In a first step, I will present a probabilistic interpretation for a nonlinear, nonlocal PDE on $(0,\infty)$ introduced to model cell polarization. More precisely, the solution of the PDE is the density at time t of a certain diffusion process. This PDE includes a nonlinear boundary attraction term, which translates on the process into a drift term depending on the density of the solution at x=0. Then, this nonlinear process can be approximated by a particle system by replacing the value of the density at 0 by an empirical mean of the local times at the edges of N copies of the process. This particle system can be seen as an N-dimensional process living in a non-regular open and with an oblique reflection on the boundary. I will discuss the existence for the particle system, for the nonlinear process and for the PDE and the links between these three objects. In particular, for a sufficiently large attraction coefficient, these objects must necessarily blow-up in finite time.

In a second step I will study the long term behavior of a class of Self-Interacting Diffusions on R, where the particle is attracted or repelled by its past trajectory, with aging and linear interactions with past positions. The novelty of this model is to introduce a weight depending on past trajectories through a memory kernel to describe the aging phenomenon. In a first step, we study the case where the memory kernel is general of convolution type. We observe three distinct asymptotic regimes when time becomes large depending on the data and the first moment of the integral kernel. We then study a more realistic non-convolution model and prove a similar transition between the first two regimes. At last, we study the particular case of an exponential memory kernel which allows to construct an explicit solution.