论文标题
Langevin动力学的逐渐收敛性在退化电位上
Gradual convergence for Langevin dynamics on a degenerate potential
论文作者
论文摘要
在本文中，我们研究了一个普通的微分方程，该方程在原点上具有退化的全局吸引子，我们在该方程式中添加了一个用小参数调节其强度的白噪声。在一般条件下，对于任何固定强度，随着时间的时间趋于无穷大，这种随机动力学的解决方案在总变化距离上呈指数快速收敛到唯一的平衡分布。我们适当地加速了随机动态，并表明前面的收敛是逐渐的，即，与每个固定的$ t \ geq 0 $关联的功能，随时间$ t $及其平衡分布的加速随机动力学之间的总变化距离且其平衡分布会收敛，因为噪声强度趋于降低零值，以减少$（0,0,1）$（0,1,1）。此外，我们证明，每个固定$ t \ geq 0 $的限制函数对应于随机微分方程的边际，时间$ t $之间的总变化距离，该方程来自无穷大及其相应的平衡分布。这完成了上述随机动力学的时间边缘之间的总变化距离的所有可能行为的分类，以及其对一个维度良好的凸电势的不变度度量。此外，对于这种单参数的随机过程和混合时间的渐近分析，没有截止现象。
In this paper, we study an ordinary differential equation with a degenerate global attractor at the origin, to which we add a white noise with a small parameter that regulates its intensity. Under general conditions, for any fixed intensity, as time tends to infinity, the solution of this stochastic dynamics converges exponentially fast in total variation distance to a unique equilibrium distribution. We suitably accelerate the random dynamics and show that the preceding convergence is gradual, that is, the function that associates to each fixed $t\geq 0$ the total variation distance between the accelerated random dynamics at time $t$ and its equilibrium distribution converges, as the noise intensity tends to zero, to a decreasing function with values in $(0,1)$. Moreover, we prove that this limit function for each fixed $t \geq 0$ corresponds to the total variation distance between the marginal, at time $t$, of a stochastic differential equation that comes down from infinity and its corresponding equilibrium distribution. This completes the classification of all possible behaviors of the total variation distance between the time marginal of the aforementioned stochastic dynamics and its invariant measure for one dimensional well-behaved convex potentials. In addition, there is no cut-off phenomenon for this one-parameter family of random processes and asymptotics of the mixing times are derived.
