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Olivier Raimond

Publications and source records attributed to Olivier Raimond.

23 records · Page 2Linked to original sources
arXiv2008

A class of non homogeneous self interacting random processes with applications to learning in games and vertex-reinforced random walks

Using an approximation by a set-valued dynamical system, this paper studies a class of non Markovian and non homogeneous stochastic processes on a finite state space. It provides an unified approach to simulated annealing type processes. It permits to study new models of vertex reinforced random walks and new models of learning in games including Markovian fictitious play.

math.PR↗
arXiv2003

Self-Interacting Diffusions : Symmetric Interactions

Let $M$ be a compact Riemannian manifold. A {\em self-interacting diffusion} on $M$ is a stochastic process solution to $$dX_t = dW_t(X_t) - \frac{1}{t}(\int_0^t \nabla V_{X_s}(X_t)ds)dt$$ where $\{W_t\}$ is a Brownian vector field on $M$ and $V_x(y) = V(x,y)$ a smooth function. Let $μ_t = \frac{1}{t} \int_0^t δ_{X_s} ds$ denote the normalized occupation measure of $X_t$. We prove that, when $V$ is symmetric, $μ_t$ converges almost surely to the critical set of a certain nonlinear free energy functional $J$. Furthermore, $J$ has generically finitely many critical points and $μ_t$ converges almost surely toward a local minimum of $J.$ Each local minimum having a positive probability to be selected.

math.PR↗
arXiv2002

Sticky flows on the circle

The purpose of this note is to give an example of stochastic flows of kernels, which naturally interpolates between the Arratia coalescing flow associated with systems of coalescing independent Brownian particles on the circle and the deterministic diffusion flow (actually, the results are given in the slightly more general framework of symmetric Levy processes for which points are not polar). The construction is performed using Dirichlet form theory and the extension of De Finetti's theorem given in the previous work by the authors. The sticky flows of kernels are associated with systems of sticky independent Levy particles on the circle, for some fixed parameter of stickyness. Some elementary asymptotic properties of the flow are also given.

math.PR↗
arXiv2001

Integration of Brownian vector fields

Using the Wiener chaos decomposition, we show that strong solutions of non Lipschitzian S.D.E.'s are given by random Markovian kernels. The example of Sobolev flows is studied in some detail, exhibiting interesting phase transitions.

math.PR↗