Associated primes of local cohomology of flat extensions with regular fibers and $\mathrm{\Sigma }$-finite $D$-modules

In this article, we study the following question raised by Mel Hochster: let $(R,m,K)$ be a local ring and $S$ be a flat extension with regular closed fiber. Is $\text{cV}(mS)\cap {\text{Ass}}_S{H}_I^i(S)$ finite for every ideal $I\subset S$ and $i\in \text{NN}?$ We prove that the answer is positive when $S$ is either a polynomial or a power series ring over $R$ and $\dim (R\mathrm{/}I\cap R)\le 1.$ In addition, we analyze when this question can be reduced to the case where $S$ is a power series ring over $R$. An important tool for our proof is the use of $\mathrm{\Sigma }$-finite $D$-modules, which are not necessarily finitely generated as $D$-modules, but whose associated primes are finite. We give examples of this class of $D$-modules and applications to local cohomology.