# 2-Wasserstein metric on convolution of probability distributions

I have two related questions. Let $$\mu$$ and $$\nu$$ be two distinct probability measures on $$\mathbb{R}^n$$ with finite second moments, and $$W_2(\cdot,\cdot)$$ be the $$2$$-Wasserstein metric. The question is: (1) As $$\mu$$ and $$\nu$$ are distinct, hence $$W_2(\mu,\nu)\neq 0$$. Is it possible that there is a sequence of distributions $$\mu_i, i\geq 1$$ on $$\mathbb{R}^n$$ such that $$\lim_{i\to\infty} W_2(\mu*\mu_i,\nu*\mu_i)=0$$? Here $$*$$ is the convolution.

While I was trying to figure out the answer using Prokhorov's theorem, I realized that I need to answer the following more elementary looking question (though I do not have an answer yet):

(2) If $$\mu$$ and $$\nu$$ are two distinct probability measures on $$\mathbb{R}^n$$, then is it possible that there is another probability measure $$\mu'$$ such that $$\mu*\mu' = \nu*\mu'$$?

Some special cases can be easily dealt with (to give a negative answer), using for example Fourier transform. However, I am unsure about the general case.

Any comments and references to these two problems are highly appreciated!

Indeed, it is easy to check that the functions $$f$$ and $$g$$ given by $$f(t):=\max(0,1-|t|)$$ and $$g(t):=\sum_{k=-\infty}^\infty f(t-2k)$$ for real $$t$$ are characteristic functions: $$f(t)=\int_{-\infty}^\infty e^{itx}\mu(dx)$$ and $$g(t)=\int_{-\infty}^\infty e^{itx}\nu(dx),$$ where $$\mu(B):=\int_B dx\,\frac{1-\cos x}{\pi x^2}$$ and $$\nu(B):=\frac{1(0\in B)}2 +\frac2{\pi^2}\,\sum_{k=-\infty}^\infty \frac{1((2k-1)\pi\in B)}{(2k-1)^2}$$ for all Borel sets $$B\subseteq\mathbb R$$.
We also have $$gf=f^2=ff.$$
So, if $$\mu':=\mu$$, then we have $$\mu\ne\nu$$ but $$\mu*\mu'=\nu*\mu'$$, as claimed.