We show that there exist Lévy processes $(X_t,t \geq 0)$ and reals $y\gt0$ such that for small $t$, the probability $P(X_t\gt y)$ has an expansion involving fractional powers or more general functions of $t$. This constrasts with previous results giving polynomial expansions under additional assumptions.
"Small time expansions for transition probabilities of some Lévy processes." Electron. Commun. Probab. 14 132 - 142, 2009. https://doi.org/10.1214/ECP.v14-1452