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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| Format: | Article |
| Language: | English |
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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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| author | Aline Kurtzmann |
| author_facet | Aline Kurtzmann |
| author_sort | Aline Kurtzmann |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-33f2005fbc8c41af96d8aa952cbd7b65 |
| institution | Kabale University |
| issn | 1687-952X 1687-9538 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Probability and Statistics |
| spelling | doaj-art-33f2005fbc8c41af96d8aa952cbd7b652025-02-03T00:59:22ZengWileyJournal of Probability and Statistics1687-952X1687-95382014-01-01201410.1155/2014/364321364321Convergence in Distribution of Some Self-Interacting DiffusionsAline Kurtzmann0Université de Lorraine, Institut Elie Cartan Lorraine, UMR 7502 CNRS, 54506 Vandœuvre-lès-Nancy, FranceThe 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.http://dx.doi.org/10.1155/2014/364321 |
| spellingShingle | Aline Kurtzmann Convergence in Distribution of Some Self-Interacting Diffusions Journal of Probability and Statistics |
| title | Convergence in Distribution of Some Self-Interacting Diffusions |
| title_full | Convergence in Distribution of Some Self-Interacting Diffusions |
| title_fullStr | Convergence in Distribution of Some Self-Interacting Diffusions |
| title_full_unstemmed | Convergence in Distribution of Some Self-Interacting Diffusions |
| title_short | Convergence in Distribution of Some Self-Interacting Diffusions |
| title_sort | convergence in distribution of some self interacting diffusions |
| url | http://dx.doi.org/10.1155/2014/364321 |
| work_keys_str_mv | AT alinekurtzmann convergenceindistributionofsomeselfinteractingdiffusions |