By Vuk Milisic
Appears in collection : Mathematical models for proliferation and propagation / Modèles mathématiques pour la prolifération et la propagation
We 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: Brownian motion with a diffusion coefficient dependent on model parameter, a process with a 3/2 order variation and a process with exponential behavior.We then study a more realistic non convolution model and prove a similar transition between the first two regimes : Brownianmotion versus process with 3/2 order variation. At last, we study the particular case of an exponential memory kernel which allows to construct an explicit solution.
By Ariane Trescases
By Pierre Roux
