Is this paper we study penalisations of diffusions satisfying some technical conditions, generalizing a result obtained by Najnudel, Roynette and Yor. If one of these diffusions has probability distribution $\mathbb{ P}$, then our result can be described as follows: for a large class of families of probability measures $(\mathbb{ Q}_t)_{ t \geq 0}$, each of them being absolutely continuous with respect to $\mathbb{ P}$, there exists a probability $\mathbb{ Q}_{ \infty}$ such that for all events $\Lambda$ depending only on the canonical trajectory up to a fixed time, $\mathbb{ Q}_t (\Lambda)$ tends to $\mathbb{ Q}_{ \infty} (\Lambda)$ when $t$ goes to infinity. In the cases we study here, the limit measure $\mathbb{ Q}_{ \infty}$ is absolutely continous with respect to a sigma-finite measure $\mathcal{ Q}$, which does not depend on the choice of the family of probabilities $(\mathbb{ Q}_t)_{ t \geq 0}$, but only on $\mathbb{ P}$. The relation between $\mathbb{ P}$ and $\mathcal{ Q}$ is obtained in a very general framework by the authors of this paper.
Categories: math.PR