Lattice convex chains in the plane

通过解析组合学的视角研究随机凸链

Random convex chains through the lens of analytic combinatorics

本文通过解析组合学方法研究随机凸链的顶点数分布，推导出其极限定理和精确渐进行为。

The mean perimeter of some random plane convex sets generated by a Brownian motion

If $C_1$ is the convex hull of the curve of the standard Brownian motion in the complex plane watched from 0 to 1, we consider the convex hulls of $C_1$ and several rotations of it and we compute the mean of the length of their perimeter by elementary calculations.

A probabilistic approach to convex $(ϕ)$-entropy decay for Markov chains

We study the exponential dissipation of entropic functionals for continuous time Markov chains and the associated convex Sobolev inequalities, including MLSI and Beckner inequalities. We propose a method that combines the Bakry Émery approach and coupling arguments, which we use as a probabilistic alternative to the discrete Bochner identities. The method is well suited to work in a non perturbative setting and we obtain new estimates for interacting random walks beyond the high temperature/weak interaction regime. In this framework, we also show that the exponential contraction of the Wassers…

Markov chains under nonlinear expectation

In this paper, we consider continuous-time Markov chains with a finite state space under nonlinear expectations. We define so-called Q-operators as an extension of Q-matrices or rate matrices to a nonlinear setup, where the nonlinearity is due to model uncertainty. The main result gives a full characterization of convex Q-operators in terms of a positive maximum principle, a dual representation by means of Q-matrices, continuous-time Markov chains under convex expectations and nonlinear ordinary differential equations. This extends a classical characterization of generators of Markov chains to…

Discrete Bochner inequalities via the Bochner-Bakry-Emery approach for Markov chains

Discrete convex Sobolev inequalities and Beckner inequalities are derived for time-continuous Markov chains on finite state spaces. Beckner inequalities interpolate between the modified logarithmic Sobolev inequality and the Poincaré inequality. Their proof is based on the Bakry-Emery approach and on discrete Bochner-type inequalities established by Caputo, Dai Pra, and Posta and recently extended by Fathi and Maas for logarithmic entropies. The abstract result for convex entropies is applied to several Markov chains, including birth-death processes, zero-range processes, Bernoulli-Laplace mod…

Stochastic billiards for sampling from the boundary of a convex set

Stochastic billiards can be used for approximate sampling from the boundary of a bounded convex set through the Markov Chain Monte Carlo (MCMC) paradigm. This paper studies how many steps of the underlying Markov chain are required to get samples (approximately) from the uniform distribution on the boundary of the set, for sets with an upper bound on the curvature of the boundary. Our main theorem implies a polynomial-time algorithm for sampling from the boundary of such sets.

双曲平面中布朗运动凸包的周长

Perimeter length of the convex hull of Brownian motion in the hyperbolic plane

研究双曲平面中布朗运动轨迹凸包周长的期望，通过将其与一维实值布朗运动的指数泛函期望关联，推导出小时间和大时间渐近行为。

Large convex holes in random point sets

A {\em convex hole} (or {\em empty convex polygon)} of a point set $P$ in the plane is a convex polygon with vertices in $P$, containing no points of $P$ in its interior. Let $R$ be a bounded convex region in the plane. We show that the expected number of vertices of the largest convex hole of a set of $n$ random points chosen independently and uniformly over $R$ is $Θ(\log{n}/(\log{\log{n}}))$, regardless of the shape of $R$.

复平面上平行裂缝半平面的Loewner链与演化族

Loewner chains and evolution families on parallel slit half-planes

本文在复平面有限多连通区域上定义并研究了Loewner链与演化族，发展了一般理论并证明其服从由测度值过程驱动的弦Komatu-Loewner微分方程，方法涉及带有缝合的布朗运动。

Compact convex sets of the plane and probability theory

The Gauss-Minkowski correspondence in $\mathbb{R}^2$ states the existence of a homeomorphism between the probability measures $μ$ on $[0,2π]$ such that $\int_0^{2π} e^{ix}dμ(x)=0$ and the compact convex sets (CCS) of the plane with perimeter~1. In this article, we bring out explicit formulas relating the border of a CCS to its probability measure. As a consequence, we show that some natural operations on CCS -- for example, the Minkowski sum -- have natural translations in terms of probability measure operations, and reciprocally, the convolution of measures translates into a new notion of con…

基于随机梯度马尔可夫链蒙特卡洛的非凸贝叶斯学习

Non-convex Bayesian Learning via Stochastic Gradient Markov Chain Monte Carlo

本文针对非凸贝叶斯学习问题，提出控制变量法和基于不可逆性的群体链副本交换以改进Langevin Monte Carlo的加速性能，并设计两种可扩展动态重要性采样算法，在非凸能量景观下证明了渐近收敛性。