350vip8888新葡的京集团350vip8888新葡的京集团2024年第38期数理大讲堂

报告人：王仁海教授， 贵州师范大学

报告时间：2024年12月16日上午10：30—11：30

地点：350vip8888新葡的京集团3号楼 3B205

主持人：赵才地教授

报告人简介：王仁海, 贵州师范大学特聘教授，博士生导师，博士毕业于西南大学(与美国New Mexico Institute of Mining and Technolog联合培养)，北京应用物理与计算数学研究所博士后，长期从事无穷维随机动力系统与随机偏微分方程的研究。主持国家自然科学基金青年基金，中国博士后科学基金特别资助和面上资助，获重庆市优秀博士学位论文，其论文发表于在Math. Ann., Math. Models Meth. Appl. Sci., Inter. Math. Research Notices, SIAM J. Math. Anal., J. Differential Equations, J. Dyn. Differential Equations, Nonlinearity, Stochastic Processes and Their Appl., Sci. China Math.等一流期刊.

报告题目： Fractional (α,p)-Laplacian Equations Driven by Superlinear Noise on R^d: Global Solvability and Invariant Measures

报告摘要：We consider a wide class of fractional (α,p)-Laplacian equations on R^d driven by infinite-dimensional superlinear noise. The model has three striking features: a general fractional (α,p)-Laplace operator defined via a symmetric and translation invariant kernel function K^α_p with α ∈ (0,1) and p > 2; a polynomial drift growing at an arbitrary rate q −1 with q > 2; and a locally Lipschitz diffusion term with superlinear growth. By using a locally monotone method and a domain approximation argument, we first establish the global-in-time well-posedness and the higher-order Itô’s energy equations when the diffusion term has a superlinear growth rate less than p/2 or q/2. We then prove the existence and derive the moment estimates of invariant measures of the equation under further assumptions on the superlinear diffusion terms. When the damping coefficient is suitably large, we show the unique-ness, ergodicity as well as the Wasserstein exponential mixing of invariant measures without adding any additional conditions on the superlinear noise.The idea of uniform tail-estimates is used to overcome the difficulties caused by the lack of compactness of Sobolev embeddings on unbounded domains. The dissipativeness of the drift terms and some appropriate stopping times are used to carefully deal with the superlinear diffusion terms. The analysis has no any restrictions on α ∈ (0,1), d ∈ N, p > 2 or q > 2. This is a joint work with Professors Bixiang Wang and Penyu Chen.