Long term behaviour of self-interacting diffusions

Le but de cette thèse est l'étude en temps long de Diffusions Auto-Interagissante sur des variétés Riemannienne de la forme $$dX_t =dB_t(X_t)- \nabla V_t(X_t)dt, $$ avec $V_t(x)= \int_0^t V(x,X_s)ds$ et $V$ est de la forme $$V(x,y)=\sum_{k=1}^n a_k e_k(x)e_k(y),\; a_k\neq 0.$$ Lorsque tous les coefficients $a_k$ sont positifs, la particule tend à fuir les zones les plus visitées (cas répulsif) et lorsque les coefficients sont négatifs, elle est attirée par les zones les plus visitées (cas attractif). Dans le cas attractif, nous considérons le cas particulier où $X$ vit sur la sphère $\mathbb{S}^{n-1}$ et $e_{k}(x)=x_{k+1}$, nous prouvons que la particule converge presque-sûrement. Lorsque $nSelf-Interacting Diffusions on compact Riemannian manifolds solving formally the stochastic differential equation $$dX_t =dB_t(X_t)- \nabla V_t(X_t)dt, $$ with $V_t(x)= \int_0^t V(x,X_s)ds$ and $$V(x,y)=\sum_{k=1}^n a_k e_k(x)e_k(y),\; a_k\neq 0.$$ When all coefficients are positive, the particle tends to go as far as possible from its past trajectory (repelling case) whereas the negativeness of the coefficients implies that it is attracted (attracting case). In the attractive case, we consider the particular case where $X$ lives on the sphere $\mathbb{S}^{n-1}$ and $e_{k}(x)=x_{k+1}$ and prove its almost-sure convergence. When $n<\infty$ and $e_1,\cdots,e_n$ are eigenfunctions of the Laplacian, we prove in the repelling case that the natural extension of the process to a Markovian one admits a unique explicit invariant probability measure. We also provide rates of convergence. On the circle $\mathbb{S}^1$, we extend the result of the existence of an invariant probability to the case $n=\infty$. Finally, on the circle and when $n=1$, we provide a condition on the function $e_1$ to ensure the existence of an unique invariant probability measure for the extended process and an other condition ensuring the almost-sure convergence of the particle. This thesis is built upon four works