This research program is divided in two parts. Part I (subdivided in projects A,B,C) is planned as a continuation of the works initiated under the grants (SNF 200021-1036251/1) and (SNF 200020-112316) obtained by the principal applicant in April 2004 and April 2006. Part II presents a new research program motivated by the recent arrival in our department of the co-applicant Béatrice de Tillière.
\subsection*{Part I}
{\bf A) {\em Self-interacting diffusions.}} The preceding SNF grant was devoted to self-interacting diffusions on a compact manifold with symmetric interactions (Bena\"{\i}m and Raimond, 2005, 2007). Our next goal is to understand and describe finer properties of self-interacting diffusions in term of the geometry of the manifold and the nature of the interaction. Visits of O. Raimond (Paris) and L. Miclo (Marseille) are scheduled to work on this project.

\noindent {\bf B) {\em Stochastic approximations and differential inclusions.}}
The first phase of the project (initiated under the preceding SNF grants) developed a mathematical theory of ``stochastic approximations associated to differential inclusions'' (Bena\"{\i}m, Hofbauer, Sorin, 2005): {\em The DI (differential inclusion) method}. It was successfully applied to problems in game theory and generalized approachability in (Bena\"{\i}m, Hofbauer, Sorin, 2007). In this second phase, we will pursue our investigation of the DI method in two directions:
\begin{description}
\item a) A new PhD thesis (supervised by the principal applicant) will aim at extending certain classical results from the ``ODE method'' (convergence with positive probability toward attractors, non convergence to unstable sets) to the DI method;
\item b) The case of ``controlled Markov chains'' will be considered with the purpose of understanding some learning processes with partial information and limited memory.
\end{description}
This project may involve the collaborations of J. Hofbauer (London), Sylvain Sorin (Paris) and O. Raimond (Orsay).
\noindent {\bf C) {\em First passage time percolation and effective resistance in $\ZZ^d$.}} This project falls under the line of (Bena\"{\i}m and Rossignol, 2006) and (Benjamini and Rossignol, 2007) which were devoted to the fluctuations of, respectively, point-to-point first passage percolation time and point-to-point effective resistance in $\ZZ^2$. Results obtained in these settings are believed not to be optimal. In order to circumvent the notorious difficulties of these models, we want first to understand them on a strip $S_k:=\ZZ\times \{0,\ldots,k\}$, where $k$ is a fixed integer.
This project will be conducted in close collaboration with R. Rossignol (Orsay).
Visits of R. Rossignol in Neuch\^atel are scheduled.