论文标题
非对称Lévy型工艺的周期性均质化
Periodic homogenization of non-symmetric Lévy-type processes
论文作者
论文摘要
在本文中，我们研究了具有无限发电机的$ \ r^d $的强马尔可夫流程的均质化问题 $$ \ sl f（x）= \ int _ {\ r^d} \ left（f（x+z）-f（x） π（dz） +\ langle b（x），\ nabla f（x）\ rangle，\ quad f \ in c^2_b（\ r^d） $$在定期媒体中，其中$π$是$ \ r^d $的一种非负措施，不收取原始$ 0 $的费用，满足$ \ int _ {\ r^d}（1 \ wedge | z | |^2）\，π（dz）<\ infty $，并且可以对lebesgue y $ $^d d^d^r^d^r^d^r^d^d。 在适当的缩放下，我们在$ \ r^d $上薄弱地显示了缩放过程与莱维过程。结果是著名作品\ cite {blp，bh}的对应物。 In particular, we completely characterize the homogenized limiting processes when $b(x) $ is a bounded continuous multivariate 1-periodic $\R^d$-valued function, $k(x,z)$ is a non-negative bounded continuous function that is multivariate 1-periodic in both $x$ and $z$ variables, and, in spherical coordinate $z=(r, θ) \in \ r _+\ times \ bs^{d-1} $，$$ \ i _ {\ {| z | |> 1 \}}} \，π（dz）= \ i _ {\ i {\ {\ {r> 1 \}}} $α\ in（0，\ infty）$和$ \ varrho_0 $是单位球体上的任何有限度量$ \ bs^{d-1} $中的$ \ r^d $。不同的现象取决于$α$的值；有五种情况： $α\ in（0，1）$，$α= 1 $，$α\ in（1，2）$，$α= 2 $和$α\ in（2，\ infty）$。
In this paper, we study homogenization problem for strong Markov processes on $\R^d$ having infinitesimal generators $$ \sL f(x)=\int_{\R^d}\left(f(x+z)-f(x)-\langle \nabla f(x), z\rangle \I_{\{|z|\le 1\}} \right) k(x,z)\, Π(dz) +\langle b(x), \nabla f(x) \rangle, \quad f\in C^2_b (\R^d) $$ in periodic media, where $Π$ is a non-negative measure on $\R^d$ that does not charge the origin $0$, satisfies $\int_{\R^d} (1 \wedge |z|^2)\, Π(dz)<\infty$, and can be singular with respect to the Lebesgue measure on $\R^d$. Under a proper scaling, we show the scaled processes converge weakly to Lévy processes on $\R^d$. The results are a counterpart of the celebrated work \cite{BLP,Bh} in the jump-diffusion setting. In particular, we completely characterize the homogenized limiting processes when $b(x) $ is a bounded continuous multivariate 1-periodic $\R^d$-valued function, $k(x,z)$ is a non-negative bounded continuous function that is multivariate 1-periodic in both $x$ and $z$ variables, and, in spherical coordinate $z=(r, θ) \in \R_+\times \bS^{d-1}$, $$ \I_{\{|z|>1\}}\,Π(dz) = \I_{\{ r>1\}} \varrho_0(dθ) \, \frac{ dr }{r^{1+α}} $$ with $α\in (0,\infty)$ and $\varrho_0$ being any finite measure on the unit sphere $\bS^{d-1}$ in $\R^d$. Different phenomena occur depending on the values of $α$; there are five cases: $α\in (0, 1)$, $α=1$, $α\in (1, 2)$, $α=2$ and $α\in (2, \infty)$.
