时　　间：2023-10-27（星期五）

地　　点：数学院南楼613

主办单位：中国科学院数学与系统科学研究院应用数学研究所

杨帆（清华大学）

题目：SIMPLE-RC: Group Network Inference with Non-Sharp Nulls and Weak Signals

摘要：The recent work of Fan, Fan, Han and Lv (2022) introduced a general framework of statistical inference on membership profiles in large networks (SIMPLE) for testing the sharp null hypothesis that a pair of given nodes share the same membership profiles. In real applications, there are often groups of nodes under investigation that may share similar membership profiles at the presence of relatively weaker signals than the setting considered in SIMPLE. To address these practical challenges, we propose a SIMPLE method with random coupling (SIMPLE-RC) for testing the non-sharp null hypothesis that a group of given nodes share similar (not necessarily identical) membership profiles under weaker signals. Utilizing the idea of random coupling, we construct our test as the maximum of the SIMPLE tests for subsampled node pairs from the group. Such technique reduces significantly the correlation among individual SIMPLE tests while largely maintaining the power, enabling delicate analysis on the asymptotic distributions of the SIMPLE-RC test. These new theoretical developments are empowered by a second-order expansion of spiked eigenvectors, built upon our work for random matrices with weak spikes. Based on joint work with Jianqing Fan, Yingying Fan and Jinchi Lv.

刘国民（南开大学）

题目: Maximum principle for optimal control of stochastic evolution equations with recursive utilities 

摘要: We consider the optimal control problem of stochastic evolution equations in a Hilbert space under a recursive utility, which is described as the solution of a backward stochastic differential equation (BSDE). A very general maximum principle is given for the optimal control, allowing the control domain not to be convex and the generator of the BSDE to vary with the second unknown variable z. The associated second-order adjoint process is characterized as a unique solution of a conditionally expected operator-valued backward stochastic integral equation. This talk is based on a joint work with Prof. Shanjian Tang.

杨寰宇（重庆大学）

题目：Geometric intermittency of continuum parabolic Anderson model with smooth Gaussion potential

摘要：We consider the parabolic Anderson model $\partial_tu=\Delta u+\xi u$ on $\mathbb{R}^d$ with smooth time-homogeneous Gaussian potential $\xi$ and initial value $u(0)=\delta_0$. Combining with the strategy of discrete parabolic Anderson model and the random polymer measure, we prove the geometric intermittency: with probability one, as $t\to\infty$, the total mass $\int_{\mathbb{R}^d} u(t,x)dx$ comes from some slow increasing, small and remote islands where the local principal eigenvalues of (Dirichlet) Anderson Hamiltonian $\Delta+\xi$ are closed to the principal eigenvalue of Anderson Hamiltonian on the box $(-t\log t, t\log t)^d$. This is a joint work with Nicolas Perkowski.

范协铨（东北大学秦皇岛分校） 

题目: Cramer type moderate deviations for martingales with applications

摘要: Let( _i,F_i )_(i≥1)be a sequence of martingale differences. Set S_n=∑_(i=1)^n _i  and ？S？_n=∑_(i=1)^n 〖E[ _i |F_(i-1)]〗. We present some Cramer's moderate deviation expansions for P(S_n/√(？S？_n )≥x) and P(S_n/√(ES_n^2 )≥x) as n ∞. Our results extend the classical Cramer result to the cases of normalized martingales S_n/√(？S？_n ) and standardized martingales S_n/√(ES_n^2 ), with martingale differences satisfying the conditional Bernstein condition. Applications to elephant random walks and autoregressive processes are also discussed. The talk is based on join work of Quansheng LIU, Ion GRAMA and Qi-Man SHAO.

李一霆（湖南大学）

题目: Central limit theorem for the linear spectral statistics of sample covariance matrix with random population

摘要：Consider the sample covariance matrix (\Sigma^{1/2}) XX^*(\Sigma^{1/2}) where X is an M by N random matrix with independent entries and \Sigma is an M by M positive definite diagonal matrix. Use L(f) to denote the linear spectral statistics of the sample covariance matrix with test function f. It is known that if \Sigma is deterministic, then the fluctuation of L(f) converges in distribution to a Gaussian distribution. We prove that if \Sigma is random and is independent of X, then L(f) multiplied by N^{-1/2} converges in distribution to a Gaussian distribution. This phenomenon implies that the randomness of \Sigma weakens the correlation among the eigenvalues of the sample covariance matrix. This is a joint work with Ji Oon Lee.

张让让（北京理工大学）

题目: Dean-Kawasaki equation with singular non-local interactions：well-posedness and large deviations 

摘要: The Dean-Kawasaki equation with non-local singular interactions and correlated noise will be introduced, which describes the evolution of the density function for a system of finitely many particles governed by Langevin dynamics. Due to the presence of non-local interaction term, the noise taking the form of a stochastic conservation law and the lack of Lipschitz continuity of the noise, the well-posedness and large deviations are proved in the framework of stochastic kinetic solution.