Density convergence on Markov diffusion chaos via Stein's method

We study the difference between the probability density of a random variable $F$ on Markov diffusion chaos and the probability density of a general target distribution $Z$. In the special case where $F$ is a chaotic random variables and $Z$ is a Pearson target, we extend our study to the $k$-th derivatives of the densities for all $k\in \mathbb{N}$. In particular, we obtain four moment theorems for the convergence of the $k$-th derivatives of the densities of $F$ to the corresponding $k$-th derivatives of the density of a Pearson target. Our work therefore significantly extends earlier works [HLN14,BDH24] which studies density convergence of random variables on Wiener chaos to respectively the normal and Gamma targets. We provide two applications of our results. The first application is about weighted sum of i.i.d. Gamma distribution where we show convergence in laws of this weighted sum to another Gamma distribution automatically implies convergence in densities. In the second application, we show that for a large class of Pearson diffusions, the density of its solution with any initial condition exponentially converge to its limiting density. Moreover, this exponential convergence holds for the $k$-th derivatives of the densities for all $k\in \mathbb{N}$.