Periodic homogenization of non-symmetric L\'evy-type processes

TL;DR

This paper investigates the homogenization of non-symmetric Lévy-type processes in periodic media, characterizing the limiting Lévy processes under various tail behaviors of the jump measure, extending classical results to jump-diffusion settings.

Contribution

It provides a comprehensive analysis of the homogenization for Lévy-type processes with singular measures, including explicit characterization of the limit processes for different tail indices.

Findings

Weak convergence to Lévy processes under proper scaling

Complete characterization of homogenized processes for various tail indices

Extension of classical homogenization results to jump processes

Abstract

In this paper, we study homogenization problem for strong Markov processes on $\R^d$ having infinitesimal generators $$ \sL f(x)=\int_{\R^d}\left(f(x+z)-f(x)-\langle \nabla f(x), z\rangle \I_{\{|z|\le 1\}} \right) k(x,z)\, \Pi (dz) +\langle b(x), \nabla f(x) \rangle, \quad f\in C^2_b (\R^d) $$ in periodic media, where $\Pi$ is a non-negative measure on $\R^d$ that does not charge the origin $0$, satisfies $\int_{\R^d} (1 \wedge |z|^2)\, \Pi (dz)<\infty$, 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 L\'evy 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 $z=(r, \theta) \in \R_+\times \bS^{d-1}$, $$ \I_{\{|z|>1\}}\,\Pi (dz) = \I_{\{ r>1\}} \varrho_0(d\theta) \, \frac{ dr }{r^{1+\alpha}} $$ with $\alpha\in (0,\infty)$ 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 $\alpha$; there are five cases: $\alpha \in (0, 1)$, $\alpha =1$, $\alpha \in (1, 2)$, $\alpha =2$ and $\alpha \in (2, \infty)$.