Strongly self-interacting processes on the circle

Abstract

The purpose of this paper is to investigate the long time behaviour for a selfinteracting diffusion and a self-interacting velocity jump process. While the diffusion case has already been studied for some particular potential function, the second one, which belongs to the family of piecewise deterministic processes, is new.Depending on the underlying potential function's shape, we prove either the almost sure convergence or the recurrence for a natural extended process given by a change a variable.

Weakly self-interacting velocity jump processes for bacterial chemotaxis and adaptive algorithms

“…we can choose a neighbourhood N ε and a time t 0 such that for all 16) and (20) again). Taking the expectation in (24), the martingale increment vanishes, and together with (18) and the fact D 2 v,v η 0 for all v ∈ H 0 , we have obtained so far…”

“…Note that a strong self-interaction, for which by contrast the drift is a function of the nonnormalized occupation measure tµ t , such as studied in [27,5] for diffusions, is studied in the case of a velocity jump process in [18]. We are interested in the long-time behaviour of the process, and in particular in the question of the influence of the weak self-interaction on this long-time behaviour: if the process tends to go back to where it has already been, is the interaction sufficient to confine it in some localized place ?…”