Consider a strong Markov process in continuous time, taking values in some Polish state
space. Recently, Douc et al. [Stoc. Proc. Appl.
119, (2009) 897–923] introduced verifiable conditions in terms of
a supermartingale property implying an explicit control of modulated moments of hitting
times. We show how this control can be translated into a control of polynomial moments of
abstract regeneration times which are obtained by using the regeneration method of
Nummelin, extended to the time-continuous context. As a consequence, if a
p-th moment of the regeneration times exists, we obtain non asymptotic
deviation bounds of the form \begin{equation*} P_{\nu} \left
(\left|\frac1t\int_0^tf(X_s){\rm d}s-\mu(f)\right|\geq\ge\right)\leq
K(p)\frac1{t^{p- 1}}\frac 1{\ge^{2(p-1)}}\|f\|_\infty^{2(p-1)} ,\quad p \geq 2.
\end{equation*} Here, f is a bounded function and
μ is the invariant measure of the process. We give several examples,
including elliptic stochastic differential equations and stochastic differential equations
driven by a jump noise.