论文标题
限制langevin动力学不变度的行为
Limit behavior of the invariant measure for Langevin dynamics
论文作者
论文摘要
在此手稿中，我们考虑$ \ mathbb {r}^d $上的langevin Dynamics具有过度抑制的矢量场，并由小振幅$ \sqrtε$，$ε> 0 $驱动。在矢量场和扩散系数上的合适假设下，众所周知，它具有独特的不变概率度量$μ^ε$。由于$ε$趋于零，我们证明了概率度量$ε^{d/2}μ^ε（\sqrtε\ mathrm {d} x）$方程式。此外，估计错误项。我们强调，一般没有针对$μ^ε$的明确公式。
In this manuscript, we consider the Langevin dynamics on $\mathbb{R}^d$ with an overdamped vector field and driven by multiplicative Brownian noise of small amplitude $\sqrtε$, $ε>0$. Under suitable assumptions on the vector field and the diffusion coefficient, it is well-known that it possesses a unique invariant probability measure $μ^ε$. As $ε$ tends to zero, we prove that the probability measure $ε^{d/2} μ^ε(\sqrtε\mathrm{d} x)$ converges in the $p$-Wasserstein distance for $p\in [1,2]$ to a Gaussian measure with zero-mean vector and non-degenerate covariance matrix which solves a Lyapunov matrix equation. Moreover, the error term is estimated. We emphasize that generically no explicit formula for $μ^ε$ can be found.
