On Penalisation Results Related with a Remarkable Class of Submartingales

#### J. Najnudel, A. Nikeghbali

2013, v.19, Issue 4, 763-790

ABSTRACT

Is this paper we study penalisations of diffusions satisfying some technical conditions, generalizing a result obtained by Najnudel, Roynette and Yor in [A Global View of Brownian Penalisations. MSJ Memoirs, v. 19, Mathematical Society of Japan, Tokyo]. If one of these diffusions has probability distribution $P$, then our result can be described as follows: for a large class of families of probability measures $(Q_t)_{t \geq 0}$, each of them being absolutely continuous with respect to $P$, there exists a probability $Q_{\infty}$ such that for all events $\Lambda$ depending only on the canonical trajectory up to a fixed time, $Q_t (\Lambda)$ tends to $Q_{\infty} (\Lambda)$ when $t$ goes to infinity. In the cases we study here, the limit measure $Q_{\infty}$ is absolutely continuous with respect to a sigma-finite measure $\mathcal{Q}$, which does not depend on the choice of the family of probabilities $(Q_t)_{t \geq 0}$, but only on $P$. The relation between $P$ and $\mathcal{Q}$ is obtained in a very general framework by the authors of this paper in [On some universal $\sigma$-finite measures and some extensions of Doob's optional stopping theorem arXiv:0906.1782].

Keywords: penalisation,convergence of probability measures,diffusion,submartingales of class sigma