报告题目: The KPZ equation, its two types of approximations and existence of global solutions
报告人: Funaki, Tadahisa,
摘 要: In this lecture, we will consider the multi-component coupled Kardar–Parisi–Zhang (KPZ) equation and its two types of approximations. One approximation is obtained as a simple replacement of the noise term by a smeared noise with a proper renormalization, while the other one introduced is suitable for studying the invariant measures. By applying the paracontrolled calculus introduced by Gubinelli et al. , we show that two approximations have the common limit under the properly adjusted choice of renormalization factors for each of these approximations. In particular, if the coupling constants of the nonlinear term of the coupled KPZ equation satisfy the so-called “trilinear” condition, the renormalization factors can be taken the same in two approximations and the difference of the limits of two approximations are explicitly computed. Moreover, under the trilinear condition, the Wiener measure twisted by the diffusion matrix becomes stationary for the limit and we show that the solution of the limit equation exists globally in time when the initial value is sampled from the stationary measure. This is shown for the associated tilt process. Combined with the strong Feller property shown by Hairer and Mattingly, this result can be extended for all initial values.
报告人概况：Funaki, Tadahisa教授， 日本东京大学教授，博士生导师。 1982年，于Nagoya University大学获得博士学位。曾任日本数学会理事长，2002年获得MSJ Analysis Prize，2007年获得MSJ Autumn Prize。目前研究兴趣为：相互作用系统的大尺度随机分析、非线性方程的推导。发表论文72篇，其中多篇论文发表在Probab. Theory Related Fields、 Ann. Probab.Comm. Math. Phys、 J. Funct. Anal.等著名期刊上。