Time-changed spectrally positive L��vy processes starting from infinity

Consider a spectrally positive L��vy process $Z$ with log-Laplace exponent $��$ and a positive continuous function $R$ on $(0,\infty)$. We investigate the entrance from $\infty$ of the process $X$ obtained by changing time in $Z$ with the inverse of the additive functional $��(t)=\int_{0}^{t}\frac{{\rm d} s}{R(Z_s)}$. We provide a necessary and sufficient condition for infinity to be an entrance boundary of the process $X$. Under this condition, the process can start from infinity and we study its speed of coming down from infinity. When the L��vy process has a negative drift $��:=-��0$ and $R$ is regularly varying at $\infty$ with index $��>��$, the process comes down from infinity and we find a renormalisation in law of its running infimum at small times.