The first chapter is concerned with some self-interacting diffusions $(X_t,t\geq 0)$
living on $\mathbb{R}^d$. These diffusions are solutions to stochastic differential equations:
$$\mathrm{d}X_t = \mathrm{d}B_t - g(t)\nabla V(X_t -\overline{\mu}_t) \mathrm{d}t,$$ where $\overline{\mu}_t$ is the empirical mean of the process $X$, $V$ is an asymptotically strictly convex potential and $g$ is a given function. We study the ergodic behavior of $X$ and prove that it is strongly related to $g$. Actually, $X$ and $\overline{\mu}_t$ have the same asymptotic behavior and we will give necessary and sufficient conditions (on $g$ and $V$) for the almost sure convergence of $X$. In chapter 2, we finish the previous study. We have still studied the ergodic behavior of $X$ and proved that it is strongly
related to $g$. We go further and give necessary and sufficient conditions (for small $g$'s) in order that $X$ converges in law to $X_\infty$ (which is related to the global minima of $V$).
In the second part, we begin to situate our study in Chapter 3. Self-interacting diffusions are solutions to SDEs with a drift term depending on the process and its normalized occupation measure $\mu_t$ (via an interaction potential $V$ and a confinement potential $W$):
$$\mathrm{d}X_t = \mathrm{d}B_t -\left( \nabla V(X_t)+\frac{1}{t} \int_0^t \nabla_x W(X_t,X_s) \mathrm{d}s \right) \mathrm{d}t \\
\mathrm{d}\mu_t = (\delta_{X_t} - \mu_t)\frac{\mathrm{d}t}{r+t}\\
X_0 = x, \mu_0=\mu. $$
We establish a relation between the asymptotic behavior of $\mu_t$ and the asymptotic behavior of a deterministic dynamical flow (defined on the space of the Borel probability measures). We extend previous results on $\mathbb{R}^d$ or more generally a smooth complete connected Riemannian manifold without boundary. We will also give some sufficient conditions for the convergence of $\mu_t$. We then illustrate, in Chapter 5, the previous study of self-interacting diffusions living in $\mathbb{R}^d$ with some examples in the two-dimensional case. The
preceding chapter contains abstract results, and therefore we describe here a simple example and illustrate some of our previous results. We will show in particular that, depending on $W$, either the empirical measure behaves like the ``Brownian motion" (constructed with respect to the measure $e^{V(x)} \mathrm{d}x$); or the empirical occupation measure converges almost surely to a probability measure, which is approximatively a Gaussian distribution ; or there is enough attraction, and then the term induced by $W$ forces $\mu_t$ to circle around and the limit set of $(\mu_t)$ is a circle of measures $\{\nu(\delta), 0\leq\delta<2\pi\}$. |
