Let M be a compact connected oriented Riemannian manifold. The purpose of this paper is to investigate the long time behavior of a degenerate stochastic differential equation on the state space M × R n ; which is obtained via a natural change of variable from a self-repelling diffusion taking the formwhere {B t } is a Brownian vector field on M , σ > 0 and V x (y) = V (x, y) is a diagonal Mercer kernel. We prove that the induced semi-group enjoys the strong Feller property and has a unique invariant probability µ given as the product of the normalized Riemannian measure on M and a Gaussian measure on R n . We then prove an exponential decay to this invariant probability in L 2 (µ) and in total variation.

In the present work we study self-interacting diffusions following an infinite dimensional approach. First we prove existence and uniqueness of a solution with Markov property. Then we study the corresponding transition semigroup and, more precisely, we prove that it has Feller property and we give an explicit form of an invariant probability of the system.Comment: Version 2: Typos are corrected. Section 6 is reorganised in order to make it more transparent; the results are unchanged. The presentation of the proof of Proposition 3 is improved. Statement of Lemma 5 is rephrased. Version 3: Acknowledgement of financial support is added. Accepted for publication in "Stochastic Partial Differential Equations: Analysis and Computations

This paper proves almost-sure convergence for the self-attracting diffusion on the unit spherey is the usual scalar product in R n , and (Wt(.)) t 0 is a Brownian motion on S n . From this follows the almost-sure convergence of the real-valued self-attracting diffusion dϑt = σdWt + a t 0 sin(ϑt − ϑs)dsdt, where (Wt) t 0 is a real Brownian motion.keywords: reinforced process, self-interacting diffusions, asymptotic pseudotrajectories, rate of convergence.MSC 60K35, 60G17, 60J60

We prove a Central Limit Theorem for the finite dimensional distributions of the displacement for the 1D self-repelling diffusion which solveswhere B is a real valued standard Brownian motion and F (x) = n k=1 a k cos(kx) with n < ∞ and a 1 , · · · , a n > 0.In dimension d ≥ 3, such a result has already been established by Horváth, Tóth and Vetö in [3] in 2012 but not for d = 1, 2. Under an integrability condition, Tarrès, Tóth and Valkó conjectured in [6] that a Central Limit Theorem result should also hold in dimension d = 1.

Consider hard balls in a bounded rotating drum. If there is no gravitation, then there is no Fermi acceleration, i.e., the energy of the balls remains bounded forever. If there is gravitation, Fermi acceleration may arise. A number of explicit formulas for the system without gravitation are given. Some of these are based on an explicit realization, which we derive, of the well-known microcanonical ensemble measure.