Convergence in Distribution of Some Self-Interacting Diffusions

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: d X t = d B t - g ( t ) ∇ V ( X t - μ ¯ t ) d t , where μ ¯ t is the empirical mean of the process X , V is an asymptotically stri...

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Bibliographic Details
Published in:Journal of probability and statistics 2014-01, Vol.2014 (2014), p.1-13
Main Author: Kurtzmann, Aline
Format: Article
Language:English
Subjects:
Asymptotic properties
Convergence
Differential equations
Diffusion
Empirical equations
Heuristic
Markov analysis
Minima
Statistics
Stochasticity
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Summary: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: d X t = d B t - g ( t ) ∇ V ( X t - μ ¯ t ) d t , where μ ¯ t is the empirical mean of the process X , V is an asymptotically strictly convex potential, and g is a given positive function. We study the asymptotic behaviour of X for three different families of functions g . If g t = k log ⁡ t with k small enough, then the process X converges in distribution towards the global minima of V , whereas if t g ( t ) → c ∈ ] 0 , + ∞ ] or if g ( t ) → g ( ∞ ) ∈ [ 0 , + ∞ [ , then X converges in distribution if and only if ∫ x e - 2 V ( x ) d x = 0 .
ISSN:1687-952X
1687-9538
DOI:10.1155/2014/364321