A major challenge in the statistical analysis of multidimensional stochastic pro- cesses consists in deriving results under conditions on the characteristics of the underlying process, which are both general and easily veriable. For exam- ple, several probabilistic tools as they are required for the in-depth analysis of estimation procedures are commonly based on the reversibility assumption or are limited to the case of diffusion operators without jumps. We extend this framework by providing bounds on the variance of integral functionals and Bernstein-type concentration inequalities for a wide class of non-reversible jump diffusions, which also includes diffusions with continuous paths. The ef- fectiveness of our findings is demonstrated by deriving fast convergence rates for the sup-norm risk for nonparametric estimation of the invariant density of the considered processes.