摘要翻译：
本文研究了存在于$\MathBB{R}^D$上的一些自相互作用扩散$(X_t,t\geq0)$。这些扩散是随机微分方程的解:\[\mathrm{d}x_t=\mathrm{d}B_t-g(t)\nabla V(x_t-\bar{\mu}t)\\mathrm{d}t\\\mathrm{d}t,\\\bar{\mu}t$是过程的经验平均值，$V$是渐近严格凸势，$g$是给定函数。我们研究了$X$的遍历性，证明了它与$G$强相关。实际上，我们证明$x$是遍历的（在极限商意义下）当且仅当$\bar{\mu}_t$收敛于a.s。我们也给出了$x$几乎肯定收敛的一些条件（在$g$和$v$上）。
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英文标题：
《Some particular self-interacting diffusions: Ergodic behaviour and
  almost sure convergence》
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作者：
S\'ebastien Chambeu, Aline Kurtzmann
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最新提交年份：
2012
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分类信息：

一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Mathematics        数学
二级分类：Statistics Theory        统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Statistics        统计学
二级分类：Statistics Theory        统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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英文摘要：
  This paper deals with some self-interacting diffusions $(X_t,t\geq 0)$ living on $\mathbb{R}^d$. These diffusions are solutions to stochastic differential equations: \[\mathrm{d}X_t=\mathrm{d}B_t-g(t)\nabla V(X_t-\bar{\mu}_t)\,\mathrm{d}t,\] where $\bar{\mu}_t$ is the empirical mean of the process $X$, $V$ is an asymptotically strictly convex potential and $g$ is a given function. We study the ergodic behaviour of $X$ and prove that it is strongly related to $g$. Actually, we show that $X$ is ergodic (in the limit quotient sense) if and only if $\bar{\mu}_t$ converges a.s. We also give some conditions (on $g$ and $V$) for the almost sure convergence of $X$.
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PDF链接：
https://arxiv.org/pdf/707.2908