YMSC Probability Seminar Modulation and amplitude equations on bounded domains for nonlinear SPDEs driven by cylindrical α-stable Lévy processes

Abstract

In the present work, we establish the approximation via modulation or amplitude equations of nonlinear stochastic partial differential equation (SPDE) driven by cylindrical α-stable Lévy processes. We study SPDEs with a cubic nonlinearity, where the deterministic equation is close to a change of stability of the trivial solution. The natural separation of time scales close to this bifurcation allows us to obtain an amplitude equation describing the essential dynamics of the bifurcating pattern, thus reducing the original infinite dimensional dynamics to a simpler finite-dimensional effective dynamics. In the presence of a multiplicative stable Lévy noise that preserves the constant trivial solution we study the impact of noise on the approximation. In contrast to Gaussian noise, where non-dominant pattern are uniformly small in time due to averaging effects, large jumps in the Lévy noise might lead to large error terms, and thus new estimates are needed to take this into account. This is the joint work with Dirk Blömker.

Speaker

2017年9月至2018年8月前往德累斯顿工业大学CSC联合培养博士。2019年6月获华中科技大学概率论与数理统计专业博士学位。随后加入华中科技大学人工智能与自动化学院从事博士后研究。而后任德国奥格斯堡大学助理研究员职位。研究方向为 Lévy 过程驱动的随机动力系统、量子力学、统计物理和随机分析。近五年在 SIAM Journal on Applied Dynamical Systems、Journal of Statistical Mechanics、 Analysis and Applications等国际重要期刊上发表 14 篇学术论文。