2023-11-7（星期二） 地点：数学院南楼613

（每个报告45分钟，包括5分钟提问时间）

 

 

 

 

张登（上海交通大学）

题目：Strichartz and local smoothing estimates for stochastic dispersive equations

摘要：Strichartz and local smoothing estimates play the key roles in the theory of dispersive equations. In this talk, we will show the pathwise Strichartz and local smoothing estimates for a general class of stochastic dispersive equations driven by linear multiplicative noise, including especially the stochastic Schroedinger and Airy equations. Several applications to stochastic nonlinear problems will then be given, including the pathwise well-posedness, the high integrability of stochastic solutions and the large deviation principle for the small noise asymptotics.

 

刘党政（中国科学技术大学）

题目: Edge statistics for deformed GinUEs

摘要: Consider sum of a complex Ginibre matrix and a non-random matrix, we prove that there are only two kinds of eigenvalue statistics at the spectral edge: Ginibre edge statistics and critical statistics. The latter is a new phenomenon, and represents a point process which seems not determinantal. This talk is based on joint work with Lu Zhang (USTC).

 

顾陈琳（清华大学）

题目：Quantitative homogenization of interacting particle systems

摘要： This talk presents that, for a class of interacting particle systems in continuous space, the finite-volume approximations of the bulk diffusion matrix converge at an algebraic rate. The models we consider are reversible with respect to the Poisson measures with constant density, and are of non-gradient type. This approach is inspired by recent progress in the quantitative homogenization of elliptic equations. Along the way, a modified Caccioppoli inequality and a multiscale Poincare inequality are developed, which are of independent interest. The talk is based on a joint work with Arianna Giunti and Jean-Christophe Mourrat.

 

陈昕昕（北京师范大学） 

题目: Conditioned branching random walk and related problems

摘要: We consider a supercritical branching random walk on the real line in the so called k-case where the whole system a.s. goes to infinity eventually, and the additive martingale converges a.s. and in mean to some non-degenerate random variable W_{\infty} under suitable moment condition. We study the asymptotical behaviors of the branching random walk conditioned on large W_{\infty}. We will also discuss several related problems on martingale limits and conditioned laws.

 

孙振尧（北京理工大学）

题目: On the regularization of reaction-diffusion equations by the Wright-Fisher white noise

摘要: Consider $[0,1]$-valued solutions to the one-dimensional stochastic partial differential equations

$\partial_t u_t= \frac{1}{2}\Delta u_t + u_t^q (1-u_t) + \sqrt{u_t(1-u_t)} \dot W \nonumber $ where $q \in [0,1]$, $\dot W$ is a space-time white noise, and $0^0 := 1$. Traditionally, the weak uniqueness of this equation is established through duality when $q = 1$; and a recent work (Comm. Math. Phys. 384 (2021), no. 2) has shown that the weak uniqueness of this equation also holds when $q \in [\frac{1}{2},1)$ provided the initial value $u_0$ has a compact interface.

In this talk, we discuss the weak uniqueness for a class of one-dimensional stochastic reaction-diffusion equations with Wright-Fisher white noise. Our results imply the week uniqueness of the aforementioned examples for every $q \in [0,1]$ without any assumptions regarding the support of the initial value. In particular, the drift term is allowed to be discontinuous at the place where the noise term degenerates.

 

杨尚杰（Bar Ilan university, Israel）

题目: Cutoff for polymer pinning dynamics in the repulsive phase

摘要：We consider the Glauber dynamics for model of polymer interacting with a substrate or wall. The state space is the set of one-dimensional nearest-neighbor paths on $\mathbb{Z}$ with nonnegative integer coordinates, starting at 0 and coming back to 0 after L ($L\in 2\mathbb{N}$) steps and the Gibbs weight of a path $\xi=(\xi_x)^{L}_{x=0}$ is given by $\lambda^{\mathcal{N}(\xi)}$, where $\lambda\geq 0$ is a parameter which models the intensity of the interaction with the substrate and $\mathcal{N}(\xi)$ is the number of zeros in $\xi$. The dynamics proceeds by updating $\xi_x$ with rate one for each $x=1,\dots, L-1$, in a heat-bath fashion. This model was introduced in Caputo et al (2008) with the aim of studying the relaxation to equilibrium of the system.

  We present new results concerning the total variation mixing time for this dynamics when $\lambda<2$, which corresponds to the phase where the effects of the wall's entropic repulsion dominates. For $\lambda \in [0, 1]$, we prove the total variation distance to equilibrium drops abruptly from 1 to 0 at time $(L^2 \log L)(1+o(1))/\pi^2$. For $\lambda\in (1,2)$, we prove that the system also exhibits cutoff at time $(L^2 \log L)(1+o(1))/\pi^2$ when considering mixing time from “extremal conditions” (that is, either the highest or lowest initial path for the natural order on the set of paths). Our results improve both previously proved upper and lower bounds in Caputo et al (2008).