Periodic homogenization of nonsymmetric Lévy-type processes

Abstract

In this paper we study homogenization problem for strong Markov processes on ${\mathbb{R}}^{\mathit{d}}$ having infinitesimal generators

in periodic media, where Π is a nonnegative measure on ${\mathbb{R}}^{\mathit{d}}$ that does not charge the origin 0, satisfies ${\int }_{{\mathbb{R}}^{\mathit{d}}}(1\wedge |\mathit{z}{|}^2)\mathrm{\Pi }(\mathit{d}\mathit{z})<\infty$ and can be singular with respect to the Lebesgue measure on ${\mathbb{R}}^{\mathit{d}}$. Under a proper scaling we show the scaled processes converge weakly to Lévy processes on ${\mathbb{R}}^{\mathit{d}}$. The results are a counterpart of the celebrated work (Asymptotic Analysis for Periodic Structures (1978) North-Holland; Ann. Probab. 13 (1985) 385–396) in the jump-diffusion setting. In particular, we completely characterize the homogenized limiting processes when $\mathit{b}(\mathit{x})$ is a bounded continuous multivariate 1-periodic ${\mathbb{R}}^{\mathit{d}}$-valued function, $\mathit{k}(\mathit{x},\mathit{z})$ is a nonnegative bounded continuous function that is multivariate 1-periodic in both x and z variables and, in spherical coordinate $\mathit{z}=(\mathit{r},\mathit{\theta })\in {\mathbb{R}}_{+}\times {\mathbb{S}}^{\mathit{d}-1}$,

$${\mathbb{1}}_{\{|\mathit{z}|>1\}}\mathrm{\Pi }(\mathit{d}\mathit{z})={\mathbb{1}}_{\{\mathit{r}>1\}}{\mathit{\varrho }}_0(\mathit{d}\mathit{\theta })\frac{\mathit{d}\mathit{r}}{{\mathit{r}}^{1+\mathit{\alpha }}}$$

with $\mathit{\alpha }\in (0,\infty )$ and ${\mathit{\varrho }}_0$ being any finite measure on the unit sphere ${\mathbb{S}}^{\mathit{d}-1}$ in ${\mathbb{R}}^{\mathit{d}}$. Different phenomena occur depending on the values of α; there are five cases: $\mathit{\alpha }\in (0,1)$, $\mathit{\alpha }=1$, $\mathit{\alpha }\in (1,2)$, $\mathit{\alpha }=2$ and $\mathit{\alpha }\in (2,\infty )$.