Velocity averaging under minimal conditions for deterministic and stochastic kinetic equations with irregular drift

This study investigates the $L^1_{\operatorname{loc}}$ compactness of velocity averages of sequences of solutions $\{u_n\}$ for a class of kinetic equations. The equations are examined within both deterministic and stochastic heterogeneous environments. The primary objective is to deduce velocity averaging results under conditions on $u_n$ and the drift ${\mathfrak f}={\mathfrak f}(t,{\boldsymbol x},{\boldsymbol \lambda})$ that are more lenient than those stipulated in previous studies. The main outcome permits the inclusion of highly irregular drift vectors ${\mathfrak f} \in L^q$ that adhere to a general non-degeneracy condition. Moreover, the sequence $\{u_n\}$ is uniformly bounded in $L^p$ -- for an exponent $p$ allowed to be strictly smaller than $2$ -- under the requirement $\frac{1}{p} + \frac{1}{q} < 1$. Resolving the matter of strong compactness in velocity averages, considering these assumptions, has remained an open problem for a long time. The cornerstone of our work's progress lies in the strategic employment of the broader concept of $H$-distributions, moving beyond the traditional reliance on $H$-measures. Notably, our study represents one of the first significant uses of $H$-distributions in this context.