Periodic homogenization for Levy-type processes
Ivana Valentić · 2020
The main goal of this thesis is to discuss periodic homogenization of a Levy-type pseudodifferential operator. Our approach to this problem is based on probabilistic techniques. More precisely, as the main result we show that the appropriately centered and scaled Levy-type process (LTP) generated by this operator converges weakly to a Brownian motion with covariance matrix given in terms of the operator coefficients. We specially focus on a class of Levy-type processes admitting “small jumps” only and a class of diffusion processes having degenerate diffusion term. These results generalize and refine the classical and well-known results related to periodic homogenization of diffusion process and of Levy-type process in balanced form. In order to resolve these problems, it is necessary to combine both probabilistic and analytical approaches and tools, such as theory of semimartingales, stochastic stability theory and theory of integro-differential equations.