Periodic homogenization of non-local operators with a convolution type kernel

The paper deals with homogenization problem for a non-local 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 behaviour 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

Gibbs point field models for extraction problems in image analysis

International audienceThis paper is a review of some probabilistic methods, based on Gibbs fields theory, applied to solve image analysis tasks. We present the mathematical background and show different applications

Homogenization of biased convolution type operators

The final publication is available at IOS Press through http://dx.doi.org/10.3233/ASY-191533This 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

Large deviations for Markov jump processes in periodic and locally periodic environments

The paper deals with a family of jump Markov process defined in a medium with a periodic or locally periodic microstructure. We assume that the generator of the process is a zero order convolution type operator with rapidly oscillating locally periodic coefficient and, under natural ellipticity and localization conditions, show that the family satisfies the large deviation principle in the path space equipped with Skorokhod topology. The corresponding rate function is defined in terms of a family of auxiliary periodic spectral problems. It is shown that the corresponding Lagrangian is a convex function of velocity that has a superlinear growth at infinity. However, neither the Lagrangian nor the corresponding Hamiltonian need not be strictly convex, we only claim their strict convexity in some neighbourhood of infinity. It then depends on the profile of the generator kernel whether the Lagrangian is strictly convex everywhere or not

Mathematical multi-scale model of water purification

In this work, we consider a mathematical model of the water treatment process and determine the effective characteristics of this model. At the microscopic length scale, we describe our model in terms of a lattice random walk in a high-contrast periodic medium with absorption. Applying then the upscaling procedure, we obtain the macroscopic model for total mass evolution. We discuss both the dynamic and the stationary regimes and show how the efficiency of the purification process depends on the characteristics of the macroscopic model