非紧空间上自相互作用扩散问题的ODE方法
The ODE method for some self-interacting diffusions on non-compact spaces
摘要：
自相互作用扩散是具有漂移项的偏微分方程的解，漂移项取决于过程及其归一化占据测度$μ_t$（通过相互作用势和限制势）。我们建立了$μ_t$的渐近行为和一个确定性的动态流（定义在Borel概率测度的空间）的渐近行为之间的关系。我们推广了以前关于$\mathbb{R}^d$或更一般的光滑完备连通无边界黎曼流形的结果。我们还将给出$μ_t$收敛的一些充分条件。最后，我们将以$\mathbb{R}^2$为例说明我们的研究。
英文摘要：
Self-interacting diffusions are solutions to SDEs with a drift term depending on the process and its normalized occupation measure $μ_t$ (via an interaction potential and a confinement potential). We establish a relation between the asymptotic behavior of $μ_t$ and the asymptotic behavior of a deterministic dynamical flow (defined on the space of the Borel probability measures). We extend previous results on $\mathbb{R}^d$ or more generally a smooth complete connected Riemannian manifold without boundary. We will also give some sufficient conditions for the convergence of $μ_t$. Finally, we will illustrate our study with an example on $\mathbb{R}^2$.
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