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Strongly reinforced vertex-reinforced-random-walk on the complete graph
Self-interacting diffusions II: Convergence in law
This paper concerns convergence in law properties of self-interacting diffusions on a compact Riemannian manifold. (C) 2003 Editions scientifiques et medicales Elsevier SAS.
Simulated Annealing, Vertex-Reinforced Random Walks and Learning in Games
A Bakry-Emery criterion for self-interacting diffusions
We give a Bakry-Emery type criterion for self-interacting diffusions on a compact manifold.
Self-interacting diffusions
This paper is concerned with a general class of self-interacting diffusions [X-t}(tgreater than or equal to0) living oil a compact Riemannian manifold M. These are solutions to stochastic differential equations of the form : dX(t) = Brownian increments + drift term depending on X-t and mu(t), the normalized occupation measure of the process. It is proved that the asymptotic behavior of {mu(t)} can be precisely related to the asymptotic behavior of a deterministic dynamical semi-flow Phi = {Phi(t)}(tgreater than or equal to0) defined on the space of the Borel probability measures on M. In particular, the limit sets of {mu(t)} are proved to be almost surely attractor free sets for Phi. These results are applied to several examples of self-attracting/repelling diffusions on the n-sphere. For instance, in the case of self-attracting diffusions, our results apply to prove that {mu(t)} can either converge toward the normalized Riemannian measure, or to a gaussian measure, depending on the value of a parameter measuring the strength of the attraction.
A class of self-interacting processes with applications to games and reinforced random walks
Self-interacting diffusions. III. Symmetric interactions
Let M be a compact Riemannian manifold. A self-interacting diffusion on M is a stochastic process solution to where {W-t} is a Brownian vector field on M and V-x(y) = V(x, y) a smooth function. Let mu(t) = 1/t integral(0)(t) delta X-s ds denote the normalized occupation measure of X-t. We prove that, when V is symmetric, mu(t) converges almost surely to the critical set of a certain nonlinear free energy functional J. Furthermore, J has generically finitely many critical points and mu(t) converges almost surely toward a local minimum of J. Each local minimum has a positive probability to be selected.
Onself attracting/repelling diffusions
We present an almost sure ergodic theorem for a class of self-interacting diffusions on a compact Riemannian manifold. (C) 2002 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.