非对称Lévy型过程的周期均匀化
Periodic homogenization of non-symmetric Lévy-type processes
摘要：
本文研究周期介质中具有无穷小生成元$$ \sL f(x)=\int_{\R^d}\left(f(x+z)-f(x)-\langle \nabla f(x), z\rangle \I_{\{|z|\le 1\}} \right) k(x,z)\, Π(dz) +\langle b(x), \nabla f(x) \rangle, \quad f\in C^2_b (\R^d) $$的$\R^d$上的强Markov过程的均匀化问题，其中$Π$是$\R^d$上的一个非负测度，它不对源$0$充电，满足$\int_{\R^d} (1 \wedge |z|^2)\, Π(dz)<\infty$，并且关于$\R^d$上的Lebesgue测度是奇异的.在适当的标度下，证明了$\R^d$上的标度过程弱收敛于Lévy过程.所得结果与著名的工作\cite{BLP,Bh}在跳跃扩散条件下的结果一致。特别是，我们完全刻画了均匀化的极限过程时，$b(x) $是一个有界的连续多元1-周期$\R^d$值函数，$k(x,z)$是一个非负的有界连续函数，是多元1-周期的$x$和$z$变量，并在球坐标$z=(r, θ) \in \R_+\times \bS^{d-1}$，$$ \I_{\{|z|>1\}}\,Π(dz) = \I_{\{ r>1\}} \varrho_0(dθ) \, \frac{ dr }{r^{1+α}} $$与$α\in (0,\infty)$和$\varrho_0$是任何有限的单位球$\bS^{d-1}$在$\R^d$。根据$α$的值，会出现不同的现象;有五种情况：$α\in (0, 1)$、$α=1$、$α\in (1, 2)$、$α=2$和$α\in (2, \infty)$。
英文摘要：
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 $α\in (2, \infty)$.
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