Median, concentration and fluctuations for Lévy processes

We estimate a median of f(Xt)f(Xt) where ff is a Lipschitz function, XX is a Lévy process and tt is an arbitrary time. This leads to concentration inequalities for f(Xt)f(Xt). In turn, corresponding fluctuation estimates are obtained under assumptions typically satisfied if the process has a regular behavior in small time and a, possibly different, regular behavior in large time.