论文标题
过度阻尼Langevin Dynamics的Eyring-Kramer退出率：边界上有鞍点的情况
Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary
论文作者
论文摘要
令$（x_t）_ {t \ ge 0} $为换档Langevin Dynamics的随机过程解决方案 $$ dx_t = - \ nabla f（x_t）\，dt +\ sqrt h \，db_t $$，让$ω\ subset \ subset \ subset \ mathbb r^d $是吸引$ f的吸引力的基础，$ f：\ mathbb r^d \ to \ to \ to \ mathbb r $。最多可以通过$（x_t）_ {x_t）_ {t \ ge 0} $的出口率从$ fules $ f $ on $ \ \poartialΩ$上的每个鞍点来参数，这是由庆祝的Eyring-eyring-kramers Laws参数来限制$ f $ on $ f $ oon $ f $ oon $ω$的$（x_t）_ {t \ ge 0} $的出口，在$（x_t）_ {t \ ge 0} $中。该结果为跳跃Markov模型提供了牢固的数学基础，用于建模分子系统的演化以及一些数值方法，这些方法使用这些基础跳跃马尔可夫模型有效地样本了过度阻尼的Langevin动力学的可稳定轨迹。
Let $(X_t)_{t\ge 0}$ be the stochastic process solution to the overdamped Langevin dynamics $$dX_t=-\nabla f(X_t) \, dt +\sqrt h \, dB_t$$ and let $Ω\subset \mathbb R^d $ be the basin of attraction of a local minimum of $f: \mathbb R^d \to \mathbb R$. Up to a small perturbation of $Ω$ to make it smooth, we prove that the exit rates of $(X_t)_{t\ge 0}$ from $Ω$ through each of the saddle points of $f$ on $\partial Ω$ can be parametrized by the celebrated Eyring-Kramers laws, in the limit $h \to 0$. This result provides firm mathematical grounds to jump Markov models which are used to model the evolution of molecular systems, as well as to some numerical methods which use these underlying jump Markov models to efficiently sample metastable trajectories of the overdamped Langevin dynamics.
