Convergence in distribution of some particular self-interacting diffusions: the simulated annealing method

2007/07/19 by Sebastien Chambeu, Chambeu, Sebastien, Aline Kurtzmann +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.0707.2910

companion paper to 0707.2908

arxiv created 2008/12/04 · arxiv updated 2009/12/01

Abstract

The present paper is concerned with some self-interacting diffusions (X_t,t≥ 0) living on ℝ^d. These diffusions are solutions to stochastic differential equations: dX_t = dB_t - g(t)∇ V(X_t - μ_t) dt where μ_t is the empirical mean of the process X, V is an asymptotically strictly convex potential and g is a given function. The authors have still studied the ergodic behavior of X and proved that it is strongly related to g. We go further and give necessary and sufficient conditions (for small g's) in order that X converges in probability to X_∞ (which is related to the global minima of V).

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