Periodic Homogenization Of Feller Processes With Jumps

The thesis is concerned about the periodic homogenization problems of Feller processes.More precisely,two issues are considered.The first issue is the homogenization of stochastic differential equations with nonlinear Poisson-type noise and the associated nonlocal parabolic partial differential equations,where the nonlinear noise intensity is in a normal scale in the noise components,compared to the small scale of the periodic structure.The second one is the homogenization of non-symmetric stable-like processes,the intensity function may involve periodic and aperiodic constituents,and has two different scales,in the noise components.Both issues contain the singular and rapidly oscillating drifts.The second problem is solved quite generally so that it can recover the results of the first one and many other papers.Moreover,we use the second results to simulate the first exit time of a two-dimensional stable-like process with variable order and reveal its convergence.The thesis is divided into five parts.After introducing the background and current progress of the periodic homogenization of Feller processes,we prove some preliminary results in the first part,for later use.Firstly,we generalize It?o formula to the functions with H¨older continuous derivatives.The operators in the Lebesgue integrals are either the generators of stochastic differential equations with multiplicative Poisson-type noise or stable-like operators.Secondly,we identify the semimartingale characteristics of the stochastic integrals with respect to random measures.In the second part,we investigate the strong well-posedness of stochastic differential equations with nonlinear Poisson-type noise and the ergodicity of the solutions.As consequences,we obtain the Feynman-Kac representation for the solutions of the associated nonlocal parabolic partial differential equations,and the well-posedness of the nonlocal Poisson equations.In the third part,we deal with the periodic homogenization for a class of nonlocal partial differential equations of parabolic-type with rapidly oscillating coefficients,related to stochastic differential equations driven by multiplicative isotropic ?-stable Lévy noise(1 < ? < 2)which is nonlinear in the noise component.The noise comes from the external environment.We use both probabilistic and analytic method to identity the limit process.Under suitable regularity assumptions,it turns out that as the scaling parameter goes to zero,the limit of the solutions satisfies a nonlocal partial differential equation with constant coefficients,which are associated to a symmetric?-stable Lévy process.In the fourth part,we look into the heat kernels estimates of the Lévy-type operators perturbed by gradients and the regularity of these operators and their generated semigroups.We also obtain the weak well-posedness of the associated stochastic differential equations and the solvability of the nonlocal Poisson equations.In the last part,we study the homogenization for a class of non-symmetric stable-like processes,and demonstrate its usage in the numerical approximation of the first exit time.The jump intensity includes periodic and aperiodic components,as well as oscillating and non-oscillating constituents.This means the noise comes from the combination of the underlying periodic structures and external environments,and is allowed to have two different spatial scales.The drift is involved and allowed to be singular.It turns out that the stable-like process converges in distribution to a Lévy process.As special cases of our results,some other homogenization problems studied in earlier works can be resolved,including the first homogenization problem in the current paper.Furthermore,we present numerical experiments to illustrate our results by demonstrating that the first exit time of the original process can be approximated by that of the limit process.Finally,a summary is presented and further questions are also discussed.