最小作用方法与有限差分方法结合的收敛性分析

全耦合多尺度随机微分方程在α-稳定过程中强收敛性和弱收敛性的收敛率

Strong and weak convergence rates for fully coupled multiscale stochastic differential equations driven by $α$-stable processes

本文研究了由α-稳定过程驱动的全耦合多尺度随机微分方程的强弱收敛率，通过非局部泊松方程构造校正方程以消除非齐次项，揭示了强弱收敛阶与Holder正则性条件的关系。

有限参与者随机微分博弈中完全耦合正倒向随机微分方程的虚拟策略收敛性

Convergence of fictitious play for fully coupled FBSDEs in finite-player stochastic differential games

研究有限参与者随机微分博弈中完全耦合正倒向随机微分方程的虚拟策略收敛性，在一定假设下证明其收敛为几何收敛，特殊博弈中为超指数收敛，通过线性二次型银行间借贷问题数值实验予以证实。

收敛到平衡分布：狄拉克场与粒子耦合

Convergence to equilibrium distribution. Dirac fields coupled to a particle

研究多个狄拉克场与一个粒子耦合系统的随机初值问题，证明在零均值、有限电荷密度、平移不变协方差及混合条件下，解的分布长时间收敛到高斯极限测度。

Convergence in law for the branching random walk seen from its tip

Considering a critical branching random walk on the real line. In a recent paper, Aidekon [3] developed a powerful method to obtain the convergence in law of its minimum after a log-factor normalization. By an adaptation of this method, we show that the point process formed by the branching random walk and its minimum converge in law to a Poisson point process colored by a certain point process. This result, confirming a conjecture of Brunet and Derrida [10], can be viewed as a discrete analog of the corresponding results for the branching brownian motion, previously established by Arguin et a…

Scaling Limit of Neural Networks with the Xavier Initialization and Convergence to a Global Minimum

We analyze single-layer neural networks with the Xavier initialization in the asymptotic regime of large numbers of hidden units and large numbers of stochastic gradient descent training steps. The evolution of the neural network during training can be viewed as a stochastic system and, using techniques from stochastic analysis, we prove the neural network converges in distribution to a random ODE with a Gaussian distribution. The limit is completely different than in the typical mean-field results for neural networks due to the $\frac{1}{\sqrt{N}}$ normalization factor in the Xavier initializ…

On the Convergence to a Statistical Equilibrium in the Crystal Coupled to a Scalar Field

We consider the dynamics of a field coupled to a harmonic crystal with $n$ components in dimension $d$, $d,n\ge 1$. The crystal and the dynamics are translation-invariant with respect to the subgroup $\Z^d$ of $\R^d$. The initial data is a random function with a finite mean density of energy which also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. Moreover, initial correlation functions are translation-invariant with respect to the discrete subgroup $\Z^d$. We study the distribution $μ_t$ of the solution at time $t\in\R$. The main result is the convergence of $μ_t$ to a Ga…

Local convergence of large random triangulations coupled with an Ising model

We prove the existence of the local weak limit of the measure obtained by sampling random triangulations of size $n$ decorated by an Ising configuration with a weight proportional to the energy of this configuration. To do so, we establish the algebraicity and the asymptotic behaviour of the partition functions of triangulations with spins for any boundary condition. In particular, we show that these partition functions all have the same phase transition at the same critical temperature. Some properties of the limiting object -- called the Infinite Ising Planar Triangulation -- are derived, in…

Diffusion approximation for fully coupled stochastic differential equations

We consider a Poisson equation in $\mathbb R^d$ for the elliptic operator corresponding to an ergodic diffusion process. Optimal regularity and smoothness with respect to the parameter are obtained under mild conditions on the coefficients. The result is then applied to establish a general diffusion approximation for fully coupled multi-time-scales stochastic differential equations with only Hölder continuous coefficients. Four different averaged equations as well as rates of convergence are obtained. Moreover, the convergence is shown to rely only on the regularities of the coefficients with …

Loopy annealing belief propagation for vertex cover and matching: convergence, LP relaxation, correctness and Bethe approximation

For the minimum cardinality vertex cover and maximum cardinality matching problems, the max-product form of belief propagation (BP) is known to perform poorly on general graphs. In this paper, we present an iterative loopy annealing BP (LABP) algorithm which is shown to converge and to solve a Linear Programming relaxation of the vertex cover or matching problem on general graphs. LABP finds (asymptotically) a minimum half-integral vertex cover (hence provides a 2-approximation) and a maximum fractional matching on any graph. We also show that LABP finds (asymptotically) a minimum size vertex …