Homogenization of biased convolution type operators

Article type: Research Article

Authors: Piatnitski, A.^{a; b; *} | Zhizhina, E.^b

Affiliations: [a] The Arctic University of Norway, Campus Narvik, P.O. Box 385, Narvik 8505, Norway | [b] Institute for Information Transmission Problems of RAS, 19, Bolshoi Karetnyi per. build. 1, 127051 Moscow, Russia. E-mails: apiatnitski@gmail.com, ejj@iitp.ru

Correspondence: [*] Corresponding author. E-mail: apiatnitski@gmail.com.

Abstract: This paper deals with homogenization of parabolic problems for integral convolution type operators with a non-symmetric jump kernel in a periodic elliptic medium. It is shown that the homogenization result holds in moving coordinates. We determine the corresponding effective velocity and prove that the limit operator is a second order parabolic operator with constant coefficients. We also consider the behaviour of the effective velocity in the case of small antisymmetric perturbations of a symmetric kernel, in particular we show that the Einstein relation holds for the studied periodic environment.

Keywords: Homogenization in moving coordinates, periodic medium, non-local operator, non-symmetric convolution kernel

DOI: 10.3233/ASY-191533

Journal: Asymptotic Analysis, vol. 115, no. 3-4, pp. 241-262, 2019