15:45
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 and a confinement potential): $$\mathrm{d}X_t = \mathrm{d}B_t -\left(
\nabla V(X_t)+ \nabla W*{\mu_t}(X_t) \right) \mathrm{d}t ; \mathrm{d}\mu_t = (\delta_{X_t}
- \mu_t)\frac{\mathrm{d}t}{r+t}; X_0 = x,\,\ \mu_0=\mu$$ where $(\mu_t)$ is the
process defined by $$\mu_t := \frac{r\mu + \int_0^t \delta_{X_s}\mathrm{d}s}{r+t}.$$
We establish a relation between the asymptotic behaviour of $\mu_t$ and the
asymptotic behaviour of a deterministic dynamical flow (defined on the space of
the Borel probability measures). We will also give some sufficient conditions
for the convergence of $\mu_t$. Finally, we will illustrate our study with an
example in the case $d=2$.