Opposite gaps defeat large-time unimodality for one-sided stable processes

Hasebe and Ueda conjectured that if $S_t$ (resp. $T_t$) is the law at time $t$ of a classical (resp. free) strictly $\alpha$-stable process and the initial law $\mu$ has a finite absolute moment of order $2+\alpha$, then $\mu*S_t$ (resp. $\mu\boxplus T_t$) is unimodal for all large $t$. We show that this fails for every $\alpha\in(0,1)$ in both settings, for the stable laws concentrated on the positive half-line. There is one probability measure $\mu_\ast$, with $\int e^{\lambda|x|}\,d\mu_\ast(x)<\infty$ for every $\lambda>0$, such that $\mu_\ast*S^{(\alpha)}_t$ and $\mu_\ast\boxplus T^{(\alpha)}_t$ are not unimodal for any $\alpha\in(0,1)$ and any $t>0$. Its atoms run to $-\infty$, against the direction of the kernel, with gaps tending to infinity. In the classical case the convolution density is strictly decreasing on intervals arbitrarily far to the left; in the free case the density of the free convolution vanishes on an interval lying between two points where it is positive, which we read off from the Huang–Wang description of the support of a free convolution with a freely infinitely divisible law.