论文标题
在数值上渐近地保存langevin方程的大偏差原则
Numerically asymptotical preservation of the large deviations principles for invariant measures of Langevin equations
论文作者
论文摘要
在本文中，我们专注于兰格文方程及其数值方法的两种大偏差原理（LDP），因为噪声强度$ε\ to 0 $和耗散强度$ν\ to \ infty $。首先，通过证明较弱的开发率和指数紧密度，我们得出的结论是，精确溶液的不变度度量$ \ {μ_{μ__{ν，ε} \} $分别满足了LDP的满足，分别满足了$ε\ to 0 $和$ν\ to \ infty $。然后，我们研究是否存在数值方法，从$ \ {μ_{ν，ε} \} $中的这两个LDP中保存，这是因为数值方法的不变率函数的函数函数函数函数的函数可点击point Compliant pointsise的速度趋向于$ \ \ {μ___{n维} $ ZERE的速度倾向。答案对线性langevin方程是阳性。对于小噪声案例，我们表明，一大类数值方法可以渐近地保留$ \ {μ_{ν，ε} \} _ {ε> 0} $的LDP为$ε\ to0 $。对于强耗散案例，我们研究随机$θ$ -Method（$θ\ in [1/2,1] $），并表明只有中点方案（$θ= 1/2 $）才能渐近地保留$ \ {μ__{c {n，ε}，ε} \} $ ndp的LDP，这些结果表明，在线性情况下，卢比为$ε\ to0 $，而LDP作为$ν\ to \ to \ infty $用于不变方法的量度具有内在差异：常见的数值方法可以分散地保留$ \ \ {n c} $ \ {n c} $ {n c} n $ \ f. $ \ {μ_{ν，ε} \} _ {ν> 0} $作为$ \ {μ_{ν> 0} $的数字方法的渐近保存$ε\ to0 $，$ \ {n {v {μ_{ν，ε} \} _ {c} _ {ν> 0} $ as $ν\ to \ infty $取决于数值方法的选择。据我们所知，这是研究随机微分方程的不变度度量与其数值方法的不变性度量之间的关系的第一个结果。
In this paper, we focus on two kinds of large deviations principles (LDPs) of the invariant measures of Langevin equations and their numerical methods, as the noise intensity $ε\to 0$ and the dissipation intensity $ν\to\infty$ respectively. First, by proving the weak LDP and the exponential tightness, we conclude that the invariant measure $\{μ_{ν,ε}\}$ of the exact solution satisfies the LDPs as $ε\to0$ and $ν\to\infty$ respectively. Then, we study whether there exist numerical methods asymptotically preserving these two LDPs of $\{μ_{ν,ε}\}$ in the sense that the rate functions of invariant measures of numerical methods converge pointwise to the rate function of $\{μ_{ν,ε}\}$ as the step-size tends to zero. The answer is positive for the linear Langevin equation. For the small noise case, we show that a large class of numerical methods can asymptotically preserve the LDP of $\{μ_{ν,ε}\}_{ε>0}$ as $ε\to0$. For the strong dissipation case, we study the stochastic $θ$-method ($θ\in[1/2,1]$) and show that only the midpoint scheme ($θ=1/2$) can asymptotically preserve the LDP of $\{μ_{ν,ε}\}_{ν>0}$ as $ν\to\infty$. These results indicate that in the linear case, the LDP as $ε\to0$ and the LDP as $ν\to\infty$ for the invariant measures of numerical methods have intrinsic differences: the common numerical methods can asymptotically preserve the LDP of $\{μ_{ν,ε}\}_{ε>0}$ as $ε\to0$ while the asymptotical preservation of numerical methods for the LDP of $\{μ_{ν,ε}\}_{ν>0}$ as $ν\to\infty$ depends on the choice of numerical methods. To the best of our knowledge, this is the first result of investigating the relationship between the LDPs of invariant measures of stochastic differential equations and those of their numerical methods.
