We give the asymptotic estimate for large $t$ of elementary solutions of one-dimensional generalized diffusion equations with regularly varying Green functions. As a corollary we obtain the precise asymptotic behavior of the semigroup $T_tf(x)$ for all $f \in L_1(dm)$ if the speed measure function $m(x)$ is regularly varying as $x \rightarrow \pm \infty$.
"Asymptotic Behavior of Elementary Solutions of One-Dimensional Generalized Diffusion Equations." Ann. Probab. 13 (3) 698 - 715, August, 1985. https://doi.org/10.1214/aop/1176992904