论文标题
Lévy噪声的广义选择问题
Generalized selection problem with Lévy noise
论文作者
论文摘要
令$ a_ \ pm> 0 $，$β\ in（0,1）$，然后让$ z^{（α）} $是一个严格的$α$α$ - 稳定的lévy流程，具有跳跃度量$ν（\ mathrm {d} z）=（c _++++\+\ \ \ \ \ \ \ \ \ \ \ \ {i} _ {i}（c） C _- \ Mathbb {i} _ {（ - \ infty，0）}（z））| z |^{ - 1-α} \，\ Mathrm {d} z $，$α\ in（1,2）$，$ C_ \ pm pm \ geq 0 $，$ c _+c _+c _++c c _++c c _--> 0 $ 0 $ 0 $ 0。模型随机微分方程的选择问题$ \ mathrm {d} \ bar x^\ varepsilon =（a _+\ \ \ \ \ \ \ \ \ \ \ \ \ \ {i} _ {[0，\ infty）}（\ bar x^\ varepsilon） - x^\ varepsilon）））| \ bar x^\ varepsilon |^β\，\ mathrm {d} t +\ varepsilon \ varepsilon \ mathrm {d} z^{（α）} $在较小的噪声限制$ \ varepsilon \ \ v varepsilon \ \ bar bar x^$ crince $ crince $ crense $ crinc.极限的非lipschitzian普通微分方程$ \ mathrm {d} \ bar x =（a _+\ \ \ \ \ \ \ \ \ \ \ \ \ {i} _ {[0，\ infty）}（\ bar x） - a _-- \ mathb {i} _ {i} _ {（\ bar） x |^β\，\ mathrm {d} t $带有概率$ \ bar p_ \ pm = \ bar p_ \ pm（α，c _+/c _-，β，a _+/a _--）$，请参阅[pilipenko and proske，stat。概率。 Lett。，132：62-73，2018]。 In this paper we solve the generalized selection problem for the stochastic differential equation $\mathrm{d} X^\varepsilon=a(X^\varepsilon)\,\mathrm{d} t+\varepsilon b(X^\varepsilon)\,\mathrm{d} Z$ whose dynamics in the vicinity of the origin in certain sense提醒模型方程的动力学。特别是我们表明，解决方案$ x^\ varepsilon $还收敛到极限不规则的不规则的普通微分方程$ \ mathrm {d} x = a（x）\，\ mathrm {d} t $，具有相同的模型选择概率$ \ bar p_ p_ p _ p _ \ pm $。这意味着，对于一大批不规则的随机微分方程，选择动力学完全由漂移的四个局部参数和跳跃度量确定。
Let $A_\pm>0$, $β\in(0,1)$, and let $Z^{(α)}$ be a strictly $α$-stable Lévy process with the jump measure $ν(\mathrm{d} z)=(C_+\mathbb{I}_{(0,\infty)}(z)+ C_-\mathbb{I}_{(-\infty,0)}(z))|z|^{-1-α}\,\mathrm{d} z$, $α\in (1,2)$, $C_\pm\geq 0$, $C_++C_->0$. The selection problem for the model stochastic differential equation $\mathrm{d} \bar X^\varepsilon=(A_+\mathbb{I}_{[0,\infty)}(\bar X^\varepsilon) - A_-\mathbb{I}_{(-\infty,0)}(\bar X^\varepsilon))|\bar X^\varepsilon|^β\,\mathrm{d} t +\varepsilon \mathrm{d} Z^{(α)}$ states that in the small noise limit $\varepsilon\to 0$, solutions $\bar X^\varepsilon$ converge weakly to the maximal or minimal solutions of the limiting non-Lipschitzian ordinary differential equation $\mathrm{d} \bar x=(A_+\mathbb{I}_{[0,\infty)}(\bar x)- A_-\mathbb{I}_{(\infty,0)}(\bar x))|\bar x|^β\,\mathrm{d} t$ with probabilities $\bar p_\pm=\bar p_\pm(α,C_+/C_-,β, A_+/A_-)$, see [Pilipenko and Proske, Stat. Probab. Lett., 132:62-73, 2018]. In this paper we solve the generalized selection problem for the stochastic differential equation $\mathrm{d} X^\varepsilon=a(X^\varepsilon)\,\mathrm{d} t+\varepsilon b(X^\varepsilon)\,\mathrm{d} Z$ whose dynamics in the vicinity of the origin in certain sense reminds of dynamics of the model equation. In particular we show that solutions $X^\varepsilon$ also converge to the maximal or minimal solutions of the limiting irregular ordinary differential equation $\mathrm{d} x=a(x) \,\mathrm{d} t$ with the same model selection probabilities $\bar p_\pm$. This means that for a large class of irregular stochastic differential equations, the selection dynamics is completely determined by four local parameters of the drift and the jump measure.
