  We propose a new scheme for the long time approximation of a diffusion when
the drift vector field is not globally Lipschitz. Under this assumption,
regular explicit Euler scheme --with constant or decreasing step-- may explode
and implicit Euler scheme are CPU-time expensive. The algorithm we introduce is
explicit and we prove that any weak limit of the weighted empirical measures of
this scheme is a stationary distribution of the stochastic differential
equation. Several examples are presented including gradient dissipative systems
and Hamiltonian dissipative systems.
