Convergence in Distribution of Some Self-Interacting Diffusions
The present paper is concerned with some self-interacting diffusions (Xt,t≥0) living on ℝd. These diffusions are solutions to stochastic differential equations: dXt=dBt-g(t)∇V(Xt-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 gi...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2014-01-01
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| Series: | Journal of Probability and Statistics |
| Online Access: | http://dx.doi.org/10.1155/2014/364321 |
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| Summary: | The present paper is concerned with some self-interacting diffusions (Xt,t≥0) living on ℝd. These diffusions are solutions to stochastic differential equations: dXt=dBt-g(t)∇V(Xt-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 positive function. We study the asymptotic behaviour of X for three different families of functions g. If gt=klogt with k small enough, then the process X converges in distribution towards the global minima of V, whereas if tg(t)→c∈]0,+∞] or if g(t)→g(∞)∈[0,+∞[, then X converges in distribution if and only if∫xe-2V(x) dx=0. |
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| ISSN: | 1687-952X 1687-9538 |