Is there a reasonable/canonical way to mollify a Borel probability measure without changing its marginals. Let $\pi \in \mathcal{P}(\mathbb{R}^2)$ with marginals $\mu,\nu$. I want to smooth out $\pi$ up to scale $\varepsilon$, which is equivalent to cutting it off at frequency $|k|\approx \varepsilon^{-1}$ in Fourier space, but I do not want to change is marginals. The resulting measure should have a smooth density. It goes without saying that the usual procedure (convolution with a smooth, positive, bump function) does not work. Of course, you can assume $\mu,\nu$ are absolutely continuous with smooth densities.



Mollifying a measure without changing its marginals
$\newcommand{\R}{\mathbb R}\newcommand{\ep}{\varepsilon}\newcommand{\tX}{\tilde X}\newcommand{\tY}{\tilde Y}\newcommand{\tpi}{\tilde\pi} $Here is how this can be done in an explicit way, at least when the densities (say $p$ and $q$) of $\mu$ and $\nu$ are everywhere $>0$.


Let $(X,Y)$ be a random point in $\R^2$ with distribution $\pi$ and thus marginals $\mu$ and $\nu$.


Let $F$ and $G$ be the c.d.f.'s of $X$ and $Y$ (respectively).


Let $\pi\_\ep$ be a distribution over $\R^2$ with a smooth density such that $\pi\_\ep$ is close to $\pi$. Let $(X\_\ep,Y\_\ep)$ be a random point in $\R^2$ with distribution $\pi\_\ep$. Let $F\_\ep$ and $G\_\ep$ be the c.d.f.'s of $X\_\ep$ and $Y\_\ep$ (respectively), so that $F\_\ep$ and $G\_\ep$ are close to $F$ and $G$, and hence $F^{-1}\circ F\_\ep$ and $G^{-1}\circ G\_\ep$ are smooth maps each close to the identity map.


Let $\tX:=F^{-1}(F\_\ep(X\_\ep))$ and $\tY:=G^{-1}(G\_\ep(Y\_\ep))$. Then the distribution (say $\tpi$) of the random point $(\tX,\tY)$ in $\R^2$ will have a smooth density and marginals $\mu$ and $\nu$, and $\tpi$ will be close to $\pi$.


