Convolution is a Lipschitz Map Consider a well-behaved space $S$ of probability distributions on $\mathbb{R}^d$, and consider the $p$th Wasserstein metric $d_p : S \times S \rightarrow \mathbb{R}_{\ge 0}$. For a fixed distribution $\nu \in S$ with a bounded support (which represents error), is the map $\mu \mapsto (\mu * \nu)$ Lipschitz with respect to the $p$th Wasserstein distance? For which $p$?
I suppose a part of the question is figuring out what such a well-behaved space $S$ could be, and I'd like to apologize ahead because I'm not an expert in analysis or probability theory.
 A: Actually, if we take $S$ to be the whole space of probability distributions on $\mathbb{R}^d$, even with no restrictions on $\nu$ (besides that $\nu$ is a probability distribution), the map $\mu \mapsto \mu * \nu$ is Lipschitz with Lipschitz constant at most 1.
Recall the definition of the $p$-th Wasserstein distance (on the space of probability measures):
$$ d_W(\mu, \mu') = 
\inf \left(\mathbb{E}[ \| X - Y \|^p]\right)^{1/p}, $$
where the infimum is taken over all joint distributions on random variables $(X,Y)$ such that $X$ is $\mu$-distributed and $Y$ is $\nu$-distributed.
Recall also that if $X$ is $\mu$-distributed and $Z$ is $\nu$-distributed and $X$ and $Z$ are independent, then $X+Z$ is $\mu * \nu$ distributed.
Thus, whenever $X$ is $\mu$-distributed and $Y$ is $\mu'$-distributed, if we take $Z$ to be a $\nu$-distributed random variable which is independent of $(X,Y)$, then $(X+Z,Y+Z)$ has marginal distributions $\mu * \nu$ and $\mu' * \nu$ respectively. Thus for any such $(X,Y)$ we get the bound
$$ d_W(\mu * \nu, \mu' * \nu)^p \le \mathbb{E}[ \| (X+Z) - (Y+Z) \|^p ] = \mathbb{E}[ \| X - Y \|^p ].$$
Taking the infimum over $(X,Y)$ then gives the inequality $d_W(\mu * \nu, \mu' * \nu) \le d_W(\mu, \mu')$.
Note that we used very little--no restrictions on $\nu$, $p$, or even the metric $\|\cdot - \cdot \|$ on $\mathbb{R}^d$ we used except that it was translation invariant. The same argument in fact shows the analoguous statement for any group endowed with an invariant metric (where convolution is defined as the law of a product of independent random variables).
