightly different volatilities. The TV distance between them is going to be $1$, but Kantorovich metric can be bounded by their average distance assuming coupling over the same driving Brownian motion, which can be estimated using supermartingale inequalities. I would expect this distance to go to $0$ as volatilities converge.