Abstract
The paper deals with a homogenization problem for a nonlocal linear operator with a kernel of convolution type in a medium with a periodic structure. We consider the natural diffusive scaling of this operator and study the limit behavior of the rescaled operators as the scaling parameter tends to 0. More precisely we show that in the topology of resolvent convergence the family of rescaled operators converges to a second order elliptic operator with constant coefficients. We also prove the convergence of the corresponding semigroups both in $L^2$ space and the space of continuous functions and show that for the related family of Markov processes the invariance principle holds.
Highlights
Recent time there is an increasing interest to the integral operators with a kernel of convolution type. These operators appear in many applications, such as models of population dynamics and the continuous contact model, where they describe the evolution of the density of a population, see for instance [3, 4, 6, 9, 10] for the details
We focus in this paper on the spacial inhomogeneous dispersal kernel depending both on the displacement y − x, and on the starting and the ending positions x, y ∈ Rd
Some of the properties of convolution type operators are similar to those of second order elliptic differential operators, there are essential differences, for example, in the form of the fundamental solution for the corresponding nonlocal parabolic equation. In this connection it is interesting to understand which asymptotic properties of differential operators are inherited by nonlocal convolution type operators, and which are not
Summary
Recent time there is an increasing interest to the integral operators with a kernel of convolution type. Some of the properties of convolution type operators are similar to those of second order elliptic differential operators, there are essential differences, for example, in the form of the fundamental solution for the corresponding nonlocal parabolic equation. In the recent work [12] the homogenization problem for a Feller diffusion process with jumps generated by an integro-differential operator has been studied under the assumptions that the corresponding generator has rapidly periodically oscillating diffusion and jump coefficients, and under additional regularity conditions. Under natural regularity assumptions on the coefficients of L, the homogenization result for operators Lε remains valid in this space of continuous functions, and that the corresponding semigroups converge. We prove that for the processes generated by Lε the invariance principle holds in the paths space
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